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Electrochemistry MAE- 212

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Electrochemistry MAE-212

Dr. Marc Madou, UCI, Winter 2014

Class V Transport in Electrochemistry (II)

- Reynolds Numbers
- Low Reynolds Numbers
- OHP, Diffusion Layer Thickness, Hydrodynamic Boundary Layer Thickness
- Mixing in low Reynolds number fluids to enhance electrochemical reactions

- The dimensionless Reynolds number is given by:
where v is the mean velocity of an object relative to the fluid (SI units: m/s), L is a characteristic linear dimension (SI: m),μ is the dynamic viscosity of the fluid [SI: Pa·s or N·s/m² or kg/(m·s)] and ν is the kinematic viscosity (ν: μ / ρ) (m²/s) and r is the density of the fluid (SI: kg/m³)

- Note that multiplying the Reynolds number by
yields:

which is the ratio of:

- Or also:

- Small systems are less turbulent than large ones (e.g., flow in very thin pipes is laminar).
- Slow flows are laminar, while fast flows are turbulent.
- More viscous materials are less turbulent (e.g., oil in a pipeline is less turbulent than water in the same pipeline).

- Creeping flow also known as “Stokes Flow” or “Low Reynolds number flow”
- Occurs when Re << 1
- ,v (often U is used),or L are very small, e.g., micro-organisms, MEMS, nano-tech, particles, bubbles
- is very large, e.g., honey, lava

- In micro-fluidics, Re<1
- In Laminarflow the viscous force is dominant over the inertial force
- Inertial forces are pretty much irrelevant

Purcell 1977

http://www.youtube.com/user/Swimmers1

Typical size of a chip

Micro and nano technology enabled

1mm

100mm

Extended lenght of DNA

Micro-channel

10mm

Microstructure and micro-drops

Cellular scale

1mm

Radius of Gyration of DNA

100nm

Colloid and polymer molecular size

10nm

- Fluids in micro-channels and nano-channels
- Here we are specifically interested in working with electrodes in confined spaces: so back to electrochemistry !!
- To maximize the supply of electro-active species to electrodes in such confined spaces one relies very often on forced convection.
- If migration is suppressed the mass transport will be under diffusion-convection control.

Micro-channels

Nano-tubes (some of the smallest channels).

- The outer Helmholtz plane (OHP) is considered to be the approximate site for electron transfer.
- Nonspecifically adsorbed ions also reside in the diffuse layer (Nernst layer) extending some distance from the electrode surface. The thickness of the diffuse layer is dependent upon ionic strength of the buffer, and for stirred aqueous solutions the thickness of the diffuse layer varies between 0.01 and 0.001 mm. And has been found to be 0.05 cm in many cases of unstirred aqueous electrolytes. The nature of the diffuse layer can have a significant impact on the rate of electron-transfer since the actual potential felt by a reactant close to the electrode is dependent upon it.
- From the above the thickness of the Nernst layer is strongly dependent on the condition of the hydrodynamic flow due to say stirring or other convective effects. The double layer on the other hand is typically less than 1 nm and is not influenced by stirring.
- So how does the hydrodynamic boundary layer influence the diffusion layer? First what is the hydrodynamic boundary layer?

- In physics and fluid mechanics, a boundary layer is the layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant.
- When fluid flows past an immersed body, a thin boundary layer will be developed near the solid body due to the no-slip condition (i.e., fluid is stuck to the solid boundary). The flow can be treated as inviscid flow outside of this boundary layer, while viscous effects are important inside of this boundary layer.

- The characteristics of flow past a flat plate with finite length L subject to different Reynolds numbers (Re = ρUL/μ) are shown in the figures on the right. At a low Reynolds number (Re = 0.1), the presence of the flat plate is felt in a relatively large area where the viscous effects are important.
- At a moderate Reynolds number (Re = 10), the viscous layer region becomes smaller. Viscous effects are only important inside of this region, and streamlines are deflected as fluid enters it.
- As the Reynolds number is increased further (Re = 107), only a thin boundary layer develops near the flat plate, and the fluid forms a narrow wake region behind the flat plate. (a) Re = 0.1, (b) Re = 10 and (c) Re = 107

- Consider now a flow past an infinite long flat plate. Also define the Reynolds number using the local distance x (i.e., the distance from the leading edge along the flat plate as the characteristic length).
- The local Reynolds number is then given by: Rex = ρUxX/μ
- The flow becomes turbulent at a critical distance xcr downstream from the leading edge. The transition from laminar to turbulent begins when the critical Reynolds number (Rexcr) reaches 5×105. The boundary layer changes from laminar to turbulent at this point.

- The concept of a boundary layer was introduced by Prandtl (1904) for steady, two-dimensional laminar flow past a flat plate using the Navier-Stokes (NS) equations. Prandtl's student, Blasius, was able to solve these equations analytically for large Reynolds number flows.

- Based on Blasius' analytical solutions, the boundary layer thickness (δ) for the laminar region is given by :
where δ is defined as the boundary layer thickness in which the velocity is 99% of the free stream velocity (i.e., y = δ, u = 0.99U).

- To compare the thickness of the Nernst Layer and the Prandtl layer we need to introduce a few more dimensionless numbers.

- (1) velocity profile; (2) Prandtl boundary layer; (3) Nernst boundary layer.
- Prandtl boundary layer thickness (hydrodynamic):δp ≈ 5(νx/U)1/2,
- Nernst boundary layer thickness (diffusion)δN ≈ D 1/3ν 1/6(x/U)1/2,
with U, fluid velocity; ν, kinematic viscosity; and D, diffusion constant.

- The Nernst diffusion layer is a sublayer of the Prandtl layer.
- As in the Prandtl layer there is no motion of the solution in the Nernst layer.
- The Nernst and the Prandtl layers are the regions where the concentration and the tangential velocity gradients are at a maximum.

- The Schmidt Number, Sc, is a dimensionless parameter representing the ratio between momentum transport and mass transport by diffusion. It is defined as:
with kinematic viscosity,n, and mass diffusivity Dc.

- Small values of the Schmidt number (<1) diffusion dominates over convection. It physically relates the relative thickness of the hydrodynamic layer and mass-transfer boundary layer.

- Prandtl Number: describes the thickness of the hydrodynamic boundary layer compared with the thermal boundary layer. It is the ratio between the molecular diffusivity of momentum to the molecular diffusivity of heat.
- Small values of the Prandtl number (< 1) in a given fluid indicates that thermal diffusion occurs at a greater rate than momentum diffusion and therefore heat conduction is more effective than convection. Conversely if the Prandtl number is large (greater than 1), momentum diffuses at a greater rate than heat and convection is more effective than conduction.

- The Schmidt number is the mass transfer equivalent of the Prandtl Number. For gases, Sc and Pr have similar values (≈0.7) and this is used as the basis for simple heat and mass transfer analogies.
- A quantitative treatment of the relative extension of dpvsdN shows that it is governed by Sc. For aqueous electrolytes n = 10-2 cm2s-1, the Schmidt number is about 1000.
- Based on
the Prandtl layer is 10 to 30 times thicker than the Nernst layer.

- Voltammograms we saw exhibit a sigmoidal (wave) shape. If the stirring rate (U) is increased, the diffusion layer thickness becomes thinner, according to:
where B and a are constants for a given system. As a result, the concentration gradient becomes steeper, thereby increasing the limiting current. Similar considerations apply to other forced convection systems, e.g., those relying on solution flow or electrode rotation. For all of these hydrodynamic systems, the sensitivity of the measurement can be enhanced by increasing the convection rate.

- In aqueous the Prandtl layer is 10-fold larger than the Nernst layer, indicating negligible convection within the diffusion layer .
- Additional means for enhancing the mass transport and thinning the diffusion layer, include the use of ultrasound, heated electrodes.

- The question we address now is how to mix reactants in a small microreactor in the absence of turbulence? The primary resistance to mixing by convection, we saw earlier, is controlled by a thin layer of stagnant fluid adjacent to a solid surface.
- In this hydrodynamic boundary layer, the flow velocity, V, varies from zero at the surface, that is, the no-slip condition, to the value in the bulk of the fluid, that is, V∞. Laminar flow around an object of length L and boundary layer δ is given by:

- At the microfluidic level, mixing is like trying to stir syrup into honey, and two liquids, traveling side-by-side through a narrow channel, only become mixed after several centimeters because mass transport in the microdomain is traditionally limited to simple diffusion . To decide which transport type dominates, diffusion or convection, one must inspect the Péclet number.

- The Péclet number represents the ratio of mass transport by convection to mass transport by diffusion
- The higher the Péclet number, the more the influence of flow dominates over molecular diffusion. In liquids, the diffusion coefficient of a small molecule typically is about 10−5 cm2/s. With a velocity of 1 mm/s, in a channel of 100-μm height, the Péclet number is on the order of 100. This elevated value suggests that the diffusion forces are acting more slowly than the hydrodynamic transport phenomena: for mixing by diffusional forces one must have Pe < 1.

- At low Reynolds numbers, stirring is like kneading dough for making bread, with stretching of fluid elements to increase the diffusional interface and folding to decrease distance over which species have to diffuse. Creating chaotic pathlines for dispersing fluid species effectively in smooth and regular flow fields is called “chaotic advection.”
- Chaotic advection results in rapid distortion and elongation of the fluid/fluid interface, increasing the interfacial area across which diffusion occurs, which increases the mean values of the gradients driving diffusion, leading to more rapid mixing.

- Mixing can also be improved on by using a fractal approach, a design akin to how nature uses passive mixing . In nature, only the smallest animals rely on diffusion for transport; animals made up of more than a few cells cannot rely on diffusion anymore to move materials within themselves. They augment transport with hearts, blood vessels, pumped lungs, digestive tubes, etc. These distribution networks typically constitute fractals. Fractals are an optimal geometry for minimizing the work lost as a result of the transfer network while maximizing the effective surface area