### Lecture 4

```Electrochemistry MAE-212
Dr. Marc Madou, UCI, Winter 2015
Class IV Transport in Electrochemistry (II)
Table of Content
 Reynolds Numbers
 Low Reynolds Numbers
 OHP, Diffusion Layer Thickness, Hydrodynamic Boundary
Layer Thickness
 Mixing in low Reynolds number fluids to enhance
electrochemical reactions
Reynolds Numbers
 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
yields:
by
which is the ratio of:
 Or also:
Reynolds Numbers
• 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).
Low Reynolds Numbers
 Creeping flow also known as “Stokes Flow” or “Low Reynolds
number flow”
 Occurs when Re << 1
 r,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
Low Reynolds Numbers (Stokes flow)
 In micro-fluidics, Re<1
 In Laminar flow the viscous force is dominant over the inertial force
 Inertial forces are pretty much irrelevant
Purcell 1977
Low Reynolds Numbers
• Micro and nano
1mm
Typical size of a chip
technology enabled
100m
Extended lenght of DNA
Micro-channel
10m
Microstructure and micro-drops
Cellular scale
1m
100nm
Colloid and polymer molecular size
10nm
Re D,max 
UDh


10 mms 500m
1
mm
2
s
5
Low Reynolds Numbers
 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).
OHP-Diffusion Layer- Hydrodynamic
Boundary Layer
 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 -0.1 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?
OHP-Diffusion Layer- 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.
OHP-Diffusion Layer- Hydrodynamic
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
OHP-Diffusion Layer- Hydrodynamic
Boundary Layer
 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.
OHP-Diffusion Layer- Hydrodynamic
Boundary Layer
 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.
OHP-Diffusion Layer- Hydrodynamic
Boundary Layer  The Nernst diffusion layer is a sublayer of the
 (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.
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.
OHP-Diffusion Layer- Hydrodynamic
Boundary Layer
 The Schmidt Number, Sc, is a dimensionless  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 (<
with kinematic viscosity,, and mass
1) in a given fluid indicates that thermal
diffusion occurs at a greater rate than
diffusivity Dc.
momentum diffusion and therefore heat
 Small values of the Schmidt number (<1)
conduction is more effective than
diffusion dominates over convection. It
convection. Conversely if the Prandtl
physically relates the relative thickness of the number is large (greater than 1),
momentum diffuses at a greater rate
hydrodynamic layer and mass-transfer
than heat and convection is more
boundary layer.
effective than conduction.
parameter representing the ratio between
momentum transport and mass transport by
diffusion. It is defined as:
OHP-Diffusion Layer- Hydrodynamic
Boundary Layer
 The Schmidt number is the mass transfer

Voltammograms we saw exhibit a sigmoidal
(wave) shape. If the stirring rate (U) is increased,
the diffusion layer thickness becomes thinner,
according to:
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.
B
d  a
 A quantitative treatment of the relative
where B and a are constantsUfor a given system. As a
extension of dp vs dN shows that it is
steeper, thereby increasing the limiting current.
governed by Sc. For aqueous electrolytes 
Similar considerations
apply to other forced
= 10-2 cm2s-1, the Schmidt number is about

convection systems, e.g., those relying on
1000.
solution flow or electrode rotation. For all of
 Based on
these hydrodynamic systems, the sensitivity of
N
the measurement can be enhanced by increasing
the convection rate.
1

dN
)  Sc 2
layer
d Pr is 10
(
the Prandtl
to 30 times thicker
than the Nernst layer.

OHP-Diffusion Layer- Hydrodynamic
Boundary Layer
 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.
Mixing in low Reynolds number fluids to
enhance electrochemical reactions
 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.
Mixing in low Reynolds number fluids to
enhance electrochemical reactions

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 100μm 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
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
 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 in low Reynolds number fluids to
enhance electrochemical reactions
 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
```