Engmeermg
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`Sclence
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`and Technology
`In Chemical
`and Other
`Industries
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`Tenn-Hawnt Company
`Exhlibit'1 1 1'7
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` Springer '
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`Exhibit 1117_0001
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`Tennant Company
`Exhibit 1117
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`

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`
`Prof. Dr. Hartmut Wendt
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`Institut fiir Chemische Technologie
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`TU Darmstadt
`PetersenstraBe 20
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`D—64287 Darmstadt
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`Germany
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`Prof. Dr. Gerhard Kreysa
`Karl Winnacker Institut
`DECHEMA e. V.
`Theodor-Heuss-Allee 25
`D-60486 Frankfurt am Main
`
`Germany
`
`ISBN 3-540—64386-9 Springer-Verlag Berlin Heidelberg New York
`
`Library of Congress Cataloging-in-Publication Data
`Wendt, Hartmut, 1933—
`Electrochemical engineering : science and technology in chemical and other industries / Hartmut Wendt,
`Gerhard Kreysa.
`p. cm.
`Includes bibliographical references.
`ISBN 3-540-64386-9 (hardcover: alk. paper)
`1. Electrochemistry, Industrial. I. Kreysa, Gerhard. 11. Title.
`TP255.W46 I999
`660‘ .297-—dc21
`
`This work is subject to copyright. All rights are reserved, whether the whole part of the material is concerned,
`specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction
`on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof
`is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current ver-
`sion, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecu-
`tion under the German Copyright Law.
`
`© Springer—Verlag Berlin Heidelberg 1999
`Printed in Germany
`
`The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply,
`even in the absence of a specific statement, that such names are exempt from the relevant protective laws and
`regulations and therefore free for general use.
`
`Typesetting: MEDIO, Berlin
`Coverdesign: Design 8: Production, Heidelberg
`
`SPIN: 10675807
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`2/3020—5 4 3 2 l 0 - Printed on acid-free paper.
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`Exhibit 1117_0002
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`Exhibit 1117_0002
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`

`

`1'
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`'
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`
`CHAPTERS
`
`Mass Transfer by Fluid Flow, Convective Diffusion and Ionic
`Electricity Transport in Electrolytes and Cells
`
`5.1
`
`Introduction
`
`The performance of electrochemical processes is not only determined by charge
`transfer and electrode kinetics, but a number of additional phenomena cause
`and rule the electrode kinetics (in the microkinetic as well as macrokinetic
`sense), the heat balance of the cell and the mass balances of all process streams
`(electrolyte, gases, solid products). Among these factors the fluid dynamic con—
`ditions, under which the electrolyte enters, passes and leaves the electrolysis cell
`or moves in it under free convection, is the most powerful process parameter
`since hydrodynamics rule mass and heat transport.
`Heat transport is important as it controls together with heat generation the
`temperature distribution in the cell. Another condition typical for electrochem-
`ical processes is charge transport through the electrolyte, which toghether with
`electrode kinetics determines in particular the current density distribution
`across the electrode surface. It may determine the overall current efficiency, con-
`version selectivity and space time yield of the process by local inhomogeneities
`of the current density for instance by locally too high current density which
`might exceed the mass transfer limited current density.
`The proper handling of these characteristic process determinants: fluid dy-
`namics, mass transport and heat transport together with proper management of
`ionic charge transport are the main subjects of electrochemical process engi—
`neering as far as the reactor i.e. the electrolyzer is concerned. But the electro-
`chemical engineer has to be aware that the cell alone is only one part of the whole
`process. Electrolyte make up, product isolation and recycling of the electrolyte
`have to be taken into consideration, too.
`
`5.2
`
`Fluid Dynamics and Convective Diffusion
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`l
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`Diffusive mass transport in an electrolyte solution without superimposed con-
`vection is relatively ineffective because of the low diffusion coefficients experi-
`enced in solvents of viscosities comparable to that of water. In usual dielectric
`
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`
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`Exhibit 1117_0003
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`

`

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`
`82
`
`5 Mass Transfer by Fluid Flow, Convective Diffusion and...
`
`solvents the diffusion coefficient of a low molecular weight species is similar to
`that in aqueous solution, which is of the order of approx. 10‘5 cm25"1.
`Moreover, diffusion alone is typically a non-steady process that means on ex-
`tension of time it proceeds with a steadily decreasing velocity because concen-
`tration gradients tend to level off. Figure 5.1 a demonstrates this, depicting
`schematically the temporal development of concentration profiles of metal ions
`in front of a cathode on which the metal ions are deposited with time-inde—
`pendent current density, i, in presence of supporting electrolyte of high con-
`centration, which allows for metal ion diffusion alone, excluding mass trans~
`port by migration.
`
`
`
`
`(b)
`
`y
`
`Fig.5.1.a Schematic showing the development of concentration profiles of metal ions in a
`stagnant electrolyte upon galvanostatic cathodic metal deposition (M1+ + ze‘—9M). Note
`that (dc/dx)x=0 remains unchanged because i is constant b Development of concentration
`profile of metal ions in stagnant electrolyte upon imposing a constant potentialtp <(pequn on
`the electrode. Declining values (dc/dy),(=0 reflect in steadily declining current densities ac-
`cording to t'“2 — law (see insert).
`
`Exhibit 1117_OOO4
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`Exhibit 1117_0004
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`

`

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`5.2 Fluid Dynamics and Convective Diffusion 83
`
`For a given current densityi the gradient (dc/dy) is fixed at the electrode sur-
`face (y=0).
`For metal ion reduction,
`
`M2+ +vee_ —> M
`
`one obtains
`
`13(dc/dy)y=0 =i/veF
`
`(5a)
`
`(5.1)
`
`and Fick’s second law describes the change of the concentration profile in front
`of the electrode with time:
`
`(5.2)
`3c/8t=D(dc2/By2)
`After the critical transition time 1: the concentration of the metal at the elec-
`
`trode is depleted to zero and the mass flow i/veF can no longer be maintained.
`Then the current will be consumed additionally by another electrode process,
`for instance by H2 evolution with cathodic and 02 evolution with anodic proc-
`esses.
`
`Therefore the electrode potential at t = T will become so negative that hydro-
`gen is evolved H"’+e'~—>1/2H2 parallel to metal deposition and hydrogen evolu-
`tion would consume then larger and larger fractions of the applied current den—
`sity with further extension of time.
`Figure 5.2 b demonstrates,what occurs if a potential is applied to the cathode
`establishing according to Nernst’s law a metal ion concentration co being lower
`than cm. Then a steadily decreasing current density for metal deposition will be
`observed. The developing concentration profile becomes flatter with time,
`(dc/dy)
`0 and i is decreasing with time according to F”.
`Impo§=ed steady convection, and steady rates of electrochemical conversion
`expressed as time-independent current densities, generate steady concentration
`profiles and accordingly the one—dimensional Fick’s second law, Eq. (5.2.), is
`substituted by the two dimensional steady state Eq. (5.3):
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`3c/Bt=0=D(32c/ay2)—8(wc)/ax
`
`(5.3)
`
`y and x are coordinates perpendicular and parallel to the electrode and w is the
`flow velocity parallel to the electrode surface. Equation (5.3) is a simplified two-di—
`mensional expression, which holds for one—dimensional flow parallel to the elec—
`trode (in x direction) and diffusive mass transfer due to metal deposition towards
`the electrode in y direction, i.e. perpendicular to the electrode surface (compare
`Fig. 5.2 a and b). Steady viscous flow along a planar surface. generates a time inde-
`pendent velocity boundary layer and this, together with electrochemical conver-
`sion at the electrode, rules the spatial concentration distribution depicted in the
`
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`
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`Exhibit 1117_0005
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`

`

` 5 Mass Transfer by Fluid Flow, Convective Diffusion and...
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`(b)
`
`:
`an-l
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`y
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`c
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`Fig.5.2. Time independent concentration profiles are generated by forced flow along the
`electrode with steady velocity distribution a and steady concentration distribution b. A dif-
`fusion layer of thickness 5N defines steady state mass transfer conditions with km: D/SN
`
`schematic concentration profile, see Fig. 5.2 b. Equation (5.3) can only be solved
`after solution of the Navier/Stokes equation discussed below, leading to formula-
`tion of the velocity field in front of the electrode.
`
`5.3
`
`Fluid Dynamics of Viscous, lncompressible Media
`
`The simplified two-dimensional Navier—Stokes equation neglecting gravity
`forces for steady flow in xy-plane(x parallel and y perpendicular to a planar elec-
`trode), which describes the balance of all forces (acceleration and friction forc-
`es), acting on a volume increment of the fluid, reads:
`
`“axway
`
`azu
`azu
`lap
`Bu
`Bu
`_ ._=___ _ __
`
`pax+v[ax2+3y2]
`
`
`
`The treatment of fluid flow in electrolyzers can in general be confined to virtu-
`ally incompressible viscous media. Moreover, viscous flow along electrodes or
`between parallel-plate electrodes can very often, as done above, be treated as
`unidirectional one-dimensional with velocity profiles extending perpendicular
`to the electrode.
`
`.4
`
`(5
`
`a)
`
`(
`
`5,41)
`
`)
`
`“avaay
`
`82v
`82v
`18p
`av
`8v
`_ _=___ _ _
`
`pay+v[ax2 +3y2]
`
`u and v are velocities in x and y directions respectively and v is the kinematic vis-
`cosity v = 11 /p . In most cases of practical electrochemical relevance, that means
`in flow, along, or between parallel plates v is to a good approximation zero.
`
`Exhibit 1117_0006
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`Exhibit 1117_0006
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`

`

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`5.3 Fluid Dynamics of Viscous, Incompressible Media 85
`
`One can introduce adimensional quantities
`
`
`iu.,v.i p
`
`U
`
`U
`
`pU2
`
`(5.4 c)
`
`where L and U are characteristic quantities of the system under consideration. L
`is for instance the total length of the plate or the distance between two plates, U
`may mean the mean flow velocity or it is the maximal flow velocity. p is the den—
`sity of the fluid and p is the pressure.
`The use of adimensional quantities simplifies the differential equations and
`allows to define important adimensional numbers, the magnitude of which
`characterizes the flow behaviour of the whole system. One obtains the Navi-
`er/Stokes equations, for instance Eq. (5.4 a), in adimensional form:
`
`ui3_u1
`Bx‘
`
`.au‘
`8p*
`1
`+v —=————+—
`By“
`3):"
`Re
`
`3211*
`
`3211*
`
`(5.4 a”)
`
`For one-dimensional steady flow between two plates in x direction this reduc-
`es still further to
`
`2
`0 = JEEWLH
`p 3x
`ayz
`and in adimensional form to
`
`)6
`2
`X-
`
`0=—ap,+ia‘:
`EX
`Re By 2
`
`From Eqs. (5.4 a‘) and (5.5‘) the Reynolds number
`
`Re = E
`v
`
`and the Euler number
`
`Eu = p/ pU2
`
`can be extracted.
`
`(5.5)
`
`(5.5‘)
`
`(5.6)
`
`(5.7)
`
`The Reynolds number can be interpreted as the ratio of shear to acceleration
`forces in the system, whereas the Euler number gives the ratio of pressure versus
`acceleration forces.
`
`The Reynolds number defines the character of fluid flow in the considered sys-
`tem with forced convection. The magnitude of the Reynolds number is indicative
`of the type of flow — laminar or turbulent -,which prevails under given flow con-
`ditions in a given device. For instance for channel flow in ducts or between plates
`a limiting Reynolds number of Re=2000~3000 determines transition from lami—
`
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`Exhibit 1117_0007
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`

`

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`
` 86
`
`5 Mass Transfer by Fluid Flow, Convective Diffusion and...
`
`nar at lower to turbulent flow at higher Reynolds number. The Reynolds number
`is by far the most important adimensional number for most cases of forced vis-
`cous fluid flow dealt with in the context of electrochemical engineering.
`
`5.3.1
`Laminar vs Turbulent Flow
`
`Laminar flow - the expression is self explaining — means fluid flow without any
`velocity component perpendicular to the main direction of flow, that means
`without vortices or eddies. The flow can be divided up into lamella, between
`which there is no mixing and which interact with each other only by transmit—
`ting and exchanging momentum by viscous forces. Figure 5.3 a,b exemplifies
`this situation for unidirectional laminar flow between two parallel, rectangular
`plates and for divergent flow from a central bore outward between two parallel
`cylindrical plates.
`Laminar flow generates smooth velocity profiles at the walls of the respective
`systems, which for channel flow and flow between parallel plates is typically par-
`abolic (Fig. 5.4 a). Double integration of Eq. (5.5) leads to parabolic velocity dis—
`tributions — for instance for Hagen—Poiseulle flow through circular ducts and
`pipes.
`
`2
`
`w = wmax 1-[LJ
`
`(5.8)
`
`Increasing flow velocities strains the velocity profile established by viscous in—
`teraction of the flow lamellas more and more until it becomes unstable so that it
`
`
`
`E \ /
`._Q__.
`/l\
`
`W WNW/A
`
`(a)
`
`(b)
`
`Fig.5.3.a, b.8chematic of laminar flow between plates: a unidirectional flow in a duct; b ra-
`dial flow from a central bore between two circular disks
`
`Exhibit 1117_0008
`
`Exhibit 1117_0008
`
`

`

`
`
`
`
`5.3 Fluid Dynamics of Viscous, Incompressible Media
`
`87
`
`
`
`Fig. 5.4.3 Parabolic velocity profile for laminar flow between plates, in tubes and rectangular
`ducts. b Steep, almost piston—like profile of mean velocities in turbulent flow between plates,
`in tubes and ducts
`
`breaks down at its steepest part that means at the wall by formation of eddies.
`These eddies move stochastically in the direction away from the wall being carried
`along with the main stream of the fluid. Coming from the wall they reach the bulk
`I of the flow and are thereby strongly accelerated in flow direction, while eddies
`from the bulk, which are approaching the walls (where they replace the volume of
`freshly generated eddies), are decelerated. This means that as soon as eddies are
`generated under turbulent flow conditions momentum exchange between bulk
`and boundary of the fluid flow becomes much more efficient than under laminar
`flow so that the distribution of mean flow velocities perpendicular to the main flow
`direction becomes remarkably more equalised — but since under any flow condi-
`tion the flow velocity at the wall is zero, the slope of the velocity profile close to the
`wall becomes much steeper (Fig. 5.4 b) than for laminar flow. This means that with
`turbulent flux drag at the wall is greatly enhanced. Three examples follow, which
`demonstrate velocity distributions in front of a plate or between plate electrodes.
`
`5.3.2
`
`Velocity Distributions for laminar Flow
`
`5 .3 .2 . 1
`
`Singular Electrode: Unidirectional Laminar FIowAIong a Plate
`
`For a plate electrode far away from a counter electrode in an electrolysis cell or
`for the initial development of the velocity profile at the inlet of a gap between a
`pair of parallel plate electrodes double integration of Eq. (5.5) applying a power
`series model for the velocity distribution results in the definition of an effective
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`Exhibit 1117_0009
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`

`

` 5 Mass Transfer by Fluid Flow, Convective Diffusion and...
`
`
`
`
`Fig.5.5.a Extending velocity profiles for laminar flow along a plate. b Development and clo-
`sure of velocity profile at the entrance of a duct or tube.
`
`thickness of the velocity profile 51,, (thickness of Prandtl’s boundary layer)
`which increases with increasing distance Z from the leading edge (Fig. 5.5 a)
`[1]”
`
`w(x) = wmax[l.5(y/SW)—O.5(y/5w)3)
`
`with
`
`5w 2 4.64(xv/ wmax)°'5 £51,,
`
`(5.9 a)
`
`(5.9 b)
`
`If these equations apply for the development of the velocity profile between two
`parallel plates with distance 2b (Fig. 5.5 b), then the condition 8w=b means com-
`plete closure of the velocity profile in the gap. Further downstream this closed ve-
`locity profile between two parallel electrodes does not change any longer.
`
`5.3.2.2
`Pair ofPlanar Electrodes
`
`Fully established laminar flow between two plates with distance 2b yields in the par—
`abolic velocity profile depicted schematically in Fig. (5.5 b) and written in Eq. (5.9 c):
`
`w=wmx [l—(y/bV)
`
`(5.9 c)
`
`1
`
`For mathematical details compare ER. G. Eckert, Heat and mass transfer, 2nd, ed. Mac
`Graw—Hill, New York, 1959 The notation 8“ defines a boundary layer thickness, which dif
`fers from the definition of the thickness of the Prandtl layer 8”: 0°(8y/5w)y=0; 8w:
`3/25“; 5w is defined by the simultaneous boundary conditions: w=0 for y=0; w=w_, for
`y>8W and (5w/5y)=0 for y28w
`
`Exhibit 1117_OO10
`
`Exhibit 1117_0010
`
`

`

`
`
`5.3 Fluid Dynamics of Viscous, Incompressible Media
`
`89
`
`
`
`Fig.5.5.c Radial flow from a central bore in a gap between circular discs results in decrease
`of mean velocities according to r‘1
`
`one calculates the volumetric flow velocity
`
`V=2me(2/3)b
`With B = width of the electrodes
`
`and the total pressure drop
`
`A = v L B 3
`P
`3 “/(bl
`with L = length of the electrodes, and it = dynamic viscosity
`
`5.3 .23
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`Circular Capillary Gap Cell
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`(5.9 d)
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`(5.9e)
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`Divergent laminar flow between two plates from a central bore with radius r0
`outward through the gap between two circular parallel plate electrodes with out-
`er radius R (see Figs. 5.3 b and 5.5 c) — in the so called circular-capillary gap cell
`— generates steadily decreasing linear flow velocity in going from the central
`bore to the periphery.
`The Navier-Stokes equations at Eqs. (5.4 a) and (5.4 b) are expressed in cylin-
`
`drical coordinates (r and y) and transform to:
`2
`2
`
`8w_p apl va_w 18w w+3_w
`(510)
`war— pr arp
`3r2
`r 8r
`r2 Byz
`
`
`
`
`
`Exhibit 1117_0011
`
`

`

`
`
`5 Mass Transfer by Fluid Flow, Convective Diffusion and...
`
`Neglecting the acceleration/deceleration term du/dr for so called creeping
`flow and setting dzw/dr2 + 1/r(dw/dr) — w/r2 according to the continuity equa-
`tion” equal to zero yields:
`2
`0=£_a_pl+\;§l
`pr
`3r p
`31-2
`
`(5.1021)
`
`Due to the continuity condition one obtains for any given distance y from the
`middle plane and radius r:
`
`wy(r)=Cy/r
`
`(5.11)
`
`One arrives at the parabolic velocity profile and radial velocity distribution of
`Eq. (5.1 l)
`
`
`
`w(r,y)= 8::r(1—y2/b2)
`
`with the pressure drop:
`
`A R=
`Pr
`
`
`3va
`41tb3
`
`(l—r/R)
`
`5.4
`
`Mass Transport by Convective Diffusion
`
`5.4.1
`Fundamentals
`
`(5.12)
`
`(5.13)
`
`
`
`Mass flux density of species dissolved in stagnant or flowing electrolyte generat-
`ed by spatial concentration gradients is described by Fick’s law:
`
`fidiff = —D grad C
`
`(5.14)
`
`Fick’s first law is the most general description of diffusional mass transfer in
`the absence of additional convective mass transfer and reads in one-dimension-
`
`al form as applied to diffusive mass transport to large, flat electrodes:
`
`{1 =—D[$]
`By y=0
`
`(5.14 a)
`
`2
`
`In Cartesian coordinates the continuity equation reads: d2“ + dz" + d2" -0
`dx2
`dy2
`dz2
`
`Exhibit 1117_0012
`
`Exhibit 1117_0012
`
`

`

`
`
`5.4 Mass Transport by Convective Diffusion
`
`91
`
`
`
`Fig. 5.6. a Defining the Nernst diffusion layer thickness 5N by linearizing the concentration
`profile. b Schematic of relative extension of velocity and concentration boundary layers 51,,
`and 8N for freely developing laminar and fully developed turbulent flow in fluids possessing
`Schmidt numbers of from several hundreds to thousands
`
`The current density of any electrochemical process consuming solute species
`by electrochemical conversion couples the rate of electrochemical conversion
`with diffusive mass transport at and towards the electrode surface:
`
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`I
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`i = r1 oveF = D(8c/3y)y=oveF
`
`(5.15)
`
`The index 0 refers to y=0 that means at the electrode surface.
`Mass transport in technical electrolyzers is always caused by the interaction
`of convection and diffusion. The calculation of concentration profiles and hence
`current densities demands the simultaneous solution of Navier Stoke’s equa-
`tions and Fick’s first and second laws, which can be achieved for systems with
`laminar flow but not for turbulent flow. But under any such conditions the mass
`transfer rate and associated current densities are given by the product of the
`mass transfer coefficient km, which is a quantity which can be easily measured,
`and the driving concentration difference Ac:
`
`iznoveF=km(cm —c0)veF=kmAcveF
`
`(5.15 a)
`
`Linearizing the concentration profiles in front of a working electrode (see
`Fig. 5.6 a, b) defines the thickness 8N of the Nernst diffusion layer, which allows
`to interpret the mass transfer coefficient km as the ratio of D and 5N3):
`
`km=D/5N
`
`(5.16 a)
`
`
`
`3 SN is similarly defined as 59,;5N=cm(5y/5c)y=o
`
`Exhibit 1117_0013
`
`

`

` 5 Mass Transfer by Fluid Flow, Convective Diffusion and...
`
`Setting c0=0 defines the mass transfer limited current density ilim:
`
`llim
`
`= kmcwveF
`
`5.4.2
`
`(5.16 b)
`
`Dimensionless Numbers Defining Mass Transport Towards Electrodes by Convective
`Diffusion
`
`To find an explicit equation for the mass transfer coefficient, km, is only possible
`for laminar flow. Under turbulent flow one can only measure mass transport co-
`efficients by measuring mass transport limited current densities. But this is a te-
`dious affair as mass transfer is influenced often by a great number of variables.
`Dimensional analysis of the problem allows to reduce considerably the number
`of variables which have to be taken into account for mass transfer determina-
`
`tions by introducing dimensionless groups which comprise several different
`characteristic quantities of the respective system like for instance mass transfer
`coefficient, velocity, density and characteristic lengths which might for instance
`be the interelectrodic distance or the electrode length in case of virtually singu—
`lar electrode.
`
`The reduction in number of variables thus obtained amounts exactly to the
`number of different fundamental dimensional quantities (length, time, mass,
`charge, voltage) which are used in total to define the complete set of variables.
`By introducing dimensionless groups one obtains then equations, “adimension-
`a1 correlations”, which relate these quantities to each other for a given flow ge-
`ometry. For laminar flow these correlations are obtained by algebraic calcula-
`tion. For turbulent flow such correlations have to be determined experimentally.
`For mass transfer under forced convection there exist at least three different
`
`dimensionless groups: The Sherwood number, Sh, which contains the mass
`transfer coefficient, the Reynolds number, Re, which contains the flow velocity
`and defines the flow condition (laminar/turbulent) and the Schmidt number, Sc,
`which characterizes the diffusive and viscous properties of the respective fluid
`and which describes the relative extension of fluid—dynamic and concentration
`boundary layer:
`
`
`
`kmL
`Sh=—
`D
`
`Re=w—L
`V
`
`Sc=v/D
`
`5.17
`
`(
`
`8)
`
`(5.17 b)
`
`(5.17'c)
`
`If forced convection does not apply, density differences between the electro-
`lyte close to the electrodes and the bulk of the solution may be generated for in-
`stance by metal deposition and dissolution or gas evolution. Then free or so
`
`Exhibit 1117_OO14
`
`Exhibit 1117_0014
`
`

`

`
`
`
`
`5.4 Mass Transport by Convective Diffusion 93
`
`called natural convection is generated and becomes very often much more in-
`tense than forced convection driven from outside by pumping the electrolyte
`through the cell. Fluid flow under free convection is characterised by the
`Grashoff number, Gr, which contains the gravity constant, g, the density differ-
`ence of the electrolyte at the electrode (p0) and the electrolyte bulk (pm).
`
`Gr=gL3(pe,m-pe,o)/pmv2
`
`(5.17 d)
`
`In general one tries to present the dependence of Sh on Re (or Gr) and Sc —
`and possibly on other adimensional quantities like the ratio l/L of characteristic
`lengths in form of a power series:
`
`Sh = CRem(or Gr“)ScP (1l /L)°‘(12 /L)r
`
`(5.18)
`
`The experimental determination of Sh is quite easy in electrochemical sys-
`tems. Measuring the mass
`transfer
`limited current density and using
`Eq. (5.16 b) one obtains immediately the surface averaged Sh from the numeri-
`cal value of ilim:
`
`
`Sh : i'limL
`cmveFD
`
`5.4.3
`
`(5.18 a)
`
`Hydrodynamic Boundary Layer and Nernst Diffusion Layer: Planar Electrodes
`
`Figure 5.6 a,b depicts schematically the relative extension of velocity and con-
`centration profiles in front of a planar electrode for (a) laminar and (b) turbulent
`flow. (Assumption: potential conditions set for mass transfer limited current
`density and freely developing, that means not yet closed, velocity boundary lay-
`er for laminar flow). For both cases at sufficiently high distance from the elec-
`trode surface the maximal fluid velocity wmax is acquired.
`Turbulent flow is distinguished from laminar flow by two main differences:
`(a) Due to the eddies, which characterise the turbulent flow condition, the flow
`in the bulk of the fluid is stochastically fluctuating in all directions around
`Wmx — which means that fluctuating velocity contributions not only parallel
`but also perpendicular to the main velocity vector are generated by eddies
`and may be observed. The intensity of these fluctuations is decaying more
`and more towards the electrode surface very close to the electrode surface in
`the region 50 of the so called viscous sublayer.
`(b) Under comparable conditions (plate dimensions, kinematic viscosity of the
`fluid) the extension of the viscous sublayer of a turbulent boundary layer is
`remarkably less extended than the velocity boundary layer for a freely devel-
`oping laminar flow. This difference bears also on mass transfer: Quite simi-
`larly the extensions of the Nernst diffusion layers connected to mass transfer
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`Exhibit 1117_0015
`
`

`

`
`
`5 Mass Transfer by Fluid Flow, Convective Diffusion and...
`
`for turbulent flow is remarkably smaller and hence mass transfer is much
`more efficient for turbulent flow than for laminar flow.
`
`A quantitative treatment [1] of the relative extension SF, of the velocity bound-
`ary layer for freely developing laminar flow and the associated Nernst diffusion
`layer thickness 8N shows that it is governed by the Schmidt number (Sc = v/D),
`which is indicative of the relative effectiveness of “momentum diffusion” (de—
`scribed by the kinematic viscosity v = n/p ) and mass diffusion (characterised
`by D). For aqueous electrolytes, with VzIO‘2 cm2 s'1 and D-~~10‘S cm2 s“, the
`Schmidt number is approximately 1000.
`For freely developing laminarflow along a plate one obtains
`
`(8N /5,,,)x m = Sc'llz
`
`(5.19 a)
`
`so that for aqueous electrolytes this ratio of the extension of both boundary lay-
`ers amounts to approximately 30 everywhere on the plate.
`For turbulentflow theoretical considerations arrive at a ratio of the thickness
`of the Nernst layer to that of the laminar sublayer of
`
`(SN/80)
`
`tur
`
`b =8c‘1/2 to Sc‘”3
`
`(5.19 b)
`
`which would be approximately from 30 to 10 for aqueous electrolytes.
`Figure 5.7 demonstrates schematically for free flow along a plate, compare
`Fig. 5.5 a, how the laminar (with thickness 39,) and (after arriving at a critical
`length) the turbulent boundary layer and laminar sublayer (with thickness 50)
`are developing. According to Eq. (5.19 a,b) the Nernst diffusion layer (with thick-
`ness 8N) would develop below 8?, and 50. In quantitative terms one obtains Sh
`(the mean Sherwood number) for laminar flow along a plate electrode of length L
`
`
`
`
`(5.20 a)
`
`(5.20 b)
`
`§L =0.67 Rey23c1’3;Re<3-105
`
`and for turbulent flow the local Sherwood number is given by
`
`ShL = 0.03 Re?8 SCI/3; Re > 3105
`
`5
`
`laminarity
`
`turbulence
`
` o
`
`X-
`
`“9.5.7. Change from laminarity to turbulence for freely developing parallel flow along a
`plate induces compression of the thickness of the velocity profile or Prandtl layer thickness
`
`Exhibit 1117_OO16
`
`Exhibit 1117_0016
`
`

`

`
`
`5,4 Mass Transport by Convective Diffusion 95
`
`In both cases the length L being used for Sh and Re is the distance from the
`leading edge.
`
`5.4.4
`
`Mass Transport Towards a Singular Planar Electrode4lUnder Laminar Forced Flow
`
`Calculation of mass transfer towards an isolated electrode under conditions of
`
`forced convection along its surface under steady state conditions and laminar
`flow starts with the solution of the Navier/Stokes equation, which describes the
`velocity distribution in y—direction (perpendicular to the electrode surface) and
`the growth of the fluid dynamic boundary layer in x direction that means down-
`stream, Eq. (5.9 a,b). One then applies the mass transfer equation neglecting dif-
`fusive mass transfer in x direction:
`'
`
`0=D(32c/5y2)—w(6c/Bx)
`
`Integration with respect to y yields
`
`D(8c/ay)o =£Tw<cm —c)dy
`
`o
`
`with the boundary conditions:
`
`c0 =0(means i=ium),
`
`cym 2cm
`
`(5.21)
`
`(5.22)
`
`(5.23)
`
`and with wx,2 equal to the already obtained velocity distribution of the laminar
`boundary layer, Eq. (5.8 ,b), one obtains:
`
`D(Bc/E)y)o=w°<,a—x 26L__[8L]
`
`0
`
`W
`
`a °° 3
`
`1
`
`3
`
`(cw—c)dy
`
`(5.24)
`
`In the next step the assumption of the development of a spatially similar
`boundary layer profile which independent of the extension of the Nernst diffusion
`layer satisfies in principle the same (reduced) mathematical description which is
`taken to be a polynomial of third order with coefficients a,b and c leads to”:
`
`c(x,y)=c.. +Ac(b(y/5C)+c(y/5c)2+d(y/55)3)and5c =5c(x) ‘
`
`(5.25)
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`4 The counter electrode is so far away that is does not influence the velocity profile. This
`holds for the very entrance of a gap or duct between any two planar electrodes, too. As
`the fluid enters this duct the fluid at the leading edge of any of the two electrode does not
`yet “feel” the presence of the counter electrode. The following example is treated more
`extensively in order to convey to the reader an impression and better understanding
`that, and how, mass transfer under laminar conditions can be calculated.
`
`Exhibit 1117_0017
`
`

`

` 5 Mass Transfer by Fluid Flow, Convective Diffusion and...
`
`(5.25 a)
`
`(5.26)
`
`dy (5.27)
`
`3
`
`y
`
`l
`
`y 2
`
`1—2 a +3 3:
`
`According to mathematically similar profiles and boundary conditions the
`mathematical formulations for the velocity distribution in the Prandtl layer and
`the concentration profile of the Nernst layer are similar only being distinguished
`by different extensions 81),, 8w and 8N , 5c of the velocity and the concentration
`boundary layers. By introducing Eq. (5.26) into Eq. (5.24) and limiting the inte-
`gration to Be one obtains
`
`
`
`Introducing the boundary conditions
`(ac/Ely)8c =0;
`(82 c/Bzy)0 =0; c5c =cm; c0 =0
`results in:
`
`0,, 3
`
`c(x,y)=c E[%j—E[%J
`
`l
`
`2
`
`and 5c =55(x)
`
`3
`
`_
`
`ZDISCmeaxCW axe! E[6—]—2[-8‘:]
`
`35‘ 3
`
`y
`W
`
`l
`
`y
`
`2
`
`Solving the integral and introducing SW for the isolated planar electrode,
`1/3
`
`compare Eq.(5.96), one arrives with 8C/8w =(E] 86“”, SN =-2—Sc and
`
`3
`
`SW =[
`
`
`280v
`
`13wmax
`
`x ]
`
`”2
`
`at
`
`_(ac/8y)y=0 =cm/8N =cmgé
`
`and equating
`
`km 2 D(ac/3y)0 Ci
`
`one obtains the local Sherwood number
`
`th =km?=[—2-183—0]m(:—:—)m Reg2Sc“3 =0.331 Rey2 Sc"3
`
`(5.28)
`
`(5.29)
`
`(5.30)
`
`5 There is a similar difference between the thickness 8c as defined by the boundary conditions
`Eq. (5.25 a) and the polynomial Eq. (5.25) and the thickness of the Nernstian diffusion layer
`--1
`
`SN mag—C]
`
`y o
`boundary layer
`
`as between SW and on which both define the extension of the velocity
`
`Exhibit 1117_OO18
`
`Exhibit 1117_0018
`
`

`

`5.4 Mass Transport by Convective Diffusion
`
`97
`
`integrating th from zero to x = L, the length of the plate, and dividing by L leads
`to Eq. (5.20 a) for the mean Sherwood number:
`
`*
`
`l
`
`Sh = %lthdx = 0.662 ReE’SSclB
`
`0
`
`(5.31)
`
`5.4.5
`Channel Flow and Mass Transfer to Electrodes of Parallel Plate Cells for Free
`and Forced Convection
`
`Free and forced convection through an interelectrodic gap between two planar
`electrodes is the most often encountered case in electrochemical engineering
`practice. Hydrometallurgical electrorefining and electrowinning of metals like
`copper, zinc and lead as well as electrochemical gas evolution at vertical elec-
`trodes are typical cases for electrolysis cells with free convection being caused
`by density differences of the electrolyte.
`
`5.4.5.1
`Free Convection at Isolated Planar Electrodes and between Two Vertical Electrodes
`
`At the anode of an electrolysis cell for copper refining the dissolution of copper
`increases the CuSO4-concentration and the electrolyte density close to the anode
`surface, whereas CuSO4-depletion and a corresponding decrease of the electro-
`lyte density has to be taken into account at the cathode (Fig. 5.8 a) due to cathodic
`copper deposition. This causes a downward flow along the anode and an upward
`flow along the cathode, Fig. 5.8 b, both flows improving the mass transfer condi-
`
`
`
`