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`WHAlearyBy
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`WBeae
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`_Engineering
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`_andTechnology
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`in Chemical|
`_andOther
`Industries —
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`Science
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`Exhibit 104ie
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`Tennant Company
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`Tennant Company
`Exhibit 1017
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`Exhibit 1017_0001
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`

`

`
`
`Prof. Dr. Hartmut Wendt
`Institut fiir Chemische Technologie
`TU Darmstadt
`Petersenstrafe 20
`D-64287 Darmstadt
`Germany
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`Prof. Dr. Gerhard Kreysa
`Karl WinnackerInstitut
`DECHEMAe.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 bibliographicalreferences.
`ISBN 3-540-64386-9 (hardcover: alk. paper)
`1. Electrochemistry, Industrial. I. Kreysa, Gerhard.II. Title.
`TP255.W46 1999
`660° .297--de21
`
`This workis subject to copyright. All rights are reserved, whether the whole partof the material is concerned,
`specifically the rights of translation,reprinting, reuseofillustrations, recitation, broadcasting, reproduction
`on microfilm orin anyother way, and storage in data banks. Duplication of this publication or parts thereof
`is permitted only underthe provisions of the German Copyright Law of September9, 1965,in its current ver-
`sion, and permission for use mustalways be obtained from Springer-Verlag.Violationsareliable 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,
`evenin the absenceofa specific statement, that such namesare exemptfrom therelevant protective laws and
`regulations and therefore free for generaluse.
`
`Typesetting: MEDIO,Berlin
`Coverdesign: Design & Production, Heidelberg
`
`SPIN: 10675807
`
`2/3020-5 4 3 2 1 0 - Printed on acid-free paper.
`
`Exhibit 1017_0002
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`Exhibit 1017_0002
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`

`

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`
`CHAPTER 5
`
`
`MassTransferby Fluid Flow, Convective Diffusion and Ionic
`Electricity Transport in Electrolytes and Cells
`
`5.1
`Introduction
`
`The performanceof electrochemical processesis not only determined by charge
`transfer and electrode kinetics, but a numberof additional phenomena cause
`and rule the electrode kinetics (in the microkinetic as well as macrokinetic
`sense), the heat balanceof the cell and the mass balancesofall process streams
`(electrolyte, gases, solid products). Amongthese factors the fluid dynamic con-
`ditions, under whichthe electrolyte enters, passes andleaves the electrolysis cell
`or moves in it under free convection, is the most powerful process parameter
`since hydrodynamics rule mass and heattransport.
`Heat transport is importantas 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
`acrossthe electrode surface. It may determine the overall currentefficiency, con-
`version selectivity and spacetime yield of the process by local inhomogeneities
`of the current density for instance by locally too high current density which
`might exceed the masstransfer limited current density.
`The proper handling of these characteristic process determinants: fluid dy-
`namics, mass transportand heat transport together with proper managementof
`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 hasto be aware that the cell aloneis only one part of the whole
`process. Electrolyte make up, productisolation and recyclingof the electrolyte
`have to be taken into consideration,too.
`
`5.2
`Fluid Dynamics and Convective Diffusion
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`Diffusive mass transportin an electrolyte solution without superimposed con-
`vectionis relatively ineffective because of the low diffusion coefficients experi-
`
`
`enced in solvents of viscosities comparable to that of water. In usual dielectric
`
`Exhibit 1017_0003
`
`

`

`
`
`82
`
`5 MassTransfer by Fluid Flow, Convective Diffusion and...
`
`Exhibit 1017_0004
`
`
`(b)
`
`solvents the diffusion coefficient of a low molecular weight species is similar to
`that in aqueoussolution, which is of the order of approx. 10~° cm?s~!.
`Moreover,diffusion aloneis typically a non-steady process that meanson 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 developmentof concentration profiles of metal ions
`in front of a cathode on which the metal ions are deposited with time-inde-
`pendentcurrent density, i, in presence of supporting electrolyte of high con-
`centration, which allows for metal ion diffusion alone, excluding mass trans-
`port by migration.
`
`p
`
`Fig. 5.1.a Schematic showing the development of concentration profiles of metal ions in a
`stagnantelectrolyte upon galvanostatic cathodic metal deposition (M** + ze-—>M). Note
`that (dc/dx),_9 remains unchanged becausei is constant b Developmentof concentration
`profile of metalionsin stagnantelectrolyte upon imposing a constantpotential @ <@_quij on
`the electrode. Declining values (dc/dy),_ reflect in steadily declining current densities ac-
`cordingto t™!/~ law (see insert).
`
`Exhibit 1017_0004
`
`

`

`
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`5.2 Fluid Dynamics and Convective Diffusion 83
`
`For a given current density i the gradient (dc/dy)is fixed at the electrode sur-
`face (y=0).
`For metal ion reduction,
`
`M** +v,° >M
`
`one obtains
`
`(5a)
`
`(5.1)
`D(dc/dy)_, =i/v,F
`and Fick’s second law describes the changeof the concentrationprofile in front
`of the electrode with time:
`
`ac/dt=D(ac?/ay?)
`
`(5.2)
`
`After the critical transition time t the concentration of the metalat the elec-
`trode is depleted to zero and the massflow i/v,F can no longer be maintained.
`Then the current will be consumed additionally by anotherelectrode process,
`for instance by H, evolution with cathodic and O, evolution with anodic proc-
`esses.
`
`Therefore the electrode potential at t = t will become so negative that hydro-
`gen is evolved H*+e"—1/2H,parallel to metal deposition and hydrogen evolu-
`tion would consumethen larger andlarger fractions of the applied current den-
`sity with further extension oftime.
`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 cy being lower
`thanc,,. Then a steadily decreasing current density for metal deposition will be
`observed. The developing concentration profile becomes flatter with time,
`(de/dy
`5 andi is decreasing with time according to t~°°.
`Imposed steady convection, and steadyrates 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):
`
`dc/dt=0 =D(d*c/dy?)—a(we)/dx
`
`(5.3)
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`y and x are coordinates perpendicular and parallel to the electrode andwis 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 viscousflow along a planar surfacegenerates a time inde-
`pendentvelocity boundary layer andthis, together with electrochemical conver-
`sion at the electrode, rules the spatial concentration distribution depicted in the
`
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`Exhibit 1017_0005
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`

`

` 5 MassTransfer by Fluid Flow, Convective Diffusion and...
`
`Exhibit 1017_0006
`
`(a)
`
`|
`Sp, —elI
`|
`
`Y
`
`iy:
`
`(b)
`
`!
`|
`on-y
`
`¢
`
`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 5y defines steady state mass transfer conditions with k= D/d5y
`
`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 Dynamicsof Viscous, Incompressible Media
`
`The treatmentof fluid flow in electrolyzers can in generalbe confinedto 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.
`The simplified two-dimensional Navier-Stokes equation neglecting gravity
`forces for steadyflow in xy-plane(x parallel and y perpendicularto a planar elec-
`trode), which describes the balanceofall forces (acceleration andfriction forc-
`es), acting on a volume increment ofthe fluid, reads:
`
`du.
`os
`
`{d*u du
`lop
`_du_
`pe eo] ee ae
`
`a 1222.2)
`"axay 132 =
`
`[dv dv
`lop.
`_dv_
`ov
`a4 ya yy) D4
`
`4
`
`(5.4)
`(5.4
`b)
`
`5.4b
`
`uand varevelocities in x and y directions respectively andv is the kinematic vis-
`cosity v = 11/p. In mostcasesofpractical electrochemical relevance, that means
`in flow, along, or between parallel plates v is to a good approximation zero.
`
`Exhibit 1017_0006
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`

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`5.3 Fluid Dynamics of Viscous, Incompressible Media 85
`
`Onecan introduce adimensional quantities
`*
`x =
`
`*
`
`Ve
`*
`u
`*
`y
`Seb We! WV SS
`
`
`P
`
`5.4
`
`+
`
`=
`
`>
`
`x L
`
`c)
`(5.4
`ou?
`P
`yp
`U
`ye
`whereL and U are characteristic quantities of the system underconsideration. L
`is for instance the total length of the plate or the distance between twoplates, U
`may mean the meanflow velocity orit is the maximal flow velocity. p is the den-
`sity of the fluid andp 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:
`
`For one-dimensionalsteady flow between twoplatesin x direction this reduc-
`es still further to
`
`(5.4 a")
`
`2
`
`oxOPy you
`p ox
`dy?
`and in adimensionalform to
`
`*
`
`7
`*
`
`
`pa,ie
`dx Redy?
`
`From Eqs. (5.4 a") and (5.5") the Reynolds number
`
`Rete
`Vv
`
`and the Euler number
`
`Eu=p/pU?
`
`(5.5)
`
`(5.5)
`
`(5.6)
`
`(5.7)
`
`can be extracted.
`The Reynolds numbercan be interpreted as theratio of shear to acceleration
`forces in the system, whereas the Euler numbergivestheratio of pressure versus
`acceleration forces.
`The Reynolds numberdefines the characteroffluid flow in the considered sys-
`tem with forced convection. The magnitude of the Reynolds numberis indicative
`of the type of flow - laminaror turbulent -, which prevails undergivenflow con-
`ditions in a given device. For instance for channelflow in ducts or between plates
`a limiting Reynolds number of Re=2000-3000 determinestransition from lami-
`
`
`
`
`Exhibit 1017_0007
`
`

`

`
`
` 86
`
`5 Mass Transfer by Fluid Flow, Convective Diffusion and...
`
`Laminarflow - the expressionis self explaining - meansfluid 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 byviscous 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 twoparallel
`cylindricalplates.
`Laminarflow generates smooth velocity profiles at the walls of the respective
`systems, which for channelflow and flow betweenparallelplatesis 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.
`
`Exhibit 1017_0008
`
`nar at lowerto turbulentflow at higher Reynolds number. The Reynolds number
`is by far the most important adimensional numberfor mostcases of forced vis-
`cous fluid flow dealt with in the context of electrochemical engineering.
`
`5.3.1
`Laminarvs Turbulent Flow
`
`2
`
`W=Wma i-(£]
`
`(5.8)
`
`Increasing flow velocities strains the velocity profile established by viscous in-
`teraction of the flow lamellas more and moreuntil it becomes unstable so thatit
`
`N\A
`af j=
`JI \
`
`KLLLA\ WL
`VLLLMLLLULLLA
`WLI CLL,
`
`(a)
`
`(b)
`
`Fig. 5.3.a, b. Schematic of laminar flow between plates: a unidirectional flow in a duct; b ra-
`dial flow from a central bore between twocircular disks
`
`Exhibit 1017_0008
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`

`

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`
`5.3 Fluid Dynamics of Viscous, Incompressible Media
`
`87
`
`Fig. 5.4.4 Parabolic velocity profile for laminar flow betweenplates, in tubes and rectangular
`ducts. b Steep, almost piston-like profile of mean velocities in turbulent flow betweenplates,
`in tubes and ducts
`
`breaks downat its steepest part that means at the wall by formation of eddies.
`These eddies movestochastically in the direction away from the wall being carried
`along with the main stream ofthefluid. Coming from the wall they reach the bulk
`of the flow and are thereby strongly accelerated in flow direction, while eddies
`from the bulk, which are approachingthe walls (where they replace the volumeof
`freshly generated eddies), are decelerated. This means that as soon as eddies are
`generated under turbulent flow conditions momentum exchange between bulk
`and boundaryof the fluid flow becomes much moreefficient than under laminar
`flow so thatthe 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 muchsteeper(Fig. 5.4 b) than for laminar flow. This meansthat with
`turbulent flux dragat 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 FlowAlong a Plate
`
`For a plate electrode far away from a counterelectrodein anelectrolysis cell or
`for the initial developmentofthe 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 ofan effective
`
`
`
`Exhibit 1017_0009
`
`

`

` 5 MassTransfer by Fluid Flow, Convective Diffusion and...
`
`
`Exhibit 1017_0010
`
`Fig. 5.5.a Extending velocity profiles for laminar flow along a plate.b Development andclo-
`sure of velocity profile at the entrance of a duct or tube.
`
`thickness of the velocity profile 5p, (thickness of Prandtl’s boundary layer)
`which increases with increasing distance Z from the leading edge (Fig. 5.5 a)
`[1]?
`
`w(x}= Wnas(15(y/3y)-05(y/5y)"|
`
`with
`
`By = 4.64(xV/ Wimae)=+B
`
`(5.9 a)
`
`(5.9 b)
`
`If these equations apply for the developmentofthe velocity profile between two
`parallel plates with distance 2b (Fig. 5.5 b), then the condition 5,=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 yieldsin the par-
`abolic velocity profile depicted schematically in Fig. (5.5 b) and written in Eq. (5.9 c):
`
`W = Winax(1-(v/0)"}
`
`(5.9 c)
`
`1
`
`For mathematical details compare E.R. G. Eckert, Heat and masstransfer, 2nd, ed. Mac
`Graw-Hill, New York, 1959 The notation 6, defines a boundarylayer thickness, which dif
`fers from the definition of the thickness of the Prandtl layer dp,=w..(8y/8w),-o; 5y=
`3/28p,; 5,, is defined by the simultaneous boundaryconditions: w=0 for y=0; w=w,, for
`y>6,, and (Sw/dy)=0 for y25,,
`
`Exhibit 1017_0010
`
`

`

`
`
`5.3 Fluid Dynamics of Viscous, Incompressible Media
`
`89
`
`Fig. 5.5.¢ Radial flow from a central bore in a gap betweencircular discs results in decrease
`of meanvelocities according to r}
`
`one calculates the volumetric flow velocity
`V = 2BWmax(2/3)b
`with B = width of the electrodes
`
`andthetotal pressure drop
`
`Ap=3VpLL/(Bb?
`
`as pL/(Bb*)
`with L = length of the electrodes, and p=dynamic viscosity
`
`5.3.2.3
`Circular Capillary Gap Cell
`
`(5.9 d)
`
`(5.9 e)
`
`Divergent laminar flow between two plates from a central bore with radius ro
`outward throughthe gap between twocircular 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 equationsat Eqs. (5.4 a) and (5.4 b) are expressedin cylin-
`drical coordinates (r and y) and transform to:
`
`dw_p opi [ze low w me9 et
`
`—— — — +—__
`ror
`y?
`gy?
`or?
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`(5.10)
`
`
`
`Or
`
`pr orp
`
`
`
`
`Exhibit 1017_0011
`
`

`

`
`
` 5 Mass Transfer by Fluid Flow, Convective Diffusion and...
`
`Massflux density of species dissolved in stagnantorflowingelectrolyte generat-
`ed by spatial concentration gradients is described by Fick’s law:
`
`Naif =-D grad c
`
`(5.14)
`
`Fick’s first law is the most general description of diffusional mass transferin
`the absenceof additional convective mass transfer and reads in one-dimension-
`al form as appliedto diffusive mass transportto large,flat electrodes:
`
`n=-1(25)
`oy cali
`
`(5.14a)
`
`2
`
`2
`2
`2
`.
`sf
`:
`;
`In Cartesian coordinates the continuity equation reads: d“u + dév + d'w =0
`dx?
`dy?
`dz?
`
`Exhibit 1017_0012
`
`Neglecting the acceleration/deceleration term du/dr for so called creeping
`flow and setting d?w/dr* + 1/r(dw/dr) ~ w/r? accordingto the continuity equa-
`tion?) equalto zero yields:
`2
`p= _EP2 oe
`pr orp
`or
`
`(5.10 a)
`
`Dueto the continuity condition one obtains for any given distance y from the
`middle plane andradiusr:
`w,(r)=C,/r
`
`(5.11)
`
`One arrives at the parabolic velocity profile and radial velocity distribution of
`Eq.(5.11)
`
`w(ty)= (1-926?)
`
`with the pressure drop:
`
`R
`3 Vvp
`"Amb? (
`=——_{1-r/R
`
`)
`
`5.4
`MassTransport by Convective Diffusion
`
`5.4.1
`Fundamentals
`
`(5.12)
`
`(5.13)
`
`Exhibit 1017_0012
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`

`

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`
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`
`
`5.4 Mass Transport by Convective Diffusion
`
`91
`
`
`
`Fig. 5.6.a Defining the Nernst diffusion layer thickness 5y by linearizing the concentration
`profile. b Schematic ofrelative extension of velocity and concentration boundary layers 5p,
`and dy for freely developing laminar and fully developed turbulentflow in fluids possessing
`Schmidt numbersof from several hundredsto thousands
`
`(5.15)
`
`ky, =D/8y
`
`(5.16 a)
`
`
`
`The current density of any electrochemical process consumingsolute species
`by electrochemical conversion couples the rate of electrochemical conversion
`with diffusive mass transport at and towardstheelectrode surface:
`i=n oV,F=D(de/9y) VE
`The index g refers to y=0 that meansat the electrode surface.
`Masstransportin 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
`laminarflow butnot for turbulentflow. But under any such conditions the mass
`transfer rate and associated current densities are given by the product of the
`masstransfer coefficient k,,, which is a quantity which can be easily measured,
`and the driving concentration difference Ac:
`(5.15a)
`i= gVeF = ky, (Cu. Co Ve = KpACVEE
`Linearizing the concentration profiles in front of a working electrode (see
`Fig. 5.6 a, b) defines the thickness 5, of the Nernst diffusion layer, which allows
`to interpret the mass transfercoefficient k,, as the ratio of D and 5y:
`
`
`3 Sy is similarly defined as 5p,35y=c..(y/5c)y<9
`
`
`
`
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`
`
`Exhibit 1017_0013
`
`

`

` 5 Mass Transfer by Fluid Flow, Convective Diffusion and...
`
`Setting cy=0 defines the mass transfer limited current density ij,,:
`
`ilim = KmC@ VeF
`
`(5.16 b)
`
`5.4.2
`Dimensionless Numbers Defining Mass Transport TowardsElectrodes by Convective
`Diffusion
`
`Exhibit 1017_0014
`
`To find an explicit equation for the mass transfer coefficient, k,,, is only possible
`for laminar flow. Under turbulent flow one can only measure masstransport co-
`efficients by measuring masstransport limited currentdensities. But this is a te-
`dious affair as mass transferis influenced often by a great numberofvariables.
`Dimensionalanalysis 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 mightfor instance
`be the interelectrodic distance or the electrode length in caseofvirtually singu-
`lar electrode.
`The reduction in numberofvariables thus obtained amounts exactly to the
`numberof different fundamental dimensional quantities (length, time, mass,
`charge, voltage) whichare usedin total to define the complete set of variables.
`By introducing dimensionless groups one obtains then equations,“adimension-
`al 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 turbulentflow such correlations have to be determined experimentally.
`For mass transfer underforced 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
`boundarylayer:
`
`sh= Km
`
`Re= we
`Vv
`
`Sc=v/D
`
`(5.17 a)
`
`(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 generatedforin-
`stance by metal deposition and dissolution or gas evolution. Then free or so
`
`Exhibit 1017_0014
`
`

`

`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`5.4 Mass Transport by Convective Diffusion 93
`
`called natural convection is generated and becomes very often much morein-
`tense than forced convection driven from outside by pumpingthe 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-
`enceof the electrolyte at the electrode (9) and theelectrolyte bulk (p,,).
`
`(5.17 d)
`Gr= gL (pe. —Peo)/P-V
`In general onetries to present the dependence of Sh on Re (or Gr) and Sc -
`and possibly on other adimensional quantities like the ratio //L of characteristic
`lengths in form of a powerseries:
`(5.18)
`Sh=CRe™(or Gr")Sc? (1,/L)*(Ip/L)
`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 ofij;,,:
`
`Sh= HimL
`c..VeFD
`
`(5.18 a)
`
`5.4.3
`Hydrodynamic Boundary Layer and Nernst Diffusion Layer: Planar Electrodes
`
`Figure 5.6 a,b depicts schematically the relative extension of velocity and con-
`centrationprofiles in front of a planarelectrode for (a) laminar and (b)turbulent
`flow. (Assumption: potential conditions set for mass transfer limited current
`density and freely developing, that means notyet closed, velocity boundarylay-
`er for laminar flow). For both cases at sufficiently high distance from the elec-
`trode surface the maximalfluid velocity w,,,, is acquired.
`Turbulent flow is distinguished from laminarflow 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 fluctuatingin all directions around
`Wymax ~ Which meansthatfluctuating velocity contributionsnot only parallel
`but also perpendicular to the main velocity vector are generated by eddies
`and may be observed. Theintensity of these fluctuations is decaying more
`and more towards the electrode surface very close to the electrode surface in
`the region 8, 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 boundarylayeris
`remarkably less extended than the velocity boundarylayer for a freely devel-
`oping laminar flow. This difference bears also on masstransfer: Quite simi-
`larly the extensionsof the Nernstdiffusion layers connected to masstransfer
`
`
`
`Exhibit 1017_0015
`
`

`

`A quantitative treatment[1] ofthe relative extension 5p, of the velocity bound-
`ary layer for freely developing laminar flow and the associated Nernstdiffusion
`layer thickness 5, showsthat 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=1/p ) and mass diffusion (characterised
`by D). For aqueouselectrolytes, with v~10-? cm*s"! and D~10-> cm?s"1, the
`Schmidt numberis approximately 1000.
`Forfreely developing laminarflow alongaplate one obtains
`(5y82+).tam = $c?
`(5.19 a)
`so that for aqueouselectrolytes this ratio of the extension of both boundarylay-
`ers amounts to approximately 30 everywhere on theplate.
`For turbulentflow theoretical considerationsarrive at a ratio of the thickness
`of the Nernstlayer to that of the laminar sublayer of
`
`
`Exhibit 1017_0016
`
`
`<a
`
`Fig.5.7. Change from laminarity to turbulence for freely developing parallel flow along a
`plate induces compression of the thicknessof the velocity profile or Prandtl layer thickness
`
`5 Mass Transfer by Fluid Flow, Convective Diffusion and...
`
`for turbulent flow is remarkably smaller and hence mass transfer is much
`moreefficient for turbulent flow than for laminar flow.
`
`(8y 180)arp = $c7¥/? to Sc“"3
`which would be approximately from 30 to 10 for aqueouselectrolytes.
`Figure 5.7 demonstrates schematically for free flow along a plate, compare
`Fig. 5.5 a, how the laminar (with thickness 5),) and(after arriving at a critical
`length) the turbulent boundarylayer and laminar sublayer (with thickness 59)
`are developing. Accordingto Eq. (5.19 a,b) the Nernst diffusion layer (with thick-
`ness 5) would develop below 5p, and 5p. In quantitative terms one obtains Sh
`(the mean Sherwood number)for laminar flow alonga plate electrode oflength L
`
`(5.19 b)
`
`Sh, =0.67 Rel? Sc¥8; Re <3-10°
`
`and for turbulent flow the local Sherwood numberis given by
`
`Sh, =0.03 Re?® Sc’?; Re>3-10°
`
`(5.20 a)
`
`(5.20 b)
`
`5
`
`laminarity
`
`turbulence
`
` 0
`
`Exhibit 1017_0016
`
`

`

`
`
`5.4 Mass Transport by Convective Diffusion 95
`
`In both cases the length L being used for Sh andReis the distance from the
`leading edge.
`
`5.4.4
`MassTransport Towardsa Singular Planar Electrode*)Under Laminar Forced Flow
`
`
`
`0=D(a%c/ay?)-w(ac/ax)
`Integration with respectto y yields
`
`D(dc/dy), =2fu —c)dy
`
`0
`
`(5.21)
`
`(5.22)
`
`with the boundary conditions:
`Cy =0 (means i=ilim ),
`Cy=o0
`and with w,,, equal to the already obtained velocity distribution of the laminar
`boundarylayer, Eq. (5.8 ,b), one obtains:
`
`(5.23)
`
`Sty
`
`(5.24)
`
`Calculation of mass transfer towards anisolated 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 boundarylayer in x direction that means down-
`stream, Eq. (5.9 a,b). One then applies the massota equation neglecting dif-
`fusive mass transfer in x direction:
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`D(ac/dy),=w.e Taree (c..-c)dy
`
`=!
`
`0
`
`3
`
`
`
`In the next step the assumption of the development of a spatially similar
`boundarylayerprofile which independentof the extension ofthe Nernstdiffusion
`layersatisfies in principle the same (reduced) mathematical description whichis
`taken to be a polynomialofthird order with coefficients a,b andc leadsto”:
`
`é(5y)=6 +e(b{y/3,)+o(y/8,)' +a(y/8,)"Janda, =5,(x) |
`
`(5.25)
`
`4 The counter electrode is so far away that is does notinfluence the velocity profile. This
`holds for the very entrance of a gap or duct between anytwoplanarelectrodes, too. As
`the fluid enters this duct the fluid at the leading edge of any ofthe two electrode does not
`yet “feel” the presence of the counterelectrode. The following exampleis 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 1017_0017
`
`

`

` 5 MassTransfer by Fluid Flow, Convective Diffusion and...
`
`Exhibit 1017_0018
`
`Introducing the boundary conditions
`(3¢/dy), =0; (3 c/d*y), =0; C5. =C.3 Cy =0
`
`results in:
`
`c(x,y)=c (x)uz) and 6,=8.(x)
`
`oo! 2
`
`1
`
`2
`
`(5.25 a)
`
`(5.26)
`
`According to mathematically similar profiles and boundary conditions the
`mathematical formulationsfor the velocity distribution in the Prandtl layer and
`the concentration profile of the Nernst layer are similar only being distinguished
`by different extensions d5p,, 6, and dy, 6, of the velocity and the concentration
`boundarylayers. By introducing Eq. (5.26) into Eq. (5.24) and limiting the inte-
`gration to 6, one obtains
`
`2
`2
`8
`
`
`
`3 y|Uy0 f)3} 3f yy) If y
`
`
`
`
`
`
`
`~D/5,= [y=] [-=) |op§1-=] S }+ =| —|bdy«ea 65.27
`
`ee aa) tz] Az)
`
`{z 218,
`
`y yen
`
`Solving the integral and introducing 6,, for the isolated planar electrode,
`3
`
`compare Eq.(5.96), one arrives with 85./3y -(2) Sc7"3, by =35. and
`
`280vVx
`6, =|—————
`13Wyrax
`
`1/2
`
`at
`
`-(8¢/dy) =€,./ 0, “ta5 5
`
`and equating
`
`k= D(dc/dy),—
`
`one obtains the local Sherwood number
`
`Sh, =SakES(Byes)Rel?Sc!? =0,331 Rel?Sc?
`
`(5.28)
`
`(5.29)
`
`(5.30)
`
`5 Thereisa similar difference betweenthe thickness 5, 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
`
`by n<.(2)
`
`Vio
`boundarylayer
`
`as between 6,, and 5p, which both define the extension of the velocity
`
`Exhibit 1017_0018
`
`

`

`5.4 Mass Transport by Convective Diffusion
`
`97
`
`integrating Sh, from zero to x = L, the lengthofthe plate, and dividing by L leads
`to Eq. (5.20 a) for the mean Sherwood number:
`
`__
`
`Sh= =[sh,dx = 0.662 Refsc!®
`
`1
`
`0
`
`(5.31)
`
`5.4.5
`Channel Flow and MassTransfer to Electrodesof 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 atIsolated Planar Electrodes and between TwoVertical Electrodes
`
`At the anodeof an electrolysis cell for copper refining the dissolution of copper
`increases the CuSO,-concentration andthe electrolyte density close to the anode
`surface, whereas CuSO,-depletion and a corresponding decrease of the electro-
`lyte density has to be taken into accountat 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-
`
`
`
`
`
`
`
`
`
`(a)
`
`
`
`
`
`Y
`
`éu-
`;
`Cu-
`
`(b) Convection|)cathodeanode
`Yi
`
`
`
`
`
`
`
`
`7
`*
`
`Cu-
`
`©)
`
`anode ] cathode
`
`4
`b
`
`Péu-
`
`7.
`
`Fig. 5.8.a-c. Schematic of developmentof natural convection driven by density differences
`in the electrolyte induced by cathodic deposition of copper from copper sulfate solutions:
`a concentration and density profiles of Cu**; b convection pattern; ¢ defining electrode
`height, h, and cell width,|
`
`Exhibit 1017_0019
`
`

`

`98
`
`5 Ma