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_r / DOCUMENT:.- Study of Electrical Analogue for Electrodialysis United States Department of the Interior Univ iv. of Tutaa Ubrarr Office of Saline Water • Research and Development Progress Report No. 238 196? Generated on 2015-10-13 04:52 GMT / http://hdl.handle.net/2027/mdp.39015078505586 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google
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_r /
DOCUMENT:.-
Study of Electrical
Analogue for Electrodialysis
United States Department of the Interior
Univiv. of Tutaa Ubrarr
Office of Saline Water • Research and Development Progress Report No. 238
196?
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Research and Development Progress Report No. 238 • February 1967
Study of Electrical
Analogue for Electrodialysis
By C. Berger, G. A. Guter, G. Belfort, Astropower Laboratory,Douglas Aircraft Company, Inc., Newport Beach, California, forOffice of Saline Water; J. A. Hunter, Acting Director; K. C.
Channabasappa, Chief, Membrane Division, M. E. Mattson,
Project Engineer
UNITED STATES DEPARTMENT OF THE INTERIOR • Stewart L Udall, Secretary
Frank C. Di Luzio, Assistant Secretary for Water Pollution Control
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Created in 1849, the Department of the Inter ior —America's Department of Natural Resources—is concerned withthe management, conservation, and development of the Nation'swater, wildlife, mineral, forest, and park and recreationalresources. It also has major responsibilities for Indian andTerritorial affairs.
As the Nation's principal conservation agency, theDepartment of the Interior works to assure that nonrenewableresources are developed and used wisely, that park and recreational resources are conserved for the future, and that renewableresources make their full contribution to the progress, prosperity, and security of the United States—now and in the future.
FOREWORD
This is the two hundred and thirty eighth of aseries of reports designed to present accounts of progressin saline water conversion with the expectation that theexchange of such data will contribute to the long-rangedevelopment of economical processes applicable to large-scale, low-cost demineralization of sea or other salinewater .
Except for minor editing, the data herein are ascontained in the reports submitted by the Astropower Laboratory,Douglas Aircraft Company, Inc. under Contract No. 14-01-0001-676. The data and conclusions given in this report areessentially those of the contractor and are not necessarilyendorsed by the Departmerit of the Interior.
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ABSTRACT
The highlight accomplishments of the program are as follows:
1. This study is a first attempt to perform an analysis ofelectrodialysis by considering the process as an electrical networkcomposed of resistive elements representative of various electrochemical subprocesses . The total effect of all subprocesses isunified into the single mathematical equation for the network. Thisstudy represents a step of major magnitude in understanding theelectrodialysis process because of the novel engineering equationsdeveloped that can be used to quantitatively analyze the electricalresistance of the stages in an electrodialysis plant. The treatment gives a breakdown of the various factors that contribute toelectrical resistance and pinpoints those factors that must beimproved to make technological improvements in the process.
2. Application of the analysis to the Webster, S. D. andBuckeye, Arizona, plants enables the resistance of the separatestages to be calculated. The average calculated values for thesix stages of these plants agree to within 94% of the averagemeasured values.
3. The major resistive factors found in the operation ofthe above plants are electrolyte resistance, ohmic polarization(due primarily to scale) and membrane potentials. Recommendationsare made to reduce the latter two factors.
4. The minor resistive elements were found to be membraneresistance, electrode polarization, and parasitic duct losses.The membrane resistance in the first stage at Webster, S. D. ,
represents about two percent of the total stack resistance. Itis recommended that polarization effects be reduced even at theexpense of increasing these minor resistive contributions, ifnecessary.
5. The electrical characteristics of the Webster andBuckeye plants were calculated based on assumed technologicaladvances which can be made in operating techniques, improvedhydrodynamics and use of exotic membranes. It was found thatreductions in electrical resistance of from 15 to 45 percentare possible using these advances.
6. The resistive elements of a hypothetical sea waterplant were also calculated by the method developed in this study.The results indicate that membrane resistance becomes an importantfactor. In the future, development of membranes for sea wateruse, low membrane resistance as well as reduction in ohmicpolarization is a justifiable goal.
11
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TABLE OF CONTENTS
Page
1.0 INTRODUCTION 1
2. 0 PROGRAM OBJECTIVES AND SUMMARY 3
2. 1 Program Objectives2. 2 Summary of Specific Accomplishments2.3 Summary of Program Results2.4 Recommendations
3
3
411
2. 5 Personnel 14
3. 0 ELECTRICAL ANALOGUE STUDIES 15
3. 1 Phase I — Subcomponent Analysis 15
3. 1. 1 Membrane Polarization 15
3. 1. 1. 1 Concentration Polarization 153. 1. 1. 2 Estimation of Ohmic Polarization 27
3.1.2 Electrode Polarization 32
3. 1. 2. 1 Concentration Overpotential3.1.2.2 Chemical Overpotential3. 1. 2. 3 Ohmic Overpotential3. 1.2.4 Cathodic Resistance Analogue3. 1. 2. 5 Anodic Resistance Analogue
3434353538
3. 1.3 Composite Cell Pair Resistance3.1.4 Parasitic Electrical Duct Losses
3842
3. 1.4. 1 Total Channel Resistance, R3.1.4.2 Cell Pair Resistance, RD3. 1. 4. 3 Resistance Analogue of Electrical
Duct Losses
4848
50
3. 1. 5 Water Transfer Processes 50
3. 1. 5. 1 Discussion 50
3.1.6 Membrane Potential 553.1.7 Membrane Selectivity 58
3. 1. 7. 1 Discussion 583. 1. 7. 2 Resistance Analogue of Membrane
Selectivity 59
3. 1. 8 Terrmerature Effects 60
iii
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Page
3. 2 Phase II — Integration of Subcomponent MathematicalElements into a General Analytical Expression 62
3.2. 1 Development of General Mathematical Analogue 623.2.2 Power Index 62
3. 3 Phase III — Application of Generalized MathematicsSolution to Specific Situations 67
3.3.1 Calculation of the Concentration Polarizationat the Membrane Surfaces 74
3.3.2 Estimation of Ohmic Polarization 863.3.3 Calculation of Electrode Polarization 893. 3. 4 Calculation of Resistance Due to the Overall
Cell Pair 943. 3. 5 Calculation of the Parasitic Duct Loss
Resistance 1043. 3. 6 Calculation of the Corrected Coulomb Efficiency
Based Solely on the Water Transfer Processes 1093. 3. 7 Calculation of Resistance Due to Membrane
Selectivity 1093. 3. 8 Calculation of Resistance Due to Membrane
Potential 110
3.3.9 Electrodialysis of Sea Water 111
3.3.9.1 Operating Characteristics 1113. 3. 9. 2 Mathematical Determination of
i .. Illoperating
3.3. 10 Electrodialysis of Projected Plants 114
3. 3. 10. 1 Concentration Polarization 1143.3.10.2 Ohmic Polarization 1153.3.10.3 Composite Cell Pair 1153.3. 10.4 Membrane Polarization 1153.3.10.5 Neglected Effects 115
3.3.11 Discussion of Calculated Values 115
STATEMENT OF INVENTION 118
REFERENCES 119
APPENDIX A - Colloidal Coagulation
APPENDIX B — Theoretical Prediction of Membrane ResistanceCombining the Statistical Theory of Meares, et al. ,
and Spiegler's Formation Factor
IV
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APPENDIX C - Hydro dynamic Flow
APPENDIX D - Nomenclature
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LIST OF ILLUSTRATIONS
Figure
14 Plot of the Leakage Fractionl-: — I Versus the NormalizedChannel Resistance Ratio (W) o 46
1 Simplified Schematic of Cell Pair Resistive Network
2 Boundary Film Thickness as a Function of VolumetricRate of Flow 17
3 Diagram of the Concentration Profile and the DiffusionLayer on Each Side of an Electrodialysis Ion ExchangeMembrane /. \ _ 19
I ( ^1 ' ) 4-Of1 I
4 Plot of the Correction Factor C. F. =7: rI (him) 25 °C JVersus Temperature of Fluids at Various ProductConcentrations 22
5 Constants A and B (From Onsager Equation) VersusTemperature °C 26
6 Relationship Between Rate of Resistance Change andReciprocal of Stack Current 29
7 Linear Relationship Between Rate of ResistanceChange and Stack Current 30
8 Development of Membrane Polarization 33
9 Comparison of Resistances in Electrode Compartments 36
10 Various Overvoltages at Electrodes 39
11 Representative Electrical Network of a MulticompartmentElectrodialysis Unit 43
12 "Reduced" Electrical Network of a MulticompartmentElectrodialysis Unit 44
13 Schematic View of Current Leakage 45
15 Dimensions of Electrodialysis Unit Required forElectrical Leakage Model 47a
16 Various Simultaneous Processes Occurring DuringElectrodialytic Separation 51
17 Water Transfer From Dialysate to Brine per GramEquivalent of Salt Transport 52
18 Water Transfer as a Function of Current Density in aCationic Membrane 53
19 Ratio Tf/T/j.., for Various Values of Water Transferred
(W ) and Dialysate Concentration at Start, (C ,). when
Cproduct= °" 01 <drinking water> 56
VI
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Figure Page
20 Equivalent Resistance Circuits for a Single Cell Pair 63
21 Equivalent Circuit for Electrodialysis Process 64
22 Electrical Schematic cf Current Resistance Lossesin a Membrane Cell Pair 66
23 Flow Sheet and Material Balance for ElectrodialysisPlant at Webster, So. Dakota 69
24 Flow Sheet and Material Balance for ElectrodialysisPlant at Buckeye, Arizona 70
25 Schematic of the Mark III (Ionics Inc. ) Spacer 75
26 Plot of Equivalent Conductance Versus AverageSolution Concentration for Various Temperatures.Derived from the Onsager Equation 79
27 Plot of Equivalent Conductance Versus AverageSolution Concentration for Various Temperatures.Derived from the Onsager Equation 80
28 Plot of Equivalent Conductance Versus Average SolutionConcentration for Various Temperatures. Derived fromthe Onsager Equation 81
29 Plot of Equivalent Conductance Versus Average SolutionConcentration for Various Temperatures. Derived fromthe Onsager Equation 82
30 Plot of Equivalent Conductance Versus Average SolutionConcentration for Various Temperatures. Derived fromthe Onsager Equation 83
31 Plot of Equivalent Conductance Versus Average SolutionConcentration for Various Temperatures. Derived fromthe Onsager Equation 84
32 Variation of Concentration Polarization With theNormalised Operating Current Density for theElectrodialysis Plants at Webster* S. D. andBuckeye, Arizona 88
33 Concentration Profiles oi the Electrode, Buffer andAdjacent Streams at Webster, South Dakota 90
34 Concentration Profiles of the Electrode and AdjacentStreams at Buckeye, Arizona 91
35 Concentration Profiles of the Dialysate and BrineStreams 98a
36 Resistivity of Ion Exchange Membranes and SodiumChloride Solutions 102
37 Mass Balance of the Dialysate and Brine Streams forStage IV, Webster, South Dakota 105
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Figure Page
38 Channel Dimensions of Brine and Dialysate Streamsas Shown by a Dialysate Gasket, from the Stack atWebster, South Dakota 108
39 Flow Sheet and Material Balance for Hypothetical SeaWater Plant With Buckeye, Arizona, Equipment andConditions 113
vin
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LIST OF TABLES
Table Page
I Cell Pair Resistance of Webster Plant 7
JJ Cell Pair Resistance of Buckeye Plant 8
III Cell Pair Resistance of Projected Plants 9
IV Cell Pair Resistance of Sea Water Plants 10
V Separator and Channel Dimensions 18
VI Comparison of Calculated and Observed OhmicResistance Change with Time 31
VII Summary of the Important Mathematical RelationshipsUsed in Phase III (Derived in Section 3. 1) — Applicationof Generalized Mathematical Equations to SpecificSituations 68
VIII Array of Data Used as Inputs for the ElectrodialysisElectrical Analogue Model for the Present Webster Plant 71
IX Array of Data Used as Inputs for the ElectrodialysisElectrical Analogue Model for the Present Buckeye Plant 72
X Array of Data Used as Inputs for the ElectrodialysisElectrical Analogue Model for the Sea Water Plant 73
XI Concentration Polarization in the Brine and DialysateStreams Using the Computed Double Integration Method 77
XII Concentration Polarization Computed Using theDouble Integration Method 87
XIII Detailed Brine, Catholyte, Anolyte and Buffer MaterialBalances for the Electrodialysis Plant at Webster,South Dakota 92
XIV Detailed Electrode Material Balances for theElectrodialysis Plant at Buckeye, Arizona 93
XV Resistance Analogue Evaluation for the Anode andCathode (Plus Buffer Stream at Webster, SouthDakota) Compartments 95
XVI Detailed Dialysate Material Balances for the Electro-dialysis Plant at Webster, South Dakota 96
XVTI Detailed Dialysate Material Balances for the Electro-dialysis Plant at Buckeye, Arizona 97
XVIII Detailed Brine Material Balances for the Electro-dialysis Plant at Buckeye, Arizona 98
XIX Summary of the Solution and Membrane ResistancesPer Cell Pair 103
XX Major Constituents of Sea Water 112
ix
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1.0 INTRODUCTION
This is the final report of work under Contract 14-01-0001-676. An
analytical engineering study of the electrodialysis process was performed and
appropriate mathematical expressions were derived and applied to the calcu
lation of the resistance of electrodialysis plants operating under a given set
of conditions. The computed values are close to the actual plant operating
values and indicate for the first time a quantitative breakdown and relative
importance of the various factors which contribute to the electrical resistance.
Previous to this study no unified treatment of the electrodialysis process
had been made. There did exist a large number of theoretical and laboratory
studies on various subprocesses of electrodialysis. Engineering studies had
also been made designed to give total operating costs of electrodialysis plants
and costs of product water. In the latter studies, stack resistance assumed a
minor role and did not require an analytical treatment. This study differs
from former studies in that it is centered on the many electrochemical proc
esses that constitute electrodialysis and contribute to stack operating charac
teristics. This study constitutes a preliminary attempt to analyze the operation
of an electrodialysis plant by reducing all associated factors to an electrical
resistance and unifies these factors by placing them in a network of resistive
elements representative of the electrodialysis process.
The major objective of this study is to develop mathematical equations
of an electrical network that is analogous to the electrodialysis process and
that can be applied to both projected and present electrodialysis plants. The
generalized equations contain parameters of operating plants and will facilitate
computing the processing costs for a given water supply and set of operating
parameters. The equations describe the resistive elements equivalent to
discrete phenomena or subprocesses such as concentration polarization, ohmic
polarization, bulk stream resistance, membrane resistance, electrical losses
through ducts, water transfer processes, and membrane potentials.
This approach has provided a step of major magnitude in the under
standing of electrodialysis plant operation. This study has resulted in an
analytical tod, applicable not only to the analysis of the operation of large
plants but the results pinpoint those technological advances in the processes
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which are required to lower plant operating costs and expand the utility of the
process.
In this report each of the various resistive elements is discussed
separately, and a resistance analogue expression is derived for each. The
area resistance equivalent for each factor is calculated. The area resistances
are then combined to give a total area resistance of a single cell pair. This
procedure is applied to the electrodialysis plants at Webster, South Dakota;
Buckeye, Arizona; a sea water plant; and a plant using assumed advanced
technology. Recommendations concerning specific aspects of the electro-
dialysis process are given as a result of the calculations for the various plants.
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2. 0 PROGRAM OBJECTIVES AND SUMMARY
2. 1 Program Objectives
The objectives of this study are as follows:
1. Formulate a general mathematical equation for electro-dialysis by developing a unified electrical analogy conceptin which critical component and subcomponent factors arerepresented in terms of an electrical resistance network.
2. Develop specific guidelines for future research and development work in improving electrodialysis technology byapplying the equation to specific situations and determininghow the variations in operating parameters and other variables influence performance and operating costs.
2. 2 Summary of Specific Accomplishments
1. Engineering equations were derived that can be used to calculate
the electrodialysis stack resistance and electrical operating costs if various
operating parameters are known such as, water compositions, temperature,
types of membranes, stack design, limiting current, operating current, flow
rates, etc.
2. The use of the derived equations gives a breakdown of and allows
a magnitude comparison of separate resistive components of the total process.
This breakdown lists electrolyte resistance, resistance due to scale formation
and membrane polarization among the major resistive elements and places
membrane resistance and membrane concentration polarization among the
relatively unimportant factors. Electrolyte resistance represents one-third
of the total cell pair resistance for brackish water and two-thirds is due to a
number of various polarization effects.
3. This study resulted in a number of recommendations for directing
technical efforts to improve the electrodialysis process. These recommenda
tions are listed below in Section 2. 3.
4. Most of the resistive elements were calculated by integrating
complex equations on a digital computer. Consequently, highly refined values
were obtained and effects of changes in stack design can be readily evaluated.
5. An empirical correlation was made between ohmic polarization,
operating time and current density. Equations were derived and
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applied to operating plants. Estimates of ohmic polarization, which may
involve phenomena such as scaling, fouling, and internal membrane changes,
are quite high and suggest ohmic polarization as one of the most important
and least understood of the membrane phenomena investigated.
2. 3 Summary of Program Results
The objectives of this program were achieved by a four-phase
study. During Phase I, Subcomponent Analysis, mathematical expressions
were developed based on electrical analogies for each of the subcomponent
factors influencing the operation of an electrodialysis system. The factors
considered were concentration polarization, ohmic polarization, membrane
selectivity, membrane resistance, parasitic electrical losses, electrode
polarization, water transfer processes, concentrate and dialysate resistance,
membrane polarization, and electrode polarization. The effects of hydro -
dynamic factors and temperature were included. The derivation of electrical
resistive equations for each of the above factors is given in Section 3. 0,
Electrical Analogue Studies. Membrane, concentrate and dialysate resistances
were combined into a single expression, designated as composite cell pair
resistance. This expression, as well as that developed for membrane con
centration polarization, parasitic duct losses, membrane selectivity, water
transfer, membrane polarization, and electrode polarization, were translated
into computer language to facilitate computations and to perform double inte
grations over the cross-sectional area of a membrane stack. A tool for
studying design effects on these various resistive elements was thus intro
duced and successfully employed.
During Phase II, Integration of Subcomponent Mathematical
Elements into a General Analytical Expression, the interrelationships between
the resistive elements was studied and their combination into a general ex
pression for a resistance network was accomplished. An expression for total
stack resistance was then written in terms of the separate resistive elements
and their combination into a resistance network. A simplified version of the
resistance network for a single cell pair is given in Figure 1. The total
current through an electrodialysis stack is the sum of i , , i_, and i_. The
currents ift and i- represent processes that do not actually conduct a current,
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however, their effect is to lower the efficiency of the desalting current, i^;
consequently, they are placed in parallel in the network. The cell pair
resistance is then easily obtained from the algebraic expression for the total
resistance of the network consisting of R, through R_. Resistances R0
through R, can be broken down to further series or parallel networks.
A power index, P., for a given stack design was then defined as
iR ACpi • -V- n)
where i is the stack current, R the stack resistance, and r\ is the current
efficiency and AC is the change in dialysate concentration. The latter consists
of the resistive elements R? and R~ in Figure 1. This index is proportional to
the power cost required for electrodialysis processing. A comparison of
power indexes is possible for stacks of various designs, when product water
rate, feed water concentration, and amount of total dissolved solids removed
are held constant.
Under Phase III, Applications of Generalized Mathematical Solu
tion to Specific Situations, the expression for the derived resistance network
was applied to specific plant situations at Webster, South Dakota; Buckeye,
Arizona; a hypothetical sea water plant; and a projected plant based on assumed
advances in electrodialysis technology. A description of the calculation and
combination of the resistive elements is given in detail in Section 3. 3 of this
report. A tabulation of results and comparison with the actual values are
given in Tables I, II, III, and IV. The separate resistive component values
are listed as well as their combined values. The calculations using assumed
advanced technology is based on the Webster, South Dakota plant design. A
comparison can thus be made of the present plant and what might be expected
if certain advances are made in membrane performance, scale elimination,
reduction of concentration polarization and membrane potentials.
Under Phase IV, recommendations on specific guidelines for
future research and development work in improving electrodialysis technology
were made. These recommendations are based on the analysis of the Webster
and Buckeye plants, projected plants, and a hypothetical sea water plant.
Recommendations are given below.
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2. 4 Recommendations
As a result of this study a number of recommendations concerning
the electro'dialysis process can be made.
1 . The elimination or substantial reduction of ohmic polariza
tion would significantly improve the economy of the process. This factor is
the least understood of all those subprocesses encountered in this study.
Although ohmic polarization is complex in nature, there appears to be no
theoretical reason for not expecting improvement in this area. Ohmic polari
zation is considered to be due to build up of hard and soft scale, flocs, and
opposing potentials that can build up within the membranes.
Improving the hydrodynamic flow at the membrane surface would
reduce the diffusion layer concentration gradient and prevent formation of the
hydroxide ion responsible for precipitation of hydroxides in the concentrate
streams. There appears to be a relationship between spacer design and scale
formation as indicated by an examination of used membranes which show
scale formation occuring at specific locations relative to the spacer mesh.
This relationship between local hydrodynamic flow and scale formation should
be investigated.
Another approach for reducing ohmic polarization is to develop
selective anion membranes to lower or limit the conduction of scale and floc
forming ions into the concentrate stream. Work on cation membranes selec
tive for calcium and magnesium has been considered in the past and appears
feasible. It is reasonable to extend this approach to anion membranes as well.
The extremes to which membrane development for reduction of
ohmic polarization can be carried is indicated by the relative effects of mem
brane resistance and ohmic polarization. Table I and Table IV give composite
cell pair resistance for the four stages at Webster, S. D. The calculations
were based on the assumption that membrane resistance for the projected
plant (Table IV) was 50 percent of the membranes now in use (Table I). How
ever, the composite cell pair resistance for Stage I differs only by two percent
because most of the resistance is offered by the electrolyte streams. In the
development of the selective membranes, it may be necessary to sacrifice
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good membrane conductivity for specific selectivity provided advantages of
reducing polarization effects can be gained.
2. Methods of reducing membrane potentials must be considered
to_ gain significant reduction in_ electrical power costs. Membrane potentials
arise due to concentration differences across the membrane coupled with
selective transport properties and are augmented by concentration gradients
in the diffusion layers adjacent to the membrane. Elimination of the latter
concentration gradients was assumed in calculating membrane potentials for
the projected plants as given in Table IV. The latter values can be considered
the lower limits obtainable with ideal flow conditions resulting in the elimina
tion of the diffusion layer. A better understanding of the influence of spacer
design on local hydrodynamic flow is required as an initial step to more effec
tive spacer design. Concentration gradients can also be discharged by intro
ducing pulsing and current reversal techniques as has already been suggested
by other investigators. Studies, however, must be made to determine the
magnitude of the advantages gained because current reversal will drastically
reduce current efficiency.
3 . Analytic al studies to optimize plant design should be made.
The breakdown of total stack resistance and capability to calculate separate
resistive elements can readily be adopted to optimization of plant design and
operating procedures. As was discussed above, the development of scale
resistant membranes may require sacrificing good membrane conductivity.
Once the characteristics of membranes are established, optimization of
membrane choice can be made. Another obvious optimization is membrane
potential. The method of feeding a multistage plant will influence both mem
brane potential and electrolyte resistance in an opposite manner. The two
stages at the Buckeye plant use series feed for each of the inlet streams
giving a greater membrane potential in the last stage than in the first due to
larger concentration differences across the membrane. If series feed were
used for the dialysate stream and parallel feed for the concentrate, as is
done at Webster, lower concentration gradients would exist across the mem
branes resulting in lower membrane potentials.
Alterations in method of feed will also change electrolyte stream
resistance and scaling tendencies. The latter is due to changes in calcium
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and magnesium ion concentration in the concentrate stream which will occur
when different feed patterns are introduced. Because many of the resistive
elements have common factors, design alterations optimum for one may not
necessarily guarantee optimization for the other factors, or their net result.
Optimization of design is possible, however, when all resistive factors are
considered simultaneously as can be done by the general network equation
derived in this study.
4. Data from operating plants must be obtained to further
refine and develop more extensive and meaningful electrical analogue expres
sions. Although OSW contractors were extremely helpful in providing data
presently available on the operations of their plants, it was found that a vast
amount of data remain unknown. Knowledge of water analyses and how it
varies with stack performance, analyses of electrode streams and water
transport data are few or lacking completely. As discussed above, the
ohmic polarization factor is one of the most important resistive components.
However, only a two-parameter equation based on empirical correlations
was found to approximate this factor. Certainly, this phenomenon must be
influenced by several factors such as spacer design and hydrodynamics, tem
perature, nature of the membrane, the presence of certain anions and cations
in addition to calcium and magnesium, flow rate, pretreatment, feed method
and suspended solids. There are far more variables that should be considered
and which require much more plant operating data to more fully understand
the ohmic polarization terms.
5 . Membrane research must be pursued from the standpoint
of reducing both short and long term polarization effects. Membrane resist
ance is a minor consideration in seeking these improvements. Polarization
effects determine cost factors exclusive of the electrical power costs. Scale
and floc formation require special operating procedures, pretreatment,
pulsing, acid backwashing, membrane breakage, and replacement. Advanced
membranes that can aid in the reduction of these costly factors need not
exhibit low membrane resistance because of the insignificance of the latter
factor compared to total operating cost. The electrical power costs approxi
mate less than ten percent of the total cost picture for electrodialysis
processing. Membrane resistance in the first stage at Webster,
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South Dakota is about one percent of the total stack resistance. Certainly,sacrifices in membrane conductivity can justifiably be made to reduce the
scaling problems.
6. The above recommendations (3) and (4) can be the subjects
of an effective advanced study using the electrical analogue approach. Much
of the required data for refinement of the study can be obtained using a
portable 1000 GPD test stack which can be operated at a number of sites
having different feed waters. Completion and refinement of the optimization
equations could then be used to optimize a test stack design for each site or
water type. Operation and testing the optimized designs at the various sites
would be a final phase of the program. The use of any advanced electro-
dialysis technologies such as inorganic membranes, and special operating
procedures such as pulsing and current reversal should definitely be a part
of this program.
2. 5 Personnel
Astropower personnel who participated in this study are
Dr. C. Berger, principal investigator , Dr. G. A. Guter and Mr. G. Belfort.
Dr. K. S. Spiegler participated in this study as consultant to Astropower.
Mr. Robert Hubata of Astropower assisted with some of the calculations.
14
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3. 0 ELECTRICAL ANALOGUE STUDIES
3. 1 Phase I — Subcomponent Analysis
A discussion and review of a number of subcomponent factors of
the electrodialysis process will be undertaken in this section. For each sub
component factor, the resistance-analogue (ohms-cm /cell pair), will be
preceded by a summary of the present state of the art. These resistance
values are combined and calculated in Sections 3. Z and 3. 3 using data obtained
from the electrodialysis plants at Webster, South Dakota, and Buckeye,
Arizona.
3. 1. 1 Membrane Polarization
Extensive experimental work has been, and is being,
done to quantitatively explain the phenomena of membrane polarization. The
dialyzing current faces a two-fold polarization effect close to the membrane
surface. A concentration gradient across the diffusion layer and scale for
mation are the respective causes of such polarization. The former is termed
concentration polarization while the latter is called ohmic polarization. Each
is separately discussed and evaluated below.
3. 1. 1. 1 Concentration Polarization
It is possible to estimate the approximate
resistance due to concentration polarization that the dialyzing current faces,
provided two important system parameters can be calculated. These are the
thickness (6) of and the concentration gradient and profile across the diffusion
layer.
(4)Several empirical approaches, such as
use of the Chilton-Colburn transfer factors and the flux equation of Fick, are
able to predict the diffusion layer thickness for nonspace-filled compartments.
Because in all practical electrodialysing plants spacers or turbulent promoters
are used, these theoretical equations are not applicable. H. P. Gregor,
et. al. have studied and measured experimentally, using various size
spacers at different compartment flow rates and Reynolds numbers, the re
lation of the diffusion layer-thickness with flow rates. Figure Z depicts this
relationship, while Table V provides the channel and spacer dimensions.
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For flow rates greater than half a gallon per minute, 6 , the diffusion layer
thickness, can be predicted from the straight line relationship obtained from
the lower curve in Figure 2, viz,
6 = 30 - 10. 7 Q microns (2)
where
Q = superficial volumetric flow rate, G. P. M. /compartment
f G. P. D. >
Q = 0.00138 -j-j-£ , gal/min, channel (3)V s J
M = membranes/ stacks
G. P. D. = gal/day of product water
To calculate the diffusion layer resistance
that the dialysing current must pass through, an average resistivity calculated
as an average salt concentration within the diffusion layer must be obtained.
Refer to Figure 3 for the diagrammatic explanation. The resistivity at the
bulk inlet (p, ) and bulk outlet (p, ) are known. The resistivity at the surfacei o
inlet and outlet is now obtained by estimating the salt concentration at these
coordinates. Most empirical and theoretical equations available to do
this are not applicable to spacer filled configurations and so the Nernst
idealized equation will be used as an approximation. Together with several
simplifying assumption
concentration gradient.
simplifying assumptions, the Nernst equation also presupposes a linear
dc _ i (f - t) _ (c - c ) . .
dx FD 6'
where
6 = diffusion layer thickness, cm
t , t = transference numbers of counter ion in membrane in bulksolution respectively
i = current density, amp/cm
F = Faradays constant, coulombs/ gm equiv.
D = Diffusion coefficient of electrolyte
c , c; = Concentration of ions at membrane surface and inbulk solution respectively, gm equiv/ cc
K
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TABLE V
SEPARATOR AND CHANNEL DIMENSIONS
ChannelThickness
Hydraulic, Radius
Area. Sq.Ft.
Mils Separator Designation Ft . x 10T-5X 10+ -' VolV
42 I/ 11 -inch mesh, 1 . 00-mm OAT(a) 1. 72 0. 58 1. 90
72 11 11 -inch mesh, 1 . 75 -mm OAT 2. 89 1.. 00 1. 38
135 11 1 -inch mesh, I/ 1-inch OAT;II 11 -inch mesh, 0 . 15 -inch strand 5. 27 1. 85 1. 41
270 \l 1-inch mesh, I/ 1-inch strand;11 11-inch mesh, 1 . 75 -mm OAT 9. 91 3. 70 1. 26
(b)
(a) Overall thickness(b) Ratio of the volume of water displaced per unit length of compartment
without a separator, to the volume of water displaced per unit length ofcompartment with a separator.
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Outlet
Brine Stream
Co- Counter-Current Current
tFlow Direction
DiffusionLayer
Dialysate StreamBulk
I Concentration Profile
Ion ExchangeMembrane
tFlowDirection
DiffusionLaYer Inlet
Figure 3. Diagram of the Concentration Profile and the DiffusionLayer on Each Side of an Electrodialysis Ion ExchangeMembrane
19
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The current density, i, must be estimated
before the Nernst equation can be used. Limiting current density (ir ) will
occur whan the exit concentration of the dilute stream (p ) shown in Figure 3,o
rea.ches z&ro.. providing the fluid velocity and diffusion layer thickness are
constant.
,5)lim 6 (f - t)
(7)E. A. Mason and T. A. Kirkham use the following experimental relation
ship to determine the operating current density with varying flow rates.
'°'6(6)
^d] lim
where
i = current density, ma /cmC, = dialysate concentration, gm equiv/ 1
v = dialysate linear velocity, cm/sec
Since Equation (6) is peculiar to the Ionics membranes (Nepton Ar-111 and
CR-61), a short experiment similar to that described by Mason and Kirkhamwould have to be undertaken to determine the operating current density forthe particular membranes under study.
The Asahi Chemical Company ' has cor
related the polarization factor with linear velocity of the dialysate stream for
their particular membranes with sea water.
f^-] = 72. 3 (v)0'947 at 20°C (7)
where
i = current density, amps/cm
C = log mean concentration between inlet and outlet,desalting strsam, gm equiv/cm^
v = dialysate linear flow velocity, cm/sec(valid for v - 3 to 10 cm/ sec)
Since the equations above all apply to conditions at room temperature (25 C)
{v
J1. will be corrected using a temperature correction derived from the
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(9)recent work done at the Sea Water Conversion Laboratory in Berkeley.
Refer to Figure 4. At a given product concentration, the correction factor at
some temperature t C will be,
C.F. = (8)
Therefore, the new, or temperature corrected, limiting current density willbe
xC.F. (9)25°C
In a hypothetical plant, the K ratio
(= i /i,. ), which appears in many of the following analogue derivations,
can be optimized, but in this treatment, because the equation will be derived
and applied to two existing electrodiaiysis plants, the K ratio per stage is
fixed as each stage capacity (salt transferred/unit time) is given. Using
Faraday's Law and a material balance, the capacity is defined as,
Since,
= FdnC.-f Em ec*uiv transferreddi sec.
FF,C,.f -d di . / 2 , , , ,
i = T Amps/m (11)oper eA rP
where
F - Faraday's constant, coul/gm equiv.
F, = volumetric flow rate, I/ sec, channela
C,. = inlet dialysate concentration, gm equiv. /If = fraction desalted
e = coulomb efficiency
A = desalination area, cms.P
Two methods, concerning the prediction of
the average resistivity using the Nernst Equation* ' diffusion layer, present
themselves. Refer to Figure 3. The first method requires the average
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10
Webster<8'
(26.7)30Buckeye
Temperature °C
40 50
C1103
Figure 4. Plot of the correction Factor C.F.=
Versus Temperature of Fluidsat Various Product Concentrations
Exit Dialysate Concentration^do
t°C
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resistivity (surface and bulk) at the inlet and outlet to be evaluated from the
general equation,
1000p = ohms -cms (12)
where
C = concentration, gms/liter
A = equivalent conductance.
This simple average resistivity can be refined
by replacing it with a double integrated value. This is the second and far
more accurate method. Here a straight-line concentration profile is not
assumed (see Equation 17).
The double integration will be performed
across the diffusion layer (x co-ordinate) and along the flow path (z co-ordinate).
m
— = - rh Jtotal
Jx=0
x, z
dz (13)
From the basic definition of resistivity and the Onsager equation, we get
(VoV t°C
x, z
t°C
1000
(A + B (AJ
1000
(A ) C - C3/2 (A + B(A )
(14)
(15)
"t°c x> z x> z °°t°C
After p is substituted in Equation {13)5 C as a function of x and z hasX j Z X p Z
to be evaluated from the boundary conditions. Note also that the surface
concentration, C , is related to the bulk stream concentration, C , as
follows:
Cn = (1 - K) C1 (16)
where
K = i /i,.oper lim
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The following general equation for C can be writtenX t Z
C = ax + bz + cxz + dX, Z (17)
The four constants, a, b, c, d can be evaluated in terms of the surface and
bulk concentrations at the inlet and outlet. If we assume a constant, K, the
four constants above are readily evaluated. The validation of this assumption
is based on the fact that both the operating and the limiting current vary
similarly with the concentration (i. e. , along the flow path, z).
The concentration polarization resistance
component is evaluated for the dialysate stream and anion exchange membrane,
Rc •
= R from Equation (13)
A slightly different surface to bulk relationship from Equation (16) exists for
the bulk stream. Refer to Figure 3.
111 IV 11= (C - C) + (C - C)= civ - cn
= CIV - (1 - K) C1 (18)
We obtain A , the equivalent solution conduct
ance, from the Onsager Equation at various temperatures and concentrations.
(A)toc
(AJt; C
o25 C
-QA + B (A
+ 0.023(t -25JT]
(14)
(19)
r2Ca
f£>Na+
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where
^ = equivalent conductance at the existing concentrationC
and t°C
(A ) = equivalent conductance at infinite dilution and t C* t°C
(f ) , (f ,) = fraction in product and feed streams in e. p. m. ofp x x ionic species X
C = average concentration of salts in the water, equivalent/liter
A, B = temperature and ionic species dependent constants(see Figure 5 for plot of A, B, versus temperature)
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CL
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aa
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3.1.1.2 Estimation of Ohmic Polarization
Ohmic polarization involves a number of
complex and interrelated factors. It is associated with scale formation and
other effects that contribute to resistance rise with operating time in the
order of hours and days. Scales differ in their chemical and physical
character as well as the processes by which they are formed. Certain scales
are loosely formed and can be removed by simply flushing the stacK. Other
scales are hard and can be removed only by removing and scrubbing membranes,
a frequent cause of breakage. Scale formation can also occur within a
membrane and greatly shorten its lifetime. Flocs and other membrane
deposits can occur. Calcium sulfate and other insolubles can come out of
solution as local concentrations rise on a membrane surface. Trace organic
materials can also dissolve in the membranes over a long period of time and
change the conductivity drastically.
Calcium carbonate scaling of anion permeable
membranes is closely associated with concentration polarization. Operation
near and above the limiting current density results in water breakdown into
H,.O and OH ions at the membrane interface and transfer of OH ions
through the anion membrane. The OH~ ions convert biocarbonate ions to
carbonate, which in turn combines with calcium to form a scale.
HCO3" + OH" »H2O
+CO3
=
Ca++ + CO^ > CaCO3 (scale)
Ohmic polarization can thus be expected to
vary with electrolyte composition, current density, limiting current density,
concentration polarization, local hydrodynamic conditions as influenced by
spacer design and flow velocity, and pretreatment procedures. It has also
been suggested that membranes can undergo an orientation polarization which
is an internal polarization by reorientation of the polar groups and thereby(8)setting up a potential opposing the applied potentials.
The task of writing a resistance analogue
expression for the ohmic polarization factor is further encumbered by a lack
of plant and experimental data on this phenomenon. Some data have been
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collected using a small test stack at the Webster plant. Figure 6 shows the
rate of resistance change with continuous operating current. The influence
of current is obvious. This data was obtained using Webster water which
had not been treated with sulfuric acid. Resistance buildup is due to
deposition of calcium carbonate and calcium sulfate scales as well as ferrichydroxide floc and inorganic materials which evade the pretreatment filters.The orientation polarization may also be operative. The data of Figure 6
were found to satisfy an equation of the general type
dR bK . ,ae (21)
where a and b are constants. This correlation is represented in Figure 7
where log (dR/dt) shows a linear relationship with I, the stack current.
The ohmic polarization values in Table I
through IV were calculated using Equation (22) which is equivalent to Equation
(23)
log R , . :: 0. 150 + 0.92K (22)b ohmic '
R . . = 1.41 e2'12K (23)ohmic
where
R , . is rate of rise in ohm cm per cell pair per hour
K is the ratio of operating current to limiting current
These constants in Equations (22) and (23) were
chosen to give the best, fit of these equations to the available data. A compari
son of calculated values and observed values for ohmic polarization is given
in Table VI. The observed values for the Webster and Buckeye Plant were
calculated by assuming that the major differences between observed cell pair
resistance and the value calculated using all factors other than ohmic polari
zation were due to the latter factor.
The equation gives surprisingly good results
based on the theory and data limitations. It should be pointed out, however,
that Equation (23) iscnly an approximation at best. It should also be noted that
28
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u *•
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Ducmj_j(0
<n
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O
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fdJ= ,,
Sso ^« wJS-uD C
Cr", "I
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TABLE VI
COMPARISON OF CALCULATED AND OBSERVED
OHMIC RESISTANCE CHANGE WITH TIME
R .... aohmic
Observed Calculated
lim
Webster Test Stack' '
203 156 105
1.52 29 35
1.21 13 18
1.00 4.9 12
Webster Plant
.64 7.6 5.5
.63 7.3 5.4
.52 5.8 4. 3
.56 4. 1 4.6
Buckeye Plant
20
18 2.1 2.1
2a) ohms cm per hour per cell pair
b) Five pair test stack, 260 cm membranes.
Data supplied by Dr. John Nordin.
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R . . is very sensitive to i/i,. . This ratio can vary from time to time
during stack operation. It is also probable that i... can change during this
time.
Three general methods are used to prevent
the increase of electrical and flow resistance in the stack.
1. Pretreatment to slow or prevent scale formation
2. pH adjustments to acid conditions, preventing the precipitation of the carbonates and the hydroxides
3. Removal of the partially or fully formed deposits by areversal of polarity and a reversal of the flow streams.
The importance of polarity reversal techniques is the ability to control the
place, form, and time of scale formation during a continuous operation. The
mechanism can be explained as follows.
Refer to Figure 8. Before the voltage is
applied (Figure 8a), the concentration profile is horizontal for both streams.
After about 30 seconds, steady state is reached (Figure 8b), and the con-
centratia's maximum and minimum is at the surfaces. When a short reversal
of polarity is imposed on the system, for a short time At , concentration
minima and maxima move (Figure 8c) a short distance away from the mem
brane surface where the fluid flow is not stagnant but can flush the precipitate
deposited at the minima and maxima out of the compartment. In a short time
the concentration gradients will smooth out on reversal to normal polarity
(Figure 8d).
3. 1. 2 Electrode Polarization*12. 13*
It is proposed to develop an electrode analogue resistance (R , , , , ), that •will describe the various resistive components
within the electrode compartment due to the bulk solution, the diffusion layer
and kinetic phenomena at the electrode surface. A short general electrode
discussion on each overpotential will now follow and thereafter, the equations
developed, will be specifically applied to the cathode and anode cases that
occur in electrodialysis.
The passage of current through each of the electrode
compartments involves three steps:
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1. The transfer of ions from the bulk of the solution to thesurface of the electrode,
2. The electrochemical reaction at the electrode.
3. The formation of the final products of the reaction and theirremoval from the electrode surface.
3.1.2.1 Concentration Overpotential
When the current is flowing the ions that dis
charge migrate towards the electrode and cause a concentration gradient
across the thin diffusion layer at the electrode surface. This phenomenon is
exactly analogous to the concentration gradient that occurs at the ion exchange
membranes. The concentration gradient leads to a change in electrode poten
tial of
noting that
we get
T?
•RTconc
In 'bulk
surface
Differentiating (24) with respect to i, and
(24)
'bulk 1 limc surface 1 - i/L.lim
(25)lim
(See Section 3,1.1.1.1.)
Rconc
RT™lim
RTF
lim RTi, . - i i,.lim lim IT - ilim
(26)
3. 1.2.2 Chemical Overpotential
The chemical Overpotential 7) , is defined
to be that potential in excess of the discharge potential for the given reaction
which must be applied to the cell in order to maintain a finite rate of discharge.
Chemical Overpotential occurs as a result of steps (2) and (3) above. The
value of 77 v for the electrode reaction is given by Tafel's Formula:
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where
chem4ni, (27)
1 ,a *» y 9 and
a - constant (depends on nature of cathode).
Differentiating (27) with respect to i;
R HI MRchem
~ (28)
3.1.2.3 Ohmic Overpotential
The ohmic overpotential (A<p , ) consists
of two parts, namely the voltage drop which occurs in the bulk solution of
constant concentration plus the voltage drop across the diffusion layer where
the concentration gradient varies linearly with the current density.
(29)
Differentiating (29) with respect to i,
R , = R . + Rfi (30)ohm sol 6
The first term R ,, is evaluated by the same procedure as in Section 3. 1.3,
providing the bulk solution chemical analysis is known.
RRsol
P
R. may be evaluated by specifying the resistivity at any point within the dif
fusion layer as a function p(x,y) and by computing the double integral of this
function as was done in Section 3. 1. 1 for the case of membrane polarization.
The relationship between each resistance component and current density is
shown in Figure 9.
3.1.2.4 Cathodic Resistance Analogue
Consider first the passage of current in the
cathode compartment. The total change in potential across the cathode
compartment is
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ohm sol diff
R
1
cone
1-K
(dimensionless)
3r
4r -_Note: K = — = dimensionless current density
lim
Figure 9. Comparison of Resistances in ElectrodeCompartments
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I- »7 v (32)'chem v '
Differentiating with respect to i, we obtain the total effective resistance R of
the cathode compartment . . .
R «-uj = (R , + R.) + R + R . (33)cathode sol o i- conc chem
ohm
R , = resistance of the bulk solution = constantsol
R- = resistance of the diffusion layer (varies with i)0 r \
RT 1 JRconc K
'lcone F i,. - i
V lim
Rchem Of F i
In the casa of the cathode compartment, the
bulk solution contains H.-SO, and NaCl at given influent and effluent concen-£4 -
+trations. Since the discharge potential for Na. upon the stainless steel cathode
is substantially higher than that for H ions in electrodialysis, we may con
sider the NaCl present to be a supporting or "neutral" electrolyte. The
transfer of ions to the surface of the electrode is therefore accomplished by
5 bulk solution.
The following electrochemical reaction occurs
the migration of H ions from the bulk solution
at the cathode:
Because of the "neutral" electrolyte, R- is
negligible in the cathode compartment, and (33) becomes
., , ,Ucathode sol conc chem
„ RT I 1 \ , RT
where
R , -• evaluated as in S&ction 3. 1. 1. 1.sol
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3.1.2.5 Anodic Resistance Analogue
At the anode, as can be seen from Figure 10,
the discharge of Cl will occur preferentially to that of H_O. The transfer of
ions to the anode surface is therefore accomplished by the migration of Cl
ions from the bulk solution. In this case no supporting electrolyte is present,
and hence R. must be included in evaluating the ohmic effect. The anodic
resistance analogue is calculated from
R = (R , + R..} + R + R ,an sol 6 , cone chem
ohm
ilim-i I a* URTfl,
(35)
where
R . = evaluated as in Section 3.1.3sol
Rc = evaluated as in Section 3. 1. 1.0
3.1.3 Composite Cell Pair Resistance
The cell pair resistance is defined as the sum of the
anion exchange membra.ne resistance R , the cation exchange membranesin,
resistance R ., and the dialysate R ,, and brine R, stream resistances, allcat 'a b
calculated at some position along the flow path length.
R\ =R + R . + R , -f R, (36)pi an cat d b
where R , R ., the membrane resistances can be obtained from literaturean' cat
or evaluated (see Section 3. 1. 3 on membrane resistance) at some average
concentration, C .cL
> lOOOy, lOOOy,_, \ a .a c . c d 'b ,,_xRJ * P t + p t, + A r_ + "A
' r: - (3?)* / z a a D D
where3. C
p , p membrane resistivities are evaluated at Cd,
A , A, equivalent conductance of dialysate and brine streams.
Integration over the membrane area (pa) available for
desalination of the cell pair resistance (R } at chosen path length, z, from^ z
38
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VT)O
u4)
i—iU
01V60at
"o
Iv
6tn
§
oo'
oo
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the bottom of the stack, will result in the total resistance per cell pair.
J Note: p = fraction of usable area)
L a = total membrane area Ja
TT-=
I Tirr cell pair/0 (38)R J (R JF G - Z
If we assume that (R ) is constant in the lateral direction (perpendicular to"z
the z direction), then the area equals,
2pa = pnz cm
pda = pndz
(Note:
fraction of usable area p = . 77 1
n = overall membrane width jsubstituting,
J (R) (39)
m1
R~- pn
P z
According to E. A. Mason and T. A. Kirkham, cell pair resistance at
coordinate z, can be expressed in terms of some average concentration CZ
1 I (J- 4k
"1
Ca=
2
lCd
CbJwhere
. ... . brine compartment thickness*b/y. dialysate compartment thickness
and
Kl
(R ) -- ^i
+K2 -K3Ca (41)
v z a
Variation of C , and C , with z, wi-l let C be a function of z.da a
C = f(z) (42)3
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Substituting Equation (43) into Equation (39),
m1 P dz ....__ = pn _ _ (44)P TKi 1O i j_ v if f /~\l
T-7—T TJS.?
- JS.- 1 (Z)\L_i \ z / £ j _j
m
= pn f -^- ^^ T (45)
O /K, + K^ f (z) - K0 [f (z)]2'}
The general form of which is:
mi . K r (,, + qd-P , o (z + >* + «>
Equation (46) can be evaluated using standard integration table, viz,
first, manipulating
m \ n1 ~ Ka f (2z + y) dz , Ka 28 J P dz ..-.__ _ ___ _i_ — -. __ _j_ - y __ ^4fj
XV ^ J /£, irl £. \ 3. IV £i»ifrp
o (z + yz + 6) \ /0x + yz + 6
integrating (from tables, as both parts are standard forms)
-L =Ka Log (z2 + yz + 6) +i^
P
where
q - 46 - y2
Hence, R , the composite cell pair resistance can be evaluated. If we sumP
the various other resistances due to membrane polarization, duct losses, etc. ,
the total result could be compared with the actual plant cell pair resistance.
Equation ;46) can also be solved with a com
puter, by replacing the integral by a summation, viz,
R
z-rri
2,— ^z t4^)
,. (z + yz + 6)z-0 x ' '
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3.1.4 Parasitic Electrical Duct Losses
Because of the inherent engineering design of electro-
dialysis units, each concentration stream within the stack is connected, via
the ducts in the manifolding, to all oth?r concentration streams. The same
applies for the dilution streams. Since the concentration stream is more
conductive to electricity than the dilute stream, it would be expected that
most of the nonproductive-leakage current would flow through the concentration
stream.
(14)Wilson in his monograph describes these losses and
bases his evaluation and equation derivations or. a hypothetical model. An
expression for the effective leakage current is obtained, using a representative
electrical network (Figure 11) which can be reduced to that shown in Figureo
12. The mathematical procedure consists of first evaluation i , leakagetb
" ncurrent through the n
"loop in terms cf -f N, total number of cells; i , stack
current; and
N = total number of cellsT>
tctal leakage resistance &
overall ceil resistance R NP
total channel resistance _ Roverall cell resistance R N
P
Refer to Figure 13.
[N/2 - n + 1] i
{50)
[r + (1 + *}]
and secondly evaluation I. , effective leakage current, by summing all theAJ
leakage loops of the stack.
n=N/2
I •* (N - 2 n f 2) .,..^
-N
--- l» * J
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11
RB
e/srg
Figure 11. Representative Electrical Network of a MulticompartmentElectrodialysis Unit
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Total Number of Cell = Ni
NoCurrent
Channel resist. = 6
Figure 12. "Reduced" Electrical Network of a KlulticompartmentElectrodialysis Unit
44
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v///.
Cell Pair
c = cationic membrane
a = anionic membrane
Figure 13. Schematic View of Current Leakage
45
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.0oo
(M
o
0)(H
00•rHk
46
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Substitute i from Equation (51) into Equation (50), we get
n=N/2
<_J
n=l
n
N/2
n=l
(52)
RL RD RB(53)
Equation (52) is an exact derivation using the
hypothetical model proposed (see Figures 11, and 12). Looking at Equation
(52), if N is large, r is negligible, compared with
n=N/2
(1 + *) £ n
n=l
If we expand the summation and cancel terms
\e _ (2/N(N + 1)
io~
3(1 + *)
and again, if N is large, (2/N) (N + 1) term reduces to 2,
3(1 + *) (54)
aSee Figure 14 for plot of — versus ^.
o
Hence, we need only calculate $ ratio in order
to estimate the current leakage as a function of the total stack current. To
calculate ty , we must evaluate R., channel resistance, and R, cell pairfj
resistance. (See Figure 15 for nomenclature. )
47
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H y h
Brine in — - — - -
Dialysate Out ^ — - — —|-J4
— »"ff **T* . T T"
s
I
;".*.".'
7-' V
T
Brine Out ^ — — _ — ' •. X- ^-«— ^. f- ;
Dialysate in » -± r- -•- •»*-!-• !"f
4i
-i-
Anode
fP
m
e/ffe
Figure 15. Dimensions of Electrodialysis Unit Requiredfor Electrical Leakage Model
47a
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3.1.4.1 Total Channel Resistance, R—"TJ-"""-' " —' - .' .••--•!-..-.".- _ •• j^
The total channel resistance is obtained fromR, and R , where
b a
2pb (Z + Y) NR, = Brine channel resistance = = , O/ stack (55)
(ff/4) Yb^
2pd U + y) NR, = Dialysate channel resistance = , O /stack (56)d J u, x v ,
a a
and R. is obtained from the following parallel resistance relationship,X/
(57)
Note that in Equations (55) and (56), besides the solution resistivities p, and
p, , all other factors are of geometric consequence. (See Nomenclature,
Appendix D. )
3. 1.4.2 Cell Pair Resistance, Rp
-The cell pair resistance (R ) is defined as™
zthe sum of the anion exchange membrane resistance R , the cation exchange
2LX1
membrane resistance R ., and the dialysate R, and brine R, streamcat a b
resistances (see Section 3. 1. 3).
where
(R ) = R + R . + R , + R,p an cat d br z
i(58)
* z
By multiplying the cell pair resistance R ,
by the number of cell pairs N, the total stack resistance is obtained. Three
methods can be employed in calculating R . Two are based on Kirkham and/ 7 \ P
Mason's* choice of an average cell pair concentration C at various pathcL
positions, z, up the stack. The first method uses the equations in Appendix B
(the porosity method) while the second method uses the following relationship,
48
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- cm (59)
Assuming the concentration profiles (straight line or a logarithmic relation
ship) of the brine and dialysate streams, then calculating the average concen
tration at pre-chosen distances (z) from the bottom of the channel, C andct
thus (R ) can be evaluated at each z. Either an arithmetic mean R can beP z P
obtained by
R = 4P 2 p bottom p top
z=o z=m
- cm (60)
or (R ) , as a function of z, can be integrated over the desalination area of"z
the membrane to obtain the cell pair resistance R , (Section 3. 1.4, the
integration method).
mj_ - r LLR
"J R
dz, cm (62)
The third method to obtain R , relies onP
membrane data supplied by the manufacturer for R and R .. The solutionrr * an catresistivities are found as follows.
1 1000 1000+
+
2
1
ACK_ b, o
1000 10002 AC,
_ b, o Acb,i
dialysate
-1' l J K,..
- cm
- cm
(63)
(64)
brine
Note: lower subscript b denotes bulk solution
lower subscript i denotes inlet of channel
lower subscript o denotes exit of channel
Rp=
Ran+ Rcat
if y , = y, = y = channel thickness, cms.
49
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3.1.4.3 Resistance Analogue of ElectricalDuct Losses
Once ty, = •=— has been evaluated, T— canp o
be obtained from Equation (54) or the plot of Equation (54) in Figure 14. The
analogue duct leakage resistance can now be evaluated, knowing i , E, the
stack current and potential difference, respectively.
E 2Rj . = = — ohms - cm /stack
ohms - cm /cell pair (65)
N
3. 1. 5 Water Transfer Processes
3. 1.5. 1 Discussion
Water transfer accompanying electrodialysis
can be divided into two classes (Figure 16), "primary hydration" of ions —
either by counter or co-ion transport — and the excess above this value. The
latter has been termed electroosmosis. The effect of osmotic water transfer
may become excessive at very low dialysate concentrations unless the brine/27\ /28^
is of low concentration. This effect is shown in Figure 17. Studies
have also been carried out with reference to the effect current density has on
the water transport occurring with a cationic membrane. Water transfer
increases significantly at very low current densities (Figure 18). The ex
planation given is that at low current densities, water transport occurs
through the larger pore holes only, while at high current densities, the water
transfer values are averaged for all pore sizes.
(14)Many writers have found that the water
transport numbers (w and w moles/Faraday) of cation and anion-exchangeC cL
membranes, respectively, are close to the primary hydration numbers,
e. g. Na w = 8o c
Cl" w = 4
50
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BrineCompartment
CationicMembrane(-)
C1~(+HZO)-
NaCl
DialysateCompartment
AnionicMembrane(+)
Na (+H-O) ClCounterionTransport withElectroosmosis
Co-ionTransport withElectroosmosis
•H2°Osmosis
Back Diffusion
BrineCompartment
Na (+HZO)
NaCl
Figure 16. Various Simultaneous Processes OccurringDuring Electrodialytic Separation
51
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B o' d «
III
III
III
QVPO
< « U -u •
O "
O
U
n)
ffl
4)
9
o
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V
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•r1
0)
U
n)
(0
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sa!
h
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Is^ c
B «
0 0)
a
rla*
8 /OH TOUJ
52
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•
SiNuH
eg
s
^!nJ
E
oh
(Xo.a
(0e
•A
VQ
C U0 C
«£c *
COC
00
00
(ABp^JH jj/sajoui)
53
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Thus, a membrane pair with NaCl present will have a total water transport
number (w. = w + w ) of 1Z moles/Faraday,tcaThe deleterious effect of water transport
from the diluate to the brine streams results in a lowering of the coulomb( 14)efficiency. Wilson et al have aptly described this effect mathematically,
using apparent coulomb efficiencies T}_ and f}_ and actual coulomb efficiency
TJ=
TI numbers of equivalents transported (transport numbers)(number of Faradays passed in process) (number of membrane pairs)
r -n 'Dp~d'77D
= e = 7,
|^1
- 18
Wt(Cd)J= ^ 4
. -(Cd) F°^ (66)
based on the dilute solution
(67)
based on the concentrated solution where
77 = true coulomb efficiency
W = moles of water transported per equivalent of NaCl transfer
(C ,) = incoming dilute solution concentration (equiv/ml)
(C, ) = outgoing dilute solution concentration (equiv/ml)o
(C.) = incoming concentrate solution concentration (equiv/ml)
i
(C, ) = outgoing concentrate solution concentration (equiv/ml)o
Fn = dilute solution outgoing flow rate (ml/ sec)
FR = concentrate solution outgoing flow rate (ml/ sec)
F = Faraday constant
n = number of membrane pairs
I = current (amp)
The membrane manufacturer supplies data
on W ,. water transport number, at various concentrations. Because different
54
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and varying concentrations exist on either side of the membrane, the average
concentration (C ) concept can be used (Equation 40).
Mandersloot mathematically describes the
detrimental effect water losses have on the coulomb efficiency with the fol
lowing equation,
JL =
where
-2WtCQ)/(l
- -2WtC)
-wtcT)
In mean(6g)
oC_ = product concentration = (C,)
C = inlet concentration = (C ,)o d .
Mandersloot has plotted TJ/TJn versus W
for various desalination ranges. This enables us to get Tj immediately from
Figure 19 or Equation (68). The true coulombic efficiency, 17, is used to
correct the coulombic efficiency (T7jO obtained from the membrane manu
facturer.
Hence, no actual analogue resistance to
represent the water transfer losses is envisaged. The correction and replace
ment of TJn by TJ will represent the effect water transfer has on the system as
a whole.
3.1.6 Membrane Potential
Membrane potentials arise from the concentration cell
potentials which can exist across both types of membranes. In the derivation
of the equations for membrane potentials a concentration cell with reversible
electrodes and with transference of ions through the membrane is considered.
The membrane potential is then obtained by subtracting the electrode poten
tials from the potential of such a cell.
To derive the anion membrane potential, a concentration
cell with electrodes reversible to the cation is considered. If a concentration
difference exists across the membrane with a? > a. the net spontaneous
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oo
rtf
•o
ctf
M
(4
(O
4)
* °C- ii
T3 *
S p
rt « &
tfl C1_JI Is?
''CO
O <->4)m C JH
O <L>
T
'/P(PD) ;M 8T -
56
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reaction which can occur is
t_ electrolyte (a_) -• t_ electrolyte (a.)
The potential of the concentration cell is
RTEcell
' ¥substituting mean activity coefficients
(a±)R T ">= 2t- IT lnirr (70)
The electrode potentials of the concentration cell with
electrodes reversible for the cation is
Eal' ¥ ln
T^T
Since t , + t =1 and (a ) = a a and the assumption is made that
(a.) (a ) (a )
2 "2 ±2
1
Equation (71) can be subtracted from (70) to obtain the anion membrane
potential, Em, a
(a )9
(72)
1
Substituting electrolyte concentrations for activities
'
Em,a = {t-- V ln
57
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where (C, ) and (C , ) are the concentrations of electrolyte at the mem-b, m a, mz z
brane boundary layer on the brine side and dialysate side respectively and at
position, z, along the flow path of the membrane.
A parallel derivation for the cation membrane potential
gives
.d, m
z
<74>
In a concentration cell operating in a spontaneous
fashion, the flow of ions is from higher to lower concentrations and a potential
is set up accordingly. The direction of flow of ions and the potential set up in
each case opposes the direction of ion flow and the potential applied when the
electrodialysis process operates. Consequently Equations (73) and (74)
represent potentials which oppose the applied electrodialysis potential.
The total resistance of a membrane stack due to mem
brane potential then becomes
N E N E) =
a m.a+
c m. c O/stack (75)m.P totalI l
where
N = number of anion membranesa
N = number of cation membranes
I = total current passing through stack
In resistance- area units per cell pair,
(R ) Am,p . p -R = -x, /V - O - crn /cell pair (76)m,p M 12 'f \ i
'B
3. 1. 7 Membrane Selectivity
3.1.7.1 Discussion
Basic definitions defining selectivity are
available in order to compare various membranes on a relative basis.
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1 +•* +
P = f tR
(77)
R+
where
P = selectivity (of a cationic membrane)
f = transference number of cations (R ) in the membraneR
+t = transference number of cations (R ) in free solutionR+
For an ideal selective membrane t , = 1 and P = 1.
RFor a complete nons elective membrane f , = t , and P = 0.
R R
The theoretical explanation of ionic transport
in permselective membranes is in the embryonic stage. Unexplained anomalies
occur. For example, at current densities above the critical value, it is
expected that hydrogen ions (H ), resulting from " water- splitting, " would
start to carry the current through the cationic membranes, but this is not the
(17)case.
Qualitative treatment of permselective mem-( 18 19)brane phenomena, uses the M.S. T. ' "fixed charged" model and Donnan
equilibrium as a basis. The application of the general classical theory to
experimental cells is one of the main tasks of the basic electrochemistry of
membranes.
In as much as the practical electrodialysis
unit is concerned, empirical experimental analysis has been the mainstay
in choosing membranes for various waters.
3.1.7.2 Resistance Analogue of MembraneSelectivity
The fact that the selectivity, P, in practice
is not equal to one suggests that some hypothetical resistance R could be
placed in parallel with the current effecting separation, so as to lower the
coulomb efficiency.
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Rs=
where IT is the co-ion or leakage current.JU
IL = Itm[D(c)]I = total stack current
t = co-ion transport number
D(c) = Donnan equilibrium expression as a function ofconcentration.
Until the Donnan expression, [D (c)] , can be
theoretically predicted, the following relationship between leakage current,
(I ), rlue to co-ion transport can be used to calculate R .
I -IT
n - —y-^ = 1- (tc_+ t*) (79)
where 77 is the coulomb efficiency, considering losses due only to co-ion
transport.
I is the total stack currentC* H.
t and t, are transport numbers of the co-ions in the cationand anion exchange membrane, respectively.
Solving Equation (79) for IT and substitutingJ_*
in Equation (78) we obtain
<80>
3. 1. 8 Temperature Effects
The performance of electrodialysis units increase with
an increase in temperature. The change in electrical resistance of the mem
brane system has been found to be similar to the change in resistance of
solutions. Many of the membrane parameters vary with temperature rise,
due to membrane swelling. Salt leakage and conductance both rise with a
temperature increase. How each of these properties affect the membranes
use and final performance is not yet quantitatively explained. How high a
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temperature the membrane can withstand before degradation occurs, depends
on the type of membrane, its preparation and duration at the given temperature.Polarization, one of the major factors limiting practical performance of
electrodialysis, is a function of many variables, including temperature.
(See Section 3. 1. 1. ) The temperature dependance of the power consumption
has been studied and with feeds more concentrated than ZOOO p. p.m. , the
power expended at 5 C rises far more rapidly than at 40 C.
With sea water, the precipitation of calcium sulphate is
a major problem. In some temperature ranges, CaSO. has a negative tem
perature coefficient of solubility.
Other problems such as vapor pressure of the streams,
internal leakage caused by expansion of the stack itself, are also influenced
by temperature changes.
C. Forgacs has suggested the possibility of pre
heating the streams so as to utilize the benefits of high-temperature operation.
He delves thoroughly into the various aspects of such a possibility and suggests
that a thorough economic analysis is required to justify such operation.
Finally, K. S. Spiegler^ has suggested that for each
1 C rise in temperature, the resistance of an electrodialysis system is
reduced by 2%. Mason and Kirkham have developed the following type of
relationship,
<VT=
l + m(T-°To)
when temperatures are measured in Fahrenheit, T = 70 F, m = 0. 01Z for
strong electrolytes and the Ionic's nepton membranes. This relationship
predicts for each 1 C
as Spiegler suggests.
predicts for each 1 C rise in temperature, the same 2% resistance reduction
Study of the temperature effects on electrodialysis is
only just beginning. Now that each subcomponent of the electrodialysis process
can quantitatively be estimated, the effect of temperature on each individual
subcomponent is possible. This will enable the parameters which are most
affected by temperature to be isolated.
61
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3. 2 Phase II — Integration of Subcomponent Mathematical Elementsinto a General Analytical Expression
3. 2. 1 Development of General Mathematical Analogue
To develop a general mathematical equation the individual
resistance analogues can be combined as parallel or series resistances as
expressed in the following equations.
R 11=
i'
i (82)cell pair 1 , 1* '
R Rseries parallelequiv equiv
11,1
Rparallel RS R]equiv
(83)
Rseries=
RAP + RE + RAE + ' ' ' ' + RCM + RCE + RCP <84>
equiv
Where each component, R,,, Rr>r\' etc- , is independently evaluated. For a
list of the nomenclature in above equations, refer to Figures 20 and 21.
Resistance due to water transfer effects are included in
the expression for the figure of merit. The resistances in the network shown
in Figures 20 and 21 can be used to calculate total overall stack resistance
(R-,-.),- is multiplied by the total number of cell pairs in the stack and addedU.T 1
to the resistances associated with the two electrodes. An example of the
calculation of each resistive component and stack resistance is given in
Section 3.3.
3.2.2 Power Index
One of the objectives of this study is to predict the
operating stack power cost (excluding pumping cost). The power required to
treat 1000 gallons of water as dialysate is given in Equation (85).
0.0015 i RKw'hrs =
62
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OOMB
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where
Kw-hrs = kilowatt-hours to treat 1000 gallons as dialysate-2i = mean stack current density, ma cm
2R = area resistance of single cell pair ohms cm per cell pair
AC = change in dialysate concentration through the stack in gramequivalents per liter
A power index, P., is defined as
iR ACP. = —E
i TJ
which can be combined with the constants in Equation (85) to give power
costs per stack.
Define (i fl/Tj) as the figure of merit. All other factors
in equation (85) are given or easily evaluated.
Refer to Figure 22 and it can be seen that the actual
total current required i , is equal to
io=
i1+ iz +
i3 (86)
(Note: i. and i,- are a measure of the amount of back diffusion and water
losses that occur and are never recovered by the system. These could be
called pseudo currents.) (R )_,, the actual measureable cell pair resistance,
is obtained from:
= ir + IT + ir (87)KI K2 K3
where the R's are obtained from the various subcomponent analysis. J7, the
total cell resistance, is defined as:
n = (R )rpN + resistance component of the electrodes
where
N - number of cell pairs.
The definition of 7J , the coulomb efficiency, includes the
various water transfer losses, whereas, in Section 3. 1.5, ?7_ is defined as the
coulomb efficiency not including these water losses. Therefore, i_ in Figure 22
65
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R
• —••
| ^
R
R3
AAArwhere
R. = duct resistance loss
R-, = actual cell pair resistance
R« = Co-ion leakage resistance
i. = duct current loss
i™ - desalinating current
i- = current carried by co-ions
i . = pseudo current loss due toback diffusion of salt
i- = pseudo current loss due towater transfer processes
Figure 22. Electrical Schematic of Current ResistanceLosses in a Membrane Cell Pair
66
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need note be calculated because by including TJ in the power equation (85), water
transfer losses are automatically taken care of. T) is obtained by the procedure
suggested in Section 3. 1.5, which requires first TJn to be calculated from:
*D' i-Tl- (89)
(Note: If the denominator in the above equation included i^, then
(90)
3. 3 Phase III — Application of Generalized Mathematics Solution toSpecific Situations
The mathematical equations derived in Section 3. 1 will be applied
to and compared with the actual operating data for two electrodialysis plants
at Webster, South Dakota and Buckeye, Arizona. Most of the plant data for
Webster was obtained from Dr. J. Nordin and M. Seko and for Buckeye
from Dr. E. J. Parsi(24) and W . E. Katz(25) of Ionics, Inc. (26) The equations
will also be used for the design of a hypothetical sea water plant, and finally
assuming various technological advances, the plants at Webster and Buckeye
will be recalculated so as to compare the projected or future plant with the
present one. Equations from Section 3. 1 will not be repeated but only referred
to in the following Sections. Table VII summarizes the important equations
to be used.
As an example, Stage IV at Webster, South Dakota, will be used
to illustrate the application of the calculations. All other stages were calculated
by the same method. One very important difference between the plants at
Buckeye and Webster is that they are operated with the dialysate and brine
streams co- and counter-current, respectively. Other differences are
recognizable in Figure 23 and Figure 24. Tables VIII, IX and X
summarize the input data used to calculate the final analogue resistances
(shown in Tables I, II, III and IV) for the Webster, Buckeye, sea
water and projected plants.
67
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TABLE VIU
ARRAY OF DATA USED AS INPUTS FOR HIE ELECTROD1ALYSIS ELF.OTRIOA 1,
ANALOGUE MODEL FOR THE PRESENT WEBSTER PLANT
Stage I Stage II Slam- III
RAY(l) = K = i /iu= 0. 6447 0. 6333 0. 5*2K7
RAY (2) = (Cj)0 = outlet dial. cone, gm eq/i 0.0136 0.0087 0. 0060
RAY (3) =(C.)i
= inlet dial. cone, gm eq// 0.0206 0.0136 0. 0087
RAY (4) = (C, ) = outlet brine cone, gm eq/i 0. 0371 0.0349 0. 0328
RAY (5) =(Cfe)i
= inlet brine cone, gm eq// 0.0301 0. 0301 0.0301
RAY(6) = E.P.M. .= total outlet equiv per million 20. 6 13.6 8. 7/ j • 1 \ tOt 3.1(dial.)
RAY (7) = m = stack path length, cm 111.76 111.76 111. 76
RAY (8) = n = stack path width, cm 111. 76 111. 76 1 11. 76
RAY (9) = p = fraction usable desalination area 0. 77 0. 77 0. 77
RAY (10) = 6 = diffusion layer thickness, cm 0. 00214 0.00214 0.00214RAY (11) = A = const, from the Onsager Eq. 38. 2 38. 2 38.2RAY (12) = B = const, from the Onsager Eq. 0. 2215 0. 2215 0. 2215
RAY (13) = t°C = temperature, centigrade 8. 89 8.89 8.89RAY(14) = FPCA = 1/2 Ca++ fraction in product (epm 0.495 0.465 0.44
units> 4.4RAY (15) = FFCA = 1/2 Ca fraction in feed 0. 507 0.495 0.465RAY (16) = FPMG = 1/2 Mg++ fraction in product 0. 321 0.324 0. 316RAY (17) = FFMG =1/2 Mg++ fraction in feed 0.313 0. 321 0. 324
RAY (18) = FPNA = Na+ fraction in product 0. 183 0. 211 0. 244
RAY (19) = FFNA = Na+ fraction in feed 0. 180 0. 183 0.211RAY (20) = FPC1 = Cl" fraction in product 0.011 0.009 0.008RAY (21) = FFC1 = Cr fraction in feed 0.013 0.011 0.009RAY (22) = FPHCO3 =
HCO3 fraction in product 0.296 0.324 0. 360
RAY (23) = FFHCO3 = HCOj fraction in feed 0.280 0.296 0. 324
RAY (24) =FPSO4
= 1/2 SO4= fraction in product 0.693 0.667 0. 634
RAY (25) =FFSO4 = 1/2 SO4= fraction in feed 0. 707 0.693 0. 667
RAY (26) = CF = current dens. temp, correction 0.85 0.85 0.85RAY (27) = M = const, in limiting current equation 72. 3 72. 3 72. 3RAY (28) = n = const, in limiting current equation 0. 947 0.947 0.947RAY (29) = Mg = membranes per stack 432.0 432.0 432.0RAY (30) = y = spacer thickness, cm 0.023 0.023 0. 023RAY (31) -I- membrane thickness, cm 0.075 0.075 0.075RAY (32) = Ap = usable area, cm2 9574. 0 9574.0 9574. 0RAY (33) =
wa= water transport, ilf (anion) 0. 168 0.2333 0. 349
RAY (34) =Wfc
= water transport, t/F (cation) 0. 24 0. 3333 0.499RAY (35) = f/0 = uncorrected coul. efficiency 0. 982 0.993 0. 968
RAY (36) = E = stack voltage, volts 240.0 200.0 145.0RAY (37) = I = current, amps 35.0 24. 0 14. 0RAY (38) = yjj = brine manifold width, cms 2.818 2.818 2. 818RAY (39) = xjj = brine manifold height, cms — — —
RAY (40) = u<j = dialysate manifold width, cm 4. 246 4. 246 4. 246RAY (41) = Vd = dialysate manifold height, cm 3.77 3. 77 3. 77
RAY (42) = t- = anion transference no. in anion 0.9806 0.9806 0. 9806exchanger
RAY (43) = t = cation trans, no. in cation exchanger 0.9518 0. 9542 0.9557RAY (44) = i = constant in equation (100) 0.3838 0. 3838 0.3838RAY (45) = j = constant in equation (101) 0.0817 0.0817 0.0817RAY (46) = k = constant in equation (100) 48. 0 48.0 48. 0RAY (47) = q = constant in equation(lOl) 160. 0 160. 0 160.0RAY (48) =
(C^i)= outlet catholyte cone. (I) gm eq/X 0. 1879 0. 1879 0. 1879
RAY (49) =(CJjOj
= inlet catholyte cone. (I) gm eq/i 0. 1884 0. 1884 0. 1884
RAY (50) =(CA)Q
= outlet anolyte cone. (I) gm eq/X 0. 1879 0. 1879 0. 1879
RAY (51) =(CA)t
= inlet anolyte cone. (I) gm eq/X 0. 1884 0. 1884 0. 1884
RAY (52) = (C^) = outlet cathode buffer conc.gmeq// 0. 03026 0.03026 0.03026
RAY (53) =(C^)4
= inlet cathode buffer conc.gm eq/i 0.04005 0. 04005 0. 04005
StiiKC IV
0. <U,74
0. 0.141
0 0060
0.03190.0301
6. 0
111. 76
1 11. 76
0. 77
0.0021438.2
0.22158.89
0.41
0.440.300.320. 290.240.0070.0080.41
0. 36
0.58
0.640.85
72. 3
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4. 2463 77
0.9806
0. 9573
0. 38380.0817
48. 0160. 0
0. 1879
0. 1884
0. 1879
0. 1884
0. 03026
0.04005
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TABLE IX
ARRAY OF DATA USED AS INPUTS FOR THE ELECTRODIALYSIS ELECTRICALANALOGUE MODEL FOR THE PRESENT BUCKEYE PLANT
Stage I Stage II
RAY(l) = K =io /iUm
= 0. 1987 0. 1819
RAY (2) = (C ) = outlet dial. cone, gm eq/ i 0.0149 0.0066
RAY (3) =(Cd)i
= inlet dial. cone, gm eq/l 0.0344 0.0149
RAY (4) = (C.) = outlet brine cone, gm eq/l 0.0538 0.0621
RAY (5) = (C, ). = inlet brine cone, gm eq/l 0.0344 0.0538
RAY (6) = E.P.M. . = total outlet equiv per million (diaL) 14. 9 6. 6
RAY (7) = m = stack path length, cm 402.5 402. 5
RAY (8) = n = stack path width, cm 8.08 8.08RAY (9) = p = fraction usable desalination area 0. 700 0. 700RAY (10) = 6 = diffusion layer thickness, cm 0. 00265 0. 00265RAY (11) = A = const, from the Onsager Eq. 61.5 61.5RAY (12) = B = const, from the Onsager Eq. 0. 2284 0.2284RAY (13) = t°C = temperature, centigradeRAY (14) = FPCA = 1/2 Ca fraction in product (epm units)
26. 7 26.70.060. 11
RAY (15) = FFCA = 1/2 Ca++ fraction in feed 0. 17 0. 11
RAY (16) = FPMG = 1/2 Mg++ fraction in product 0.02 0. 01
RAY (17) = FFMG = 1/2 Mg++ fraction in feed 0.04 0.02RAY (18) = FPNA = Na+ fraction in product 0. 88 0. 94RAY (19) = FFNA = Na+ fraction in feed 0.80 0.88RAY (20) = FPC1 = Cl~ fraction in product 0. 89 0. 88RAY (21) = FFC1 = Cl" fraction in feed 0. 87 0.89RAY (22) = FPHCO3 =
HCO^ fraction in product 0.04 0.05RAY (23) = FFHCO, = HCOj fraction in feed 0. 04 0. 04RAY (24) = FPSO4 = 1/2 SO4= fraction in product 0.06 0.04RAY (25) =
FFSO4= 1/2 SO4= fraction in feed 0. 09 0.06RAY (26) = CF = current dens. temp, correction 1.05 1.05RAY (27) = M = const, in limiting current equation 590.0 590.0RAY (28) = n = const, in limiting current equation 0.6 0.6RAY (29) = MS = membranes per stack 550.0 550. 0
RAY (30) = y = spacer thickness, cm 0. 1016 0. 1016RAY (31) = i = membrane thickness, cm 0.06 0.06RAY (32) = Ap = usable area, cm2 3252.0 3252. 0
RAY (33) =wa
= water transport, l/F (anion) 0. 205 0.21RAY (34) =
wfa= water transport, i/T (cation) 0. 28 0. 30
RAY (35) =TJj-.
= uncorrected coul. efficiency 0.925 0.965RAY (36) = E = stack voltage, volts 400.0 355. 0
RAY (37) = I = current, amps 40. 1 21. 73
RAY (38) = y, = brine manifold width, cms 5.0 5.0RAY (39) =
xb= brine manifold height, cms 4.0 4.0
RAY (40) = u, = dialysate manifold width, cm 5. 0 5.0RAY (41) = V , = dialysate manifold height, cm 4.0 4.0
RAY (42) = ta = anion transference no. in anion exchanger 0. 9922 0.9961
RAY (43) =t+
= cation transf. no. in cation exchanger 0.9939 0.9939RAY (44) = i = constant in equation (100) 0. 1060 0. 1060RAY (45) = j = constant in equation (101) 0. 1014 0. 1014RAY (46) = k = constant in equation £100) 125.0 125. 0
RAY (47) = q = constant in equation (101) 67.0 67.0
RAY (48) = (CS.) = outlet catholyte cone. (I) gm eq/l 0. 1692 0. 169
RAY (49) = (C^;)i = inlet catholyte cone. (I) gm eq/l 0. 1694 0. 1692
RAY (50) =(CE)
= outlet anolyte cone. (I) gm eq/ 1 0. 1189 0. 1175
RAY (51) = (C,,). = inlet anolyte cone. (I) gm eq/l 0. 1192 0. 1189
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TABLE X
ARRAY OF DATA USED AS INPUTS FOR THE ELECTRODIALYSIS ELECTRICALANALOGUE MODEL FOR THE SEA WATER PLANT
Stage I Stage II Stage III Stage IV Stage V
RAY(l) = K = i /i.. =op hm
0. 0208 0.0208 0. 0208 0. 0208 0. 0208
RAY (2) = (C ) = outlet dial. cone, gm eq/£ 0. 242 0.098 0.04 0. 0162 0. 0066
RAY (3) = (C,)i = inlet dial. cone, gm eq/i 0. 594 0.242 0 098 0.04 0.0162
RAY (4) = (C, ) = outlet brine cone, gm eq/J 0. 652 0.444 0.358 0. 3238 0.3096
RAY (5) =(Cb>i
= inlet brine cone, gm eq/i 0.300 0. 300 0. 300 0. 300 0. 300
RAY (6) = E.P.M. ,= total outlet equiv per million(dial. )
° d242. 0 98.0 40.0 16.2 6.6
RAY (7) = m = stack path length, cm 4025.0 4025.0 4025. 0 4025. 0 4025. 0
RAY (8) = n = stack path width, cm 8.08 8.08 8.08 8.08 8.08RAY (9) = p = fraction usable desalination area 0. 7000 0. 7000 0. 7000 0. 7000 0. 7000RAY (10) = 6 = diffusion layer thickness, cm 0 00265 0.00265 0. 00265 0. 00265 0. 00265RAY (11) = A = const, from the Onsager Eq. 61.5 61.5 61.5 61.5 61.5RAY (12) = B = const, from the Onsager Eq. 0. 2284 0.2284 0. 2284 0. 2284 0. 2284RAY (13) = t°C = temperature, centrigrade 26.7 26.7 26. 7 26.7 26. 7
RAY (14) = FPCA = 1/2 Ca++ fraction in product (epm 0. 0207 0. 0265 0.0168 0. 0105 0. 0061units)
RAY (15) = FFCA = 1/2 Ca"l"f fraction in feed 0.0336 0.0207 0. 0265 0.0168 0.0105RAY (16) = FPMG =1/2 Mg++ fraction in product 0. 0992 0. 0561 0.0320 0. 0185 0. 0106RAY (17) = FFMG = 1/2 Mg++ fraction in feed 0. 1762 0. 0992 0.0561 0.032 0.0185RAY (18) = FPNA = Na+ fraction in product 0.8801 0.9173 0.9513 0. 9710 0. 9833RAY (19) = FFNA = Na+ fraction in feed 0. 7901 0.8801 0.9173 0.9513 0.9710RAY (20) = FPC1 = Cl" fraction in product 0. 9008 0.9031 0.9025 0. 9012 0.9031RAY (21) = FFC1 = Cr fraction in feed 0. 9031 0. 9008 0. 9031 0.9025 0.9012RAY (22) = FPHCO3 = HCOj fraction in product 0. 0062 0. 0041 0. 0050 0.0061 0. 0039RAY (23) = FFHCO3 = HCOj fraction in feed 0. 0039 0. 0062 0. 0041 0. 005 0. 0061
RAY (24) =FPSO4 =1/2 SO4
= fraction in product 0. 0930 0. 0928 0.0925 0.0926 O.tf929RAY (25) =
FFSO4= 1/2 SO4
= fraction in feed 0. 0929 0. 0930 0. 0928 0.0925 0.0926RAY (26) = CF = current dens. temp, correction 1.05 1. 05 1.05 1.05 1.-05
RAY (27) = M = const, in limiting current equation 590.0 590.0 590.0 590.0 590.0RAY (28) = n = const, in limiting current equation 0. 6 0.6 0. 6 0. 6 0. 6RAY (29) = Mg = membranes per stack 550. 0 550.0 550.0 550.0 550.0RAY (30) = y = spacer thickness, cm 0. 1016 0. 1016 0. 1016 0. 1016 0. 1016RAY (31) = i = membrane thickness, cm 0.06 0.06 0.06 0.06 0. 06RAY (32) = Ap = usable area, cm2 32520.0 32520. 0 32520. 0 32520. 0 32520.0RAY (33) =
wa= water transport, ilT (anion) 0. 17 0. 18 0. 19 0. 205 0.21
RAY (34) =wb
= water transport, i/F (cation) 0. 22 0. 24 0. 26 0. 28 0.30RAY (35) =
Thj= uncorrected coul. efficiency 0. 95 0.95 0.95 0.95 0. 95
RAY (36) = E = stack voltage, volts 4500. 0 2500. 0 1300.0 700.0 300. 0RAY (37) = I = current, amps 0.02298 0. 00934 0. 003794 0.001544 0. 000627RAY (38) = yfc = brine manifold width, cms 5. 0 5.0 5.0 5. 0 5.0RAY (39) = xjj = brine manifold height, cms 4. 0 4. 0 4. 0 4. 0 4. 0RAY (40) =
ud= dialysate manifold width, cm 5.0 5.0 5. 0 5. 0 5. 0
RAY (41) = Vd = dialysate manifold height, cm 4. 0 4.0 4.0 4. 0 4.0
RAY (42) = ta = anion transference no. in anion 0. 995 0. 995 0.995 0.995 0.995exchanger
RAY (43) =t+
= cation trans, no. in cation exchanger 0.995 0.995 0.995 0. 995 0. 995
RAY (44) = i = constant in equation (100) 0. 1060 0. 1060 0. 1060 0. 1060 0. 1060RAY (45) = j = constant in equation (101) 0. 1014 0. 1014 0. 1014 0. 1014 0. 1014RAY (46) = k = constant in equation (100) 125.0 125.0 125.0 125.0 125.0RAY (47) = q = constant in equation (101) 67. 0 67. 0 67. 0 67. 0 67. O
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3.3.1 Calculation of the Concentration Polarization at theMembrane Surfaces
(a) 6 , diffusion layer thickness
At the Webster, South Dakota plant, the volumetric
flow rate is approximately 250,000 gal/day, stack and the membranes per stack
are 432. Substituting these values into equation (3 )
Q = .00138 x432
= 0.799, gal/min, channel.
Substituting this value of Q into equation ( 2 ) we get the diffusion layer thickness
6 = [30-10.7(0.799)] 10"4 = 21. 4 x 10"4, cms.
At the Buckeye, Arizona, plant, the volumetric flow rate is approximately
90 gal/min through each of the three parallel stacks per stage. The number
of membranes per stack is 550. Substituting into equation ( 3 ),
^ nnt oo 90 X 60 X 24 ooc o i / i_ 1Q = .00138 --FTTJ- = . 3252 gal/min, channel.
Substituting this value of Q into equation ( 2 ), we get the diffusion layer thickness,
= [30 - 10. 7 (.3252)] 10"4 = 26. 52 x 10"4 cms.
(b) i, operating current density
At Webster, Q = 0. 779 gal/min, channel, and the
cross-sectional area to flow, A = 0.075 x 98 cm , the dialysate linear flow
velocity, as shown in equation (7). (Note the dimensions for the Buckeye
flow channel are given in Figure 25. )
V = 3. 7853 x 103 xQ , ,_ /-50-
A= 6- 69 cm/sec
and
- _ (Cdh" ^o .006 - .0041 nn_ ,
C -.)./(C.)
=
ln(.006/.004l)= • °°5 gm eq/cc
d i do
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ManifoldPorts
Baffels
The top half of the spacer; Y = 17. 7 "
Approximate path length = 8 x Y + 17 = 158.5" (402.5 cms)Path width = x = 1.59" (4.04 cms)
158.5 x 1.59 252% area utilization =
18 x 20= 70.0%
Figure 25. Schematic of the Mark III (Ionics Inc. ) Spacer75
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Substituting into equation ( 7 )
i.. = (0.005 x 10"3) x 72.3 (6. 69)°'94? = 0. 002165 amp/ cm2llITl
from Figure 4,(C.F.)9oc
= 0.9285
2= .002176 x . 9285 = 0. 00202 amps/cm
From equation (10) and (11)
..«
Q x 3, 785 3capiclty . -55--
= 0.031544 x Q x M x (C,)i -(Cd)Q gm equiv/sec
Capacity x F - Cap x 193, OOP
oper Axexn AxexMP PS
For Stage IV, Webster
Capacity = 0.031544 x . 799 x 432 (.0060 - .0041)
= 0.0207gms salt transf/ sec
0.0207 x 193,000 n nni . , 2i = ..,--. ,.—0 . . . 00 = 0.0011 amp/ cmoper 9574 x 0.914 x 432 *
Current Ratio = K = 0.001/0.00202 = 0.5674
(c) p mean, mean resistivity of the diffusion layer
The two methods mentioned in Section 3. 1. 1. 1. 1
are outlined. The second is used in Table XI.
(1) The Simple Mean Method
P is evaluated as the mean resistivity^mean 7
of the diffusion at the entrance and exit of the stream under consideration.
The dialysate stream in this case.
(C, ). = bulk solution inlet of dialysate = . 006 N
(Cb)Q= bulk solution exit of dialysate = .0051 N
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From the Nernst Equation (4), the surface concentration is:
<Cm>i*
<Cb>i-
(C ). = (CJ. - 1mi b i
ft - t)
(21.4 x 10"4) (.806 - . 39) 103
(96,500) (.905 x 10"4)
= (C, ). - 1.0205 (i .. )b i operating
2at operating current i = .0011 amp/ cm
inlet (C ). = .006 - 1.0205(. 0011) = . 004878 Nmioutlet (C )
= .0041 - 1.0205(. 0011) = . 002978 N™™™"™^"^^~ III O
C onc entr ation Inlet Outlet
surface . 004878 .002978
bulk .006 .0041
Figures 26, 27, 28 and 29 for Webster and
Figures 30 and 31 for Buckeye are all constructed using the Onsager Equation
(14) and Equations (19) and (20). The equivalent ionic conductances are obtained
from C. F. Prutton and S. H. Maron, "Fundamental Principles of PhysicalChemistry," p. 462.
From Figure 5, the following data is obtained,
t = 8. 89 C (Webster) A = 38.2
t = 26. 7°C (Buckeye) A = 61.5
B = 0.2215
B = 0.2284
Equivalent Conductance Inlet Outlet
Surface 74.0 75.0
Bulk 71.8 72. 0
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First Stage atWebster, South Dakota
50
0.0001 0.001 0.01
Average Solution Concentration (gm equiv/1)
Figure 26. Plot of Equivalent Conductance Versus Average SolutionConcentration for Various Temperatures. Derived fromthe Onsager Equation.
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Second Stage atWebster, South Dakota
50
0. 0001 0.001 0.01Average Solution Concentration (gm equiv/1)
0. 1
Figure 27. Plot of Equivalent Conductance Versus Average SolutionConcentration for Various Temperatures. Derived fromthe Onsager Equation.
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Then the average resistivity of the diffusion layer at the entrance and exit
(P ). = 1/2*1000
004878 x 74.0 t 1000
(p )= 1/2*™ rzir o 'ave
mean
I 1000002978 x 75.0
1397.0 + 3932.4
0060 x 71.8
| 1000
[.0041 x 72.0|
2664. 6 fl-cm
'= 1397.0 O-cm
= 3932.4
-4.R =p 6 = (2664. 6 x 21.4 x 10 )= 4. 702 fi-cm/diffusion layerc.p. 'mean v y
(2) The Integrated Method
In Equation (17), using the four boundary
conditions,
BC.: If x = 0, then C = C1 x, z o, o
BC.,: Ifx = 6, z = 0, then C = C.2 x, z 0 1 o
BC-: If x = 0, z = m, then C = C3 x, z o, m
BC . : If x = 6 , z = m, then C = C,4 x, z 6. m
it is possible to evaluate a, b, c, and d in terms of the concentration, viz. ,
a =(C. - C )6, o 0,0
b =
c =
(C - C )o, m o, o
m
(C. + C )- (C. + C T )
0 , m 0,0 6 , o o, L/6 m
d = C o, o
Thus, C can be evaluated at any x, and z from Equation (17).x, z
C = ax + bz + cxz + d . . . .X, Z(17)
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Hence, from Equation (15), we find
p = function of (x, z)X , Z
At the same time, (A )top is evaluated at any coordinate x from Equations
(14), (19) and (20). Equation (13), the integration can be easily evaluated by
summation and iteration on a computer.
(91)
m ••— —
1 1 V A ZR. . , m Ltotal nz=0
6
/ P AXZ-i x, zz=0
(Note: 150 iterations for both coordinates were used. )
At K = 0.5674, (R )„. ... = 8.557 O-cm2/diffusion layer.c. p. otage IV ——
The second method (8. 557 compared to 4. 7022O-cm /diffusion layer) is more accurate because it does not assume a linear
concentration relationship as does the first. Hence, Table XI is constructed
using the second or double integration method. In order to see the effect of
varying the K ration, Table XII and Figure 32 were constructed. When
optimization of the electrodialysis plant is undertaken, variation of resistance
(in this case concentration polarization) could be determined as a function of
K or the operating current density, i .
To obtain the overall concentration polariza
tion effect of a cell pair, there exists two brine and two dialysate diffusion
layers. For the brine stream, the concentration gradient increases from the
bulk to the surface and equation (16) is replaced by (18) in the computation.
Table XI summarizes the data obtained for
the Webster and Buckeye plants.
3.3.2 Estimation of Ohmic Polarization
The resistance rise due to ohmic polarization was
estimated using an Equation (22).
log R , =0. 150 + 0. 92K (22)s ohmic * '
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1 1. 00
10.00
} Webster, S.D.
1. 000
II III IVStage Number
Figure 32. Variation of Concentration Polarization With the NormalizedOperating Current Density for the Electrodialysis Plants atWebster, S.D. and Buckeye, Arizona.
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For the fourth stage at Webster, South Dakota, K is 0. 5674
log R , . = 0.150 + 0. 92 x 0.5674 = 0.672ohrnic
R , . =4. 699 ohms cm per cell pair per hour
R , . for a 20 hour period 4. 699 x 20 = 93. 98 ohms cm per cell pair.
3. 3. 3 Calculation of Electrode Polarization
In addition to the electrode streams, the buffer
streams at Webster will also be considered in this section. Figures 33
and 34 depict the approximate concentrations and schematic profiles of
both the electrode and buffer streams for Webster and Buckeye. Material
balances are shown in Table XIII and Table XIV. .
Cathodic and Anodic Resistances
Equation (33) will be used to evaluate the cathodic and
anodic resistance analogue, which will include the catholyte anolyte and buffer
bulk stream, the four diffusion layers and the electrode surface resistances.
(See Figure 33.) The full equation is,
R *, =(R .), ,, + (R ,),,,+ 2(R) + 2(R~)cath. /v sol'buffer sol'catholyte 6 0or
|or
anode ^ anolyte J ohmic
+ R , + R (33)chem conc
where
(R..) = positive sloping diffusion layer
(R.) = negative sloping diffusion layer
M''sol m
B
(46)
f az +
J 7T~o (z + yz + 6)
(Section 3.1.3 and 3. 3. 4)
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<v =in
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I 6
dz
o
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o
(13)
(Section 3. 1. 1. 1. 1 and 3. 3. 1)
conc-RTF
RT
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1
- i
chem . 5F V i
(Section 3. 1.2)
(26)
(28)
Using the above five equations and referring to the evaluation of the limiting
and operating current density in Section 3. 3. 1 (b), Table XV is constructed.
3.3.4 Calculation of Resistance Due to the Overall Cell Pair
Dialysate and Brine Material balances for Webster and
Buckeye are given in Tables XVI, XVII, XVHI and XIII.
Assume a straight line relationship from inlet to outlet
of brine and dialysate concentrations.
r "CB+CD
if k = 1 (40)
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streams, both of which will be described below so as to obtain a general
equation in the form of equation (46).
I. Counter Current Flow (Figure 35a)
C.
Az + Bz + CDz + E
(92)
(93)
(94)
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z = o z = L
(a) Counter-Current Flow (Webster)
c.1
z = o z = L
(b) Co-current Flow (Buckeye)
Figure 35. Concentration Profiles of the Dialysateand Brine Streams.
98a
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DA
'BE + CDDA
CEDA
/K7AD - K72AB\6 =—- —
/K.D2 + (AE-BD)K7 - (2AC+B2)K,e = -M 1
3
K3A
<P
12 DEKj + (BE + CD) KZ- 2 BCK3
K3A
., + JEL/t,, .. - C K,i'
K —2
K3A
a = F/G
m^ 2llf (z3 + az + ]8z + y)dx__ ._ ^ _ _ ___
pd
(z + 6z + 8z + z + Tj)
II. Co-Current Flow (Figure 35b)
CD= -ax + C. (96)
CD= +az + C. (97)
C = F + Gz2 (98)cL
m
2
^ = K" | <-4
+ »> ^ ,99)
P o (« + yz + M)
The nomenclature for the above eight equations is as follows:
where/AE + CB"
(8y
99
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- 2 KFKSG
K"K3G
where
A = 2(a')2
B = 2a'(Ci +CB)o
C = 2C. (CB)
D = 2 a1
E = C. +(CB)o
F = C.
c =
and
a = —— slope of C vs. z chart\mJ
m = total path length (of desalination area), cms
n = total width of chamber, cms
p = fraction of usable desalination area
= slope of C vs. z chart, gm. equiv/1 cm
= (C,,) - C. = C. - (C»)Bo i i Do
C. = initial feed concentration, gm. equiv/1
(C_) = final brine concentration, gm. equiv/1
a1 = I
<CD'o)= final dialysate concentration, gm. equiv/1
100
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K,,K0,K,, = constants obtained from R C vs. C ,1 i 3 p a a
see following derivation.
In order to evaluate K., K-, and K-, Equation (43) will be evaluated for three
different z-coordinate positions — a quarter, a half, and three quarters of the
path length, m. Thus, three simultaneous equations are obtained and the three
unknowns can be evaluated provided (R ) is evaluated at each coordinate from
the following equation,
(R ) = (P . +p ) t + < TT-T fr- + -rr— r 75- > 103y (37)p'z lKcat *an' m \ (Av)toc x CD' y \ '
^ D
Pcat=
^a^+ k
where
t = membrane thickness, cmsm
y = spacer thickness, cms
Ca,CD,CB = obtained from Equations (9Z), (93) and (94)
(A )for~= obtained from Section 3. 1. 1. 1. 1
v 't°C
i, j,k, q = obtained from manufacturer for given membrane.
(See Figure 36 for the constant values.)
The results of the above method is presented in Table XIX. It must be
emphasized that using equation (47) and (48) result in the exact answer, but
are more tedious to use than the summation Equation (49).
For Stage IV Webster,
KI = 1.9687 1
K, = 14.3605 ) and R = 249- 79 O-cm2/cell pairf P
K0 = 86.3433
101
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10,000
10.0
0.001 0.01 0.1
Solution Concentration - C (gm equiv/liter)
1.0
Figure 36. Resistivity of Ion Exhange Membranes andSodium Chloride Solutions
102
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a
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3.3.5 Calculation of the Parasitic Duct Loss Resistance
(a) Cell Pair Resistance, R
+R+R. + R, (58)cat d TJ van
From Equation (56), p and p are evaluated using the data from Stage IV
Webster, S. D. Refer to mass balance in Figure 37 for concentrations and
Figure 29 for equivalent conductance at 48 F.
(i) Dialysate
ol(63)
= 2.828 x 103 O-cm
where b refers to bulk stream, o to stream outlet and i the stream inlet.
(ii) Brine
P =j fioodl fiooo"| L i Jf looo ~| f looo 1\,/,v\ \ C \, C , .( 2 \ 64. 8 x . 0427 65.0 x .0406 f
b4'
\LJb,o L Jb.rf H. J L JJ= 0. 370 x 103 O-cm
We now evaluate
Rd
'
where
msn = 44 in. x 44 in. = 111.76 x 111.76 cm
y = . 075 cm
p = fraction of area for ion transfer = . 77
Substituting into above equation
(58)
= '-^p- 2.828xl03 + 0.370 x 103
(111. 76) x 0. 77 L -J
= 0.02494 O/cell pair
104
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Membrane
Brine Stream
Inlet0406 N (2699 ppm)
I
Outlet
(CB)= .0427 N (2826 ppm)
Dialysate Stream
t
Outlet
(CD). .0041 N (293 ppm)
Appm = 127
IInlet
(CD)= .0060 N (420 ppm)
eitfi
Figure 37. Mass Balance of the Dialysate and Brine Streamsfor Stage IV, Webster, South Dakota
105
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From Equation (100) and (lOl) and Figure 29,
R + R = (p + p )-J-
cat an 'cat 'an AP
R , + R = (Cl
+ k) + (Cj
+ q)c at an a x a ^
1
AP
From Figure 37, the average concentration C , defined by Equation (40), canct
be calculated at the inlet and outlet. Because C varies with path coordinate3,
z, the equation above was integrated from Z .= 0 to Z = 11 1. 76 cm. Forexplanation purposes, in this example of the calculation for Stage IV Webster,
the use of the inlet and outlet average concentration will suffice.
' °. 01052
Outlet (Ca)0= . „. 00744 gm eq/(
Substituting the correct data in the above equation for Webster, S. D. ,
r~ •?Q •?Q
R , + R =< (.01052)'3838 + 48.01 + |(. 00744)'
"01 '
+ 160|cat an N ' |X '
= 0.0005018 O/cell pair
Substituting into Equation (58)
+ R_) + (R d
= 0.000,5018 + 0.02494
= 0.0254418 O/cell pair
or in area units,
R = 0.0254418 x 9574 = 243, 579 fi-cm2/cell pair
This approximate method provides a fairly accurate result, i.e. , 243.579
(approximate method), and 249,579 (exact method). (See Table I.)
(b) Channel Resistance, RT______________________ j_j1_1 + _1
R£
Rd Rb
106
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Refer to Figure 38 for channel dimensions
Dialysate
Substituting the following values into Equation (56)
where (see Figure 38)
U , = 4. 246 cma
V , = 3. 770 cma
pd= 2.828 x 103 fl-cm
SL = 0.075 cm
y =0. 023 cm
N =216 cell pairs
R 2.828 x 103 x 2 x (.075 + .023) 216=
,Kd
"4. 246 x 3. 77
'
and substituting the following values in Equation (55)
p^= 0. 370 x 103 fl-cm
y, = 2.818 cms
•= - 0. 370 x 103 x 2 x (. 075 + .023) 216*
TT/4(2.818)2
Substituting equations (55) and (56) into Equation (57)
^+3
0.1337 + 0.3983= 1'880x
R
n ,t .
O/stack
"/stack
= 83.329 x 10 O/cell pair
We can now evaluate, * = -^ = 83'32C.9:° = 342. 101\ ^47 . D ( 7
From Figure 14, or Equation (54),
u 1•:— I = fraction of current leakage
= 0.0011, i.e., less than 1% current leakage
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Substituting into Equation (65), if we use data for
stack voltage E = 240 volts
and current i =35 amps
° • 6418'°.dOU35/ 23)This result is for the fourth stack at Webster (Sekov
' p. 591). Obviously,
the higher R, . the better because it will be in parallel with the stack current.d . Jo
In area-resistance units,
6418x9574 -,„. A-n ~ 2, ,.R, = -—yyr- = 284,470 Q-cm /cell paird« Jc £•l o
3.3.6 Calculation of the Corrected Coulomb Efficiency BasedSolely on the Water Transfer Processes
In Equation (68) is substituted the following data for
Stage IV Webster, S. D.
CO..= co + w, = 0.446 + 0.666 = 1.1321/Ftab
CQ=
(Cd).= .006N
CT= (Cd>o= -°°41N
tj_ = 0.914 from manufacturer.
The result is: TJ= 0.9078.
3.3.7 Calculation of Resistance Due to Membrane Selectivity
The values from the first stack at Webster, South
Dakota plant are substituted into Equation (80) below, if
E = 240 voltss
I =35 amps
t* = (1 - ta) = (1-0.9809) = 0.0191
t^= (1 -
t£) = (1-0.9573) = 0.0427
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R = ='
= 207. 12 ohms/ stack (80)s T,. c , .aI (t + t ;
or?f)7 1 ? v Q1^74. 7
R = (R, in Figure 22) =71 A
= 9180.4 O-cm /cell pair3 J ^ 1 O
3. 3. 8 Calculation of Resistance Due to Membrane Potential
Substituting Equations (73) and (74) into (75) and then
into Equation (76) we get
r (c ) "\R = £££ An b.m.z ) x (t
j+ ta -
1) 0-cm2/cellpair (76)m. p i £ \ \\j i I t -
1 o d, m z I
Since (C, ) , the brine surface concentration and (C , ) , the dialysateb,mz d, m z '
surface concentration both vary along the flow path, z, the above equation
was integrated to obtain an average (C, )/(C , ) ratio. To do this, theb , m d , m
concentration profile of both the dialysate and brine streams from inlet to
exit would have to be known. As a starting point, the simplified expressions(31) (32)of Shaffer and Mintz or Spiegler's ideas led us to assume the following
concentration profile along the flow path:
Dialysate: (CQ)z=
(C]-))i exP (^ ln CDo/'C;Di ) <102)
In the bulk solution,
Brine: (C,) = (C,,). exp (— In C^ /C_. ] (103)D z xi i \m JDO JDI /
Hence, Equation (76) becomes,
* m(C )
Rm.P= -TF- -Z I ln(C?l «P &^ CBo/CBi
0 Ul
r
&^
L
Replacing the integration by summation and using the data from Table VIIIfor Webster Stage IV,
R = 260. 16 ohms-cm /cell pairm. p l
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3.3.9 Electrodialysis of Sea Water
3.3.9.1 Operating Characteristics
The power requirements for a hypothetical sea
water conversion plant were calculated. In order to effect a direct comparison
with the power requirements for desalinating brackish water , the same stack
design as used in the Buckeye, Arizona plant was assumed. Since the salt
concentration of sea water is roughly 18 times that of the brackish water at
Buckeye, it was necessary to increase the number of stages in the hypothetical
plant to 5, and to increase the total effective membrane area within each stage
to 10 times that of a single Buckeye stage. This entailed a corresponding
reduction in the ratio of i , . /i.. , as explained below. All otheroperating lim £
operating characteristics, such as flow velocity and current efficiency, were
taken to be the same as those used at Buckeye. Refer to Table VIII, IX and X
for operating data, and Figure 39 for material balance.
3.3.9-2 Mathematical Determination of ioperating
An exponential relationship between the amount
of salt to be removed and the number of stages required was derived and applied
to the Buckeye plant in order to determine the value of i at any given stage.
Let C = dialysate concentration (e.p. m.) after passing through n stages;
C = initial concentration of sea water (593.685 e.p.m.). Then
•^= e'ne (105)
o
•where ,. ,
K(A ) Ml/ (C.F.) n6 =
*2 F 0 (31.544)= °Perating constant.
(Obtained from Section 3. 1. 1, Equations ( 5 J7T8 ), ( 9 ) and ( 10). )
For the two-stage desalination of brackish
water at the present Buckeye plant, we find that 6=0. 8238 (using K = 0. 1903
and A =3252 cm ). If we now divide 0 by K A , we obtain
Qn -n6' K A— = e p
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where 0' = 8/K A = . 00133. We require a product water salt conc. of
6. 645 e. p.m. To achieve this desalination ratio we shall employ five stages
wherein the total usable area per membrane = 10 A = 32,520 cm . Hence
(106) becomes
6. 645 -5(. 00133) 32,520 K593. 685
"e
or
89.343 = e2l6K
In 89. 343 - 4.49 = 216 K
K = iji2. = .0208 = i /i.. .216 op lim
We may now use Equation (105)to predict i at each stage:
i = K i,. = .0208 (C.F.) M l/°' 6log mean C (107)op lim H n '
M = constant = 590
V - linear flow velocity = 12.495 cm/sec
c -cn-1
log mean C = -,—=—7-* at the n'th stage (eq/cc).
n in N> / \** -.n n- 1 2i = operating current density (amps/cm ).
Refer to Table IV for results of the Sea Water Plant.
3.3.10 Electrodialysis of Projected Plants
After calculating the resistance analogues and comparing
the results for each stage at Webster and Buckeye, an attempt is now made to
predict what the resistance analogue of a future or projected plant would be,
providing several detrimental factors are favorably improved during the next
few years. A short description of each projected assumption is described
below. All unmentioned factors are held contant for both the plants discussed.
Table III shows the results. To comprehend the effect of the following
assumption, Table III should be compared to Tables I and II
3.3.10.1 Concentration Polarization
Equation ( 2 ) is changed to
6 = 20 - 10. 7 Q microns
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Reasoning is based on the fact that future development in spacer design,
hydrodynamic control and membrane surface phenomena would at least half
the diffusion layer thickness.
3.3.10.2 Ohmic Polarization
It is thought that advances in membrane
technology, operating and pretreatment procedures could possibly lower the
ohmic resistance tenfold.
3.3.10.3 Composite Cell Pair
Membrane advances would result in at least
the lowering of membrane resistances by fifty percent. Hence, the intercepts
for the membranes in Figure 36 are halved.
3.3.10.4 Membrane Polarization
Provided considerable advances take place in
minimizing the diffusion layer thickness, $, the limiting case would occur when
the concentration gradient across the layer is negligible (i.e. , 6" 0). Hence,
instead of using the membrane surface concentrations, we could, in the limit
use the bulk concentrations in Equations (73) and (74). This would lower the
final membrane resistances as shown in Table III.
3.3.10.5 Neglected Effects
Duct leakage at present is negligible (lessthan 1% — see Section 3. 3.5). Membrane selectivity although not a hundred
percent is at present very close (within the 98 to 100 percent range). Although
electrode polarization does contribute some resistance to the stack, it is very
small, and has less effect the more cell pairs used per stack. It is conceivable
that many times more cell pairs per stack could in future be used, and so
minimizing the electrode effects.
3.3.11 Discussion of Calculated Values
The area resistance for the factors listed in Tables Ithrough IV were calculated as described in the above sections.
The composite cell pair resistance consists of membrane
resistance and electrolyte resistance. The relative importance of these two
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factors is shown in Table XIX. The composite cell pair resistance in general
increases with inlet feed concentration as can be noticed in the Tables I - IV
in progressing from the first stage to the latter stages. Webster water contains
the least electrolyte and gives the highest value for composite cell pair resist
ance. A comparison of membrane resistance to electrolyte resistance at
various positions along the flow path are shown in Table XIX. The membrane
resistance in the Webster plant is low and insignificant, as the feedwater
changes in composition to that in the sea water plant, the membrane resistance
becomes greater than the electrolyte resistance and is twenty percent of the
total cell pair resistance In the Webster plant membrane resistance is only
two percent of the total stack resistance. The calculation for the projected
plants assumed that the membrane resistance was one -half that used in the
present plants. As indicated in Table III, the composite cell pair resistances
are not appreciably changed by this membrane improvement.
The concentration polarization values represent the
resistance of the diffusion layers adjacent to the membrane surface. Compared
to other factors, this resistance is minor and represents at most five percent
of total cell pair resistance.
The resistance due to ohmic polarization is a major
factor in total cell pair resistance. It represents over 40 percent of the
total resistance in Stage I at Webster. It is estimated that in a sea water
plant this factor could account for as high as sixty percent of the total. In
calculating ohmic polarization values for the projected plants, assumptions
were made that ohmic polarization could be reduced to ten percent of its
present value.
Duct leakage and selectivity resistances are parallel
to the desalting resistances in the network equations. The high resistance
values found for these factors are desirable and make only minor contributions
to total stack resistance. In the projected plants the assumption was made
that the influence of these factors could be reduced to an even greater extent,
makint their contributions completely insignificant.
Membrane polarization is a major factor contributing
as much as 87 percent of the total resistance in the last stage of a sea water
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plant. The resistance values for the projected plants are based on the assump
tion that concentration polarization in the diffusion layer can be reduced to
50 percent of its present value. The lower value of the membrane polarization
for the projected Webster plant is due partly to lower concentration gradients
in the diffusion layer which in turn is brought about by a higher flow velocity.
Electrode polarization is a minor factor, making up
about one-tenth of one percent of the total resistance.
The power index was calculated for each stage using
Equation ( 85A). These values are reported in the last column of Tables Ithrough IV. The index is proportional to the electrical power required to
treat 1000 gallons of water. This index decreases from first to last stage
for all plants, and increases from plant to plant as the salinity of the feed
water increases as is expected. The index can be used to compare stack or
plant design only when the same feed water is used and when the same change
in salinity is obtained. For example, a direct comparison between the
Webster plant and the improved Webster plant can be made with the power
index figures which are lower for the latter. The power index can therefore
be used to optimize stack and plant design.
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STATEMENT OF INVENTION
The Douglas Astropower Laboratory of the Douglas Aircraft Company
Missile and Space Systems Division does hereby certify that to the best of its
knowledge and belief no inventions resulted from performance under this
contract.
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REFERENCES
1. Mason and Rust; An Engineering Evaluation of the Electrodialysis ProcessAdapted to Computer Methods for Waste Deslination, Office of Saline WaterReport 134, February (1965)
~~~ ~~~~~ ~
2. Cooke, B. A. ; "Some Phenomena Associated with Concentration Polarization in Electrodialysis," First International Symposium on Water Desalination, October 3-9, 1965, SWD/18.
3. Weiner, S. A. ; Sea Water Conversion Laboratory, Report No. 65-6,University of California, Berkeley.
4. Rosenberg, N. W. and Tirrell, C. E. ; "Limiting Currents in MembraneCells," Industrial k Eng. Chemistry, April 57, 49, 4, 781.
5. Gregor, H. P.; Bruins, Paul F. and Rothenberg, M. ; "Boundary LayerDimensions in Dialysis," I & ER Process Design &t Development 4,No. 1, January 1965, p. 3.
6. Spiegler, K. S. ; "Saline Water Conversion Research in Israel," SalineWater Conversion II, Advances in Chemistry Series 38, p. 179 (1963).
7. Mason, E. A. and Kirkham, T. A.; C.E.P. Symposium Series No. 24,55 (1959), p. 173.
8. Private Communication with Dr. John Nordin from the Demonstration Unitat Webster, South Dakota.
9. Weiner, S. A. ; "The Effect of Temperature on Electrodialytic Deminerali-zation, " Bull. Chem. Soc. , 38, Japan (1965), p. 2002.
10. Spiegler, K. S. ; "Determination of Resistance Factors in Porous Diaphragmsand Electrodes," Journal of the Electrochemical Soc. , 113, 161 February1966.
11. Forgacs, C. ; Dechema, Monogr. 47, 601(1962).
12. Levich, V. ; Physiochemical Hydrodynamics, Chapter 6, Prentice-Hall,1962.
13. Mantell; Electrochemical Engineering, McGraw-Hill, New York, 1966,pp. pp. 53 f.
14. Wilson, J. R. ; Demineralization by Electrodialysis, p. 268, Butterworth,London (I960).
15. Mandersloot, W. G. B. ; Journal of the Electrochemical Soc. Ill, No. 7,838 (1964).
119
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16. Bergsma, Dr. F. ; Dechema Monogr. 47, 449 (1962).
17. Solt, G. S. ; First Internation Symposium on Water Deslination, October3-9, 1965, Washington, D. C. SWD/3.
18. Teorell, T. ; Proc. Soc. Expt, Biol. Med. _3-3, 282 (1935).
19. Meyer, K. H. andSievers, J. F. ; Helv. Chim. Acta 19. 649 (1936) andHelv. Chim. Acta 20, 634 (1937).
20. The Encyclopedia of Electrochemistry; Edited by C. A. Hampel, p. 802,Reinhold (1964).
21. Forgacs, C. ; First International Symposium on Water Desalination,October 3-9, 1965, Washington D. C., SWD/83.
22. Spiegler, K. S. ; Electrochemical Operations in Ion-Exchange Technology,ed. by F. C. Nachod - F. Schubert, Academic Press, Inc. 1956, p. 165.
23. Seko, M. ; Dechema, Monogr. 47, 575 (1962).
24. Private Communication with E. J. Par si , ionics, Inc.
25. Private Communication with W. E. Katz, ionics, Inc.
26. Kirkham, T. A. ; Paper presented April 4, 1963, Division of Water andWaste Chemistry, A.C.C., Los Angeles, California.
27. Mandersloot, W. G. B. ; First International Symposium on Water Desalination, Oct. 3-9, 1965, Washington, D. C. , SWD/40.
28. An Investigation of the Transport Properties of Ion Exchange Membranes,Arthur D. Little, Inc., Report No. C-66143, Contract 14-01-0001-372.
29. Graydon, W. F. ; Canadian J. Chem. 66, 1549 (1962).
30. Wilson, J. R. ; Demineralization by Electrodialysis, p. 268, Butterworth,London (I960).
31. Shaffer and Mintz in Principles of Desalination, ed. by K. S. Spiegler,Academic Press, 1966, N. Y.
32. Private Communication with K. S. Spiegler.
120
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APPENDIX A
COLLOIDAL COAGULATION
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APPENDIX A
COLLOIDAL COAGULATION
Conceivably, scale formation can occur within the stack by the flocculation of(Al)colloidal particles. Consequently, it is of importance to consider the
kinetics of colloidal coagulation. Using the authoritative monography by(A2)V. G. Levich, equations predicting the rate of coagulation of colloidal
particles were derived for three types of fluid conditions
A. 1 Brownian Motion (Stagnant Fluid)
Using the diffusion equation,
where
n = concentration of colloidal particles in the fluid(n = bulk concentration)
t = time-4 -7r = a = radius of the colloidal particle (10 - 10 cms)
D = diffusion coefficient of dispersed particles
Smoluchowski was able to derive the rate of coagulation per unit volume of
dispersion as
N = 8rrD a n2 (2A)
where
D, obtained from Stokes -Einstein formula =
ft = Boltzman's Constant
T = Temperature, degrees absolute
A. Z Gradient Coagulation (Laminar Flow)
In the electrodialysis unit, rectanular flow channels exist and the
laminar region will manifest itself near the membrane surface even though
turbulent flow at the center of the channel is predominant. The following
theory is developed without the presence of spacers or turbulent promoters
in the channels. For overall laminar flow only, the rate of coagulation per
unit volume is given by:
A-l
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NGrad TT no
where F = velocity gradient of the fluid,dv
dx
(3A)
and is easy to obtain if we(A3)know the velocity profile in a narrow slit. (See Figure A-l.)
-i
(0~ - 0 «)
and
dvT = B = Maximum at x = ± B (5A)
where
2B = distance between membranes
L = length of membrane
W = width of membrane
p - p, = includes both pressure and the gravityforces exerted onto the fluid
=(P0
-PL)
- Pg(L)
= Ap - pgL
(6A)
Ap
p
p
= density of fluid= pressure at some point z
)j, = viscosity of fluid
A. 3 Coagulation of Colloids in Turbulent Flows
We assume that a diffusion of particles occurs toward a sphere of
radius r, such that the distribution of the particles may be characterized by
the diffusion equation below, (refer to Figure A-2)
div (D grad n) = 0 (7A)
where
D. = effective diffusion coefficient-5R = coagulation radius « 10 cm
\ = mic rosc ale of turbulence « 10o
a = diameter of particle, cm
-2"cm
A-2
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CationExchangeMembrane
Dialysate FlowDirection
A AA A
VConcentrateFlowDirection
An ionExchangeMembrane
Figure A-l. Schematic Diagram of the Electrodialysis Flow Channel
A- 3
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\\
Figure A-2. Coagulation in Spherical Coordinates
A-4
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e = energy dissipated/unit volume and is characteristicfor a given flow, ergs/cm sec
p = density of the fluid, gms/cm
V = — = kinematic viscosityp
(3= constant
r = radial distance in spherical coordinates
Intergrating equation (7A), substituting for D , , for the case where the
eddies (\) are smaller (in size) than the turbulent microscale (X ).
D*=
Dturb ~ ^- X ~ P
- X2 for X <XQ
(8A)
We get
where r < X (9A)v dr '
1 o
Intergrating equation (9A) with boundary conditions
n = 0 at r = R
n = n as r -» ooo
and allowing for continuity of n over the surface r = X we find,
n 3
n K —j- (1 - =j.), r <XQ (10A)
i . « R X
o2Flux of particles across one square centimeter (1 cm ) of surface of the
sphere of coagulation, R, is equal to:
r = R , = R
Differentiating equation (10A) and let r = R, and substitute the result into
equation (HA). Also substitute D , from equation (8A) into equation (11A).
3n R —J = r— « in Kp t/ — (12A)
••*
-i-NJ ^
A-5
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if
2R3-= —
.5- is very small then j is a constant.
Xo
The total number of encounters per unit time, brought about by turbulence
agitation, or the rate of coagulation is
N. , = (4TTR2) n j = 12TT& ,/—- R3 n2 (13A)turb oj M V v o \ • i
Expressing e in terms of fluid flow velocity v, and large scale motions L3
where e >j- , equation (13A) becomesL
3
(14A)'turb ~./ T
~^T
If one can separate the phenomena of saturation precipitation from colloidal
coagulation and precipitation on the membrane surface, then the importance
of the latter effect on the Rohmjc polarization can be evaluated. Note that in
equation (14A), coagulation rate is proportional to the velocity to the one and
a half power. Once we know the size of the colloidal particles in the stream
and can determine the velocity and turbulence in the channel, an increase in
the turbulence will cause a thinner diffusion layer (beneficial) and an increase
in the rate of coagulation (detrimental) — or in terms of resistances, R willdecrease and R , , will increase. A trade-off point exists,
ohmic pol '
A. 4 References
Al Cooke, B. A. and Mandersloot, W. G. B., Trans. Inst. Chem.Engrs., _37, 14 (1959).
A2 Levich, V. G. , Physiochemical Hydrodynamics, p. 207,Prentice -Hall (1962).
A3 Bird, R. B. ; Stewart, W. E. and Lightfoot, E. N. ; TransportPhenomena, p. 62, John Wiley (1962).
A-6
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APPENDIX B
THEORETICAL PREDICTION OF MEMBRANE RESISTANCE COMBININGTHE STATISTICAL THEORY OF MEARES, ET AL. ,
AND SPIEGLER'S FORMATION FACTOR
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APPENDIX B
THEORETICAL PREDICTION OF MEMBRANE RESISTANCE COMBININGTHE STATISTICAL THEORY OF MEARES, ET AL. ,
AND SPIEGLER'S FORMATION FACTOR
B. 1 Discussion
The purely theoretical prediction of a membrane's resistance in some
known salt solution is the result of the following discussion. It is significant
to note, that the work presented below combines the statistical theory of(B 1)J. S. Makie and P. Meares, v and the recent work on porous diaphragms
/•O 7\by K. S. Spiegler. An added section on nonrigid membranes is also pre
sented.
In order to determine the resistance that a membrane will exhibit
when placed between two electrodes, which are passing current, it is neces
sary to predict its resistivity. Recognizing that the streams, flowing counter-
currently, are of different and varying composition, the membrane resistivity
will also vary from top to bottom of the channel.
Although it is somewhat naive to choose an average composition be
tween the streams and then evaluate the membrane's resistivity when saturated
with this solution, at present no better approach is available. Mason and/•p QX
Kirkham have used this approach defining an average normality, Ca, at
any point along the flow path, as
where
C , = dialysate concentration
C, = brine concentrationb
, brine compartment thickness _ D
dialysate compartment thickness t ,
Assume that the average concentration, C is linear with path length,ct
z, from channel bottom to top of stack. Define a mean C between inlet and3,
outlet average C as followscL
B-l
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* a' meanbottom
2cdcb(2B)
top
(B2)It has recently been shown by K. S. Spiegler, XJ~""/ that the porous path
through a diaphragm (rigid) by ions and gas molecules are similar, although
certain corrections for ion-exchange membranes must be made. Both mem
brane swelling and the lowering of the resistivity due to adsorbed water would
necessitate such corrections.
B. 1. 1 Rigid Membranes
A glass diaphragm is an example of this classification. One
of the distinguishing factors between rigid and nonrigid (organic ion exchange)
membranes, is that the porosity, e, is constant for the former but variable
for the latter. A discussion of this phenomenon is given in the following
section.(B2) defines a formation (resistance) factor:
where
K. S. Spiegler
F, = o1/ = e2/E S e
p = resistivity of the material when the pores arefilled with a conductive liquid, ohms -cms.
(3B)
= resistivity of liquid without the diaphragm, ohms -cms
real path lengthgeometric path length9 = tortuosity of the pore model =
J. S. Makie and P. Meares1 + V
e = dry porosity ( = non-solid volume fraction ofthe diaphragm)
Starting from the lattice model for polymer solutions,(Bl) derived,
9 =1 - V (4B)
where
V = volume fraction of the polymer
V = (1 - e), by definition.
substituting into equation (4B)
B-2
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a - t1 + (*- e>j (2 - e)
9LI - (1 -e)J
— (5B)
Substituting equation (5B) into equation (3B) and solving for p ,
(6B)\c/ eJ
therefore
(2 .2F_ = * ~*e) = formation factor. (7B)Hi J
Refer to Figure B-l for plot of F_ versus e.jii
B.I. 2 Nonrigid Membranes
The porosity, e, as described in the above equations,
is constant for rigid membranes or diaphragms, but when swelling occurs,
e, will increase, as is the case with organic ion exchange membranes. To
compensate for this phenomena, e will be increased by the wet to dry volume
ratio, Rw or by adding a swelling porosity component e . The lowering ofw s
the membrane resistivity due to the presence of adsorbed water within the
pores would in essence also result in a higher value of e. An empirical
adjustment factor (A_) will be used in the latter effect.
The modified membrane porosity, e is given by
(8B)»»
where
D s wet volumert.
W e dry volume
For the adsorbed water effect, instead of using Fp, as defined by equation
(3B), a modified FJ, will be defined, where
B-3
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10,000.0
1,000.0
100.0
w
10.0
0.2 0.4 0.6
Porosity, e
0.8 1.0
1,Figure B-l. Plot of the Ratio of the Membrane Resistivity (n ) to the Solution
Resistivity ( ) Versus Porosity (e )
B-4
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where empirically
AD <9B>
- e(3 W (10B)
Hence, if e, dry porosity and Rw, wet to dry volume ratio are obtained from1
the manufacturer, using e , modified porosity, we can determine F^, forma
tion factor from Figure B-l and finally p at various points along the flow path.
B. 2 Resistance Analogue of Membrane Resistance
After evaluating p , membrane resistivity at a given concentration,
the cationic or, anionic membrane resistance analogue can easily be evaluated
from the definition
II = 0 i» ohms -cm . (HB)cat or anionic
where
I - thickness of the membrane, cms
B.3 References
Bl Makie, J. S. and Meares, P., Proc. Roy. Soc. (London) A 232(1955), p. 498.
B2 Spiegler, K. S. , "Determination of Resistance Factors inPorous Diaphragms and Electrodes, " Journal of the Electro-chemical Soc. , 113, 161 February 19&ftST
——————
B3 Mason, E. A. and Kirkham, T. A., C.E.P. Symposium SeriesNo. 24, 55 (1959), p. 173.
B-5
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APPENDIX C
HYDRODYNAMIC FLOW
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APPENDIX C
HYDRODYNAMIC FLOW
The equations of continuity and motion apply to turbulent flow, but it is nigh
impossible to solve them because the turbulent eddies fluctuate widely about
their mean velocity and pressure. By averaging the equations of change,
"time smoothed" equations result, which in turn need empirical expressions
for the "turbulent momentum flux. " The spacer network is inserted between
the ion exchange membranes so as to increase turbulence and lower polari
zation near the membrane surface. The complex spacer pattern such as the
expanded spacer design (see Figures C-l, C-2, C-3), will be used to deter
mine local Reynold's numbers so as to predict which areas within the network
are more susceptible to scale deposit as a result of stagnant or slow liquid
spots. Since the statistical theory of turbulence is in its embryonic stage
and the complexity of the problem is enormous, no attempt here will be made
to set up flux equations across the network and membrane. A semi -qualitative
description will be attempted, using Reynold's number to describe the kind of
flow at various local points. The following assumptions are made.
1. The fluid flow direction will be perpendicular to section A-A in
Figures C-l and C-2.2. The flow is equally distributed across the spacer network.
3. The velocity, calculated fromthe volumetric flow rate across
some chosen cross-sectional area, is representative of the fully
developed average velocity of the profile in the prechosen cross-
section.
4. Equivalent diameter, D , is equal to four times the hydraulic
radius, Ru, which is the ratio of the stream cross- sectionalrlarea to the wetted perimeter.
5. Dimensions of the expanded spacer as shown in Figures C-4,
C-5 and C-6 are obtained from Figure 8 of "Electrodialysis
Equipment and Membranes, " by H. J. Cohan, 53rd Annual
Meeting of AIChE, Symposium on Saline Water Conversion,
Dec. 4-7, I960.
C-l
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\Fluid FlowDirection
Figure C-l. Isometric View of Expanded Spacer Design
C-2
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Figure C-6. Number of Expanded Spacer Contact Points
Per Unit Area, n =
20x 1
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The following ratio (N~ • —) will be evaluated for three different cases:
.tx c p
the first case for the empty channel without a spacer (Figure C-5a); the
second case with the fluid flowing inside the hexagon design of the spacer,
perpendicular to plane A-A (Figures C-l, C-2 and C-5b); thirdly when the
fluid passes under and over the strut at plane C-C (Figures C-l, C-3 and
C-5c).
The equations below are now applied to the data obtained from Webster,
South Dakota.*02*
Volumetric flow rate = Q gal/min channel
'3,785^ _ 3, , ,• Q cm /sec, channel
160 ;
D VReynolds number is defined: N0 = p
x\.e p,
Comparison ratio used: N_ • —I = D V (1C)
where
V = local velocity cm/secD = equivalent diameter, cms
_. , area of stream cross-section= r R = 4H wetted perimeter
Refer to Table C-I.
From column (7) Table C-I we conclude that, the flux equations and their
vector directions must be evaluated (not possible with present technological
tools) to determine the turbulence inducing effects of the spacer design. This
conclusion is based on the unreasonable prediction from column (7) that the
NR is higher without the spacer than with. A statistical approach would be
necessary so as to account for the random turbulent eddies, but with the
present extremely complex configuration this is not possible. This is an
area where considerably more experimental and theoretical study is needed.
C-8
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REFERENCES
Cl Cohan, H. J. , "Electrodialysis Equipment and Membranes, " 53rdAnnual Meeting of AIChE, Symposium on Saline Water Conversion,December 4-7, I960.
C2 Seko, M. , Dechema. Monogr. 47, 575 (1962).
C-10
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APPENDIX D
NOMENCLATURE
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APPENDIX D
NOMENCLATURE
Symbol
A
a
A,D
B
C
a mean
d or
(C )d or b b'o or i
Cd. Cb
d or b mo or i
S
EPM
F
Definition
temperature dependent constant in the Onsager equation
actual membrane cross-sectional area, cm (=mxn)
empirical adjustment factor for F'i,
temperature dependent constant in the Onsager equation
average concentration of a cell pair, gm equiv/liter
mean of C at the bottom and top of the stack channel,8L
gm equiv/liter
bulk dialysate or brine concentration at some point x alongthe fluid path length from the bottom of the channel upwards
dialysate or brine bulk concentration (outlet or inlet), gmequiv/liter
shortened notation of dialysate and brine bulk solutionconcentration, gm equiv/liter
dialysate or brine membrane surface concentration(outlet or inlet) gm equiv/liter
Diffusion coefficient
porosity (non-solid volume fraction of the membrane)
T]ni coulombic efficiency (not including the water transfer
losses)
stack voltage , volts
total equivalents of cation per million in feed water
Faraday's No. , coulombs /gm equiv
concentration stream exit flow rate, ml/sec
dilute stream exit flow rate, ml/sec
formation factor for a rigid membrane
D-l
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F' = formation (resistance)factor for a flexible membraneft
GPD = gal /day of product water
I = stack current, amps
i = stack current density, amps /cm
i = current leakage due to duct losses, m?* cm
k = ratio of brine to dialysate compartment thickness
K,,K_,K, = constants in the generalized R versus C equation± £ j DA
^ j = anionic or cationic exchange membrane thickness, cmscL C
m = overall membrane length, cms
Mc = membranes /stackS
N = number of cell pairs /stack
n = overall membrane width, cms
p = fraction of membrane area available for desalination
ppm = parts per million of dissolved salt
Q = volumetric flow rate, gal/min, channel
R = brine channel resistance, Q / stack
R ..R = cationic or anionic exchange membrane resistancescat' an respectively, Q-
R = concentration polarization resistance,£}
- cm
= dialysate channel resistance, fj/stack
d. & . = parasitic duct loss resistance, j^/stack
R = equivalent channel resistance, Q/stack
R , = rate of change in ohmic polarization resistance of stack,
ohms hr
R = equivalent cell pair resistance, Q/cell pair
D-2
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(R ) = cell pair resistance at some specific distance x, up fromthe bottom (x = O) of the stack, j)
R = wet to dry volume ratio of the membrane
S = 1000 at given temperature for NaC 1 solutionA
"
0T = temperature, C
t = time of operation, hours
t , t- = transport number, of counter ion in anion and cationexchange membranes respectively
(* f\.t , t = transport numbers of the co-ions in the cation and anion
exchange membrane respectively, where tc= 1 - tc and
t* = 1 - t*
d or b = transport number of counter ion in anion and cationexchange solution (dialysate or brine)
u, = dialysate channel width (rectangular cross-section) cms
v, = dialysate channel height, cm
V = volume fraction of the polymer (solid)
w = ave equivalent wt of dissolved salts, gm equiv/liter
primary hydrespectively
w , w = primary hydration number for sodium and chloride ions
w = primary hydration number for both sodium and chlorideions
i
x = fraction open area of spacer
= brine channel diameter (circular cross -section), cms
= dialysate or brine compartment thickness, cms
6 = diffusion layer thickness, cm
p = tortuosity of the pore model
A = equivalent conductixaice at concentration , , (C, )' a or b b ' o or i
D-3
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equivalent conductisjBrce at infinite dilution
o or id or b "ave m or b =
c aP P
P(Ca)mean
d bP 'P
d or b mean
average resistivity (dialysate or brine at the inlet oroutlet) of solution at membrane surface (m) and the bulksolution (b), Q - cm
resistivity of the cationic or anionic membranesrespectively, Q
- cm
resistivity of the solution at (Ca) , n- cmmean
resistivity of the dialysate and brine bulk solutionrespectively, Q- cm
mean resistivity of p at the inlet and outlet of
dialysate or brine streams, Q - cm
°ohmic rate of change in ohmic resistivity, ohms cm per cellpair per hour ^
Idimensionless channel resistance, = •=—
=^-
ftU. S. GOVERNMENT PRINTING OFFICE : 1967O - 256-935 D-4
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UNIVERSITY OF MICHIGAN
39015078505586
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