UNIVERSITY OF HAWAI'llIBRARY...outputW orOlI'and output P 205 Discussion 207 Conclusion 208...

$: UNIVERSITY OF HAWAI'llIBRARY...outputW orOlI'and output P 205 Discussion 207 Conclusion 208 References 208 Chapter 7: The Properties ofMinerals X-raydiffraction analysis 239 Surface$
UNIVERSITY OF HAWAI'llIBRARY

KINETICS OF ADSORPTIONIDESORPTION OF NITRATE AND PHOSPHATE ATTHE MINERALIWATER INTERFACES BY SYSTEM IDENTIFICATION

APPROACH

A DISSERTATION SUBMITTED TO THE GRADUATE DIVISION OF THEUNIVERSITY OF HAWAll IN PARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

IN

AGRONOMY AND SOIL SCIENCE

MAY 2004

ByXiufu Shuai

Dissertation Committee:

Russell S. Yost, ChairpersonRichard E. GreenClark C. K. Liu

Vassilis L. SyrmosGoro Uehara

ACKNOWLEGEMENTS

Many individuals deserve special thanks for the help and support to make this

dissertation complete.

I especially thank my advisor, Dr. Russell S. Yost, for his leadership, friendship,

encouragement, patience, and continuing support over the seven years. I appreciate his

unselfish sharing of his time with me and take great pleasure in sharing this work with

him.

I thank my dissertation committee members, Drs. Carl I. Evensen (former

committer member), Richard E. Green, Clark C. K. Liu, Vassilis L. Syrmos, Goro

Uehara, for their critical comments and helpful suggestions on my dissertation.

I thank Dr. Jaw-Kai Wang and his Aquifer Culture Inc. to support me on using

HPLC, other device, and space in his lab. I thank Dr. Jingyu Chen for his friendship and

help in HPLC and chemistry. I thank Mrs. Sally Koba for her help and coordinator in lab

work. I thank Prof Ningshou Xu for his friendship, teaching and help on system

identification. I thank Dr. Istvan Kollar on teaching software of system identification. I

thank Dr. Vassilis L. Syrmos for his inspiring and wonderful courses in linear system

theory and his help on system identification. I thank Mr. Jin Yin, Mr. Chaopin Zhu and

Mr. Wei Zheng for the help on system theory and system identification. I thank Dr.

David D. Bleecker and Mr. Sixiang Nie for their helps on mathematics. I thank Dr. Moto

Kumagai, Dr. Lin Hong, and Mr. John (Liangzhong) Zhuang for the help on the nanopure

deionized water. I thank Dr. Mike Garcia, Dr. Kent Ross, and Dr. Yucheng Pan for their

helps on grounding the minerals. I thank Mr. Scott Edward for the help on pH detector. I

thank Dr. Jane Schoonmaker for her help on the X-ray analysis. I thank Dr. James A.

III

Silva and Mr. John F. Fong for their helps on phosphorus analysis. I thank: Dr. Jiacai Liu,

Dr. Xiushen Miao, and Mr. Zhaohui Wang for their helps on my experiments.

I am grateful to Dr. Donald R. Nielsen and Mrs. Joanne Nielsen for their

continuous friendship, love and encouragement over the ten years. I am grateful to my

wife Zhijun Zhou, my daughter Michelle Yuche Shuai, and my parents for their love,

support and patience on me.

IV

ABSTRACT

The currently available surface complexation models, such as the Two-Plane

Model and Triple Layer Model, are based on experiments at equilibrium status, and thus,

need to be validated by the experiments of kinetics. A set of novel column experiments

were designed and carried out based on the system identification approach. The input

signals, the sinusoidal change of concentrations of solute in the influent solutions, were

designed to excite the adsorption/desorption at the mineral systems at both pH 4 and pH

10, and the corresponding output signals, the dynamic concentrations of solute in eflluent

solution, were obtained. Mathematical models in the frequency domain, transfer

functions, were derived according to the various surface complexation models. Complex

curve fitting of transfer functions was used to identify the proper model.

The columns were separately packed with variable charge minerals including

bauxite, goethite, hematite, and kaolinite. The tracers, acetone, nitrate and phosphate,

were sequentially used to study their adsorption/desorption at the mineral/water interface.

When acetone was used as inert tracer, the transfer function of Convection

Dispersion Equation (CDE) was derived and simplified into two linear equations, and the

dispersion coefficients and water velocities were estimated by least squares methods.

In the study of nitrate and phosphate (P) adsorption/desorption at the

mineral/water interface, the transport and reaction were coupled together. The algorithm

of complex curve fitting adjusted the weights of the real and imaginary parts of the

logarithmic transfer function, and estimated the model parameters with Gauss-Newton

nonlinear procedure. The adsorption/desorption of nitrate and H+ or Off for the mineral

systems at both pH 4 and pH 10 were linear or approximately linear. The relationships of

v

the concentrations ofW or OIr and nitrate in the eflluent solutions were linear. Similar

results were obtained for the study ofP adsorption/desorption at the mineral/water

interface. The proper mechanisms for nitrate adsorption/desorption at mineral/water

interfaces were Triple-Layer Model at pH 4 and Two-Plane Model at pH 10. The proper

mechanisms for phosphate adsorption/desorption at mineral/water interfaces were Triple

Layer Model at pH 4 for all four minerals and gibbsite and goethite at pH 10, and Two

Plane Model for hematite and kaolinite at pH 10.

VI

TABLE OF CONTENTS

Acknowledgement .iii

Abstract '" v

List of Tables xiii

L" fF" ..1st 0 19ures '" '" , '" ..XVll

Chapter 1: Parameter Estimation for the Convection-Dispersion Model

for Non-reactive Transport Process via Transfer Function Approach

Abstract '" , 1

Introduction 2

Materials and Methods .4

Chemicals 4

Minerals , 4

Setup ofExperiments , 5

Design of input signals , '" 5

Input Signal Experiments , , , '" 6

Output Signal Experiments 8

Data Retrieval. 8

Mathematical Models in Frequency Domain

Transfer Function of the Transport Process 9

Estimating Solvent Velocity and Dispersion Coefficient 10

Result

Relationship between Acetone Concentration and Absorbance 11

Input signals in time domain 11

Vll

Spectral component of input signals 12

Variance of input signals among repeated experiments 12

Output signals and their spectral analysis 13

Estimate ofsolvent velocity and dispersion coefficients 14

Conclusion 15

References 16

Chapter 2: Parameter Estimation of a Transfer Function

Abstract " 36

Introduction '" '" 36

Mathematical Methods

Approach I. 40

Approach II. 42

Results

Example 1 43

Example 2 '" '" 44

Conclusion , 44

References 45

Chapter 3: Kinetics ofNitrate AdsorptionlDesorption at the MinerallWater

Interface by System Identification Approach

Abstract 55

Introduction 56

Vlll

Materials and Methods

Chemicals 62

Minerals '" 62

Setup ofExperiments 63

Input signal Design 63

Input and output signal Experiments 63

Mathematical models and algorithms for parameter estimation

Transfer functions derived from Two-Plane Model 64

Transfer functions derived from Three-Plane ModeL 65

Algorithm for model selection and parameter estimation 66

Result

Linear relationship between nitrate concentration and absorbance 66

I . I' .nput sIgna s In tlme 66

Spectral component of the input signals 66

Variance of input signals among repeated experiment 67

O . I' .utput sIgna s In tlme 68

Spectral component ofoutput signals '" 68

Estimates of parameters in the transfer function for systems at pH 10....69

Estimates ofparameters in the transfer function for systems at pH 4 ......69

Equilibrium constants of electrolyte adsorption/desorption at pH 4 70

Conclusion 71

Appendix 1 72

Appendix 2 73

IX

References 76

Chapter 4: Dynamics of Aqueous nitrate and W IOIr Concentrations

in Effluent Solutions from Columns of Variable Charged Minerals

Abstract 117

Introduction 118


Experimental setup 120

Result

Dynamical changes ofH+ or OIr concentrations in influent 121

Dynamical changes ofW or OIr concentrations in effluent 121

Spectral component of output W or OIr 122

Relationship of amplitudes and phases between

output H+ or OIr and output nitrate '" 122

Discussion '" '" .,. " .124

Conclusion 125

References 126

Chapter 5: Kinetics ofPhosphorus Adsorption/Desorption at the MinerallWater


Abstract 157

Introduction 158


x

Chemicals 160

Minerals " .160


Design of input signals 161

Experiments for studying P input 161

Experiments for studying P output 162

Mathematical models and algorithm for parameter estimation

Transfer functions for mineral systems 162

Parameter estimation ofthe transfer function 163

Result

Input signals in time-domain " .164

Spectral component of the input signals 164

Variance analysis of input signals among repeated experiments 164

Output P and their spectral analysis 165

Model selection and parameter estimation 166

Conclusion " .166

Reference 167

Chapter 6: Dynamics ofPhosphate and H+/OIf Concentrations in

Eflluent Solutions from Columns ofVariable Charged Minerals

Abstract. 199

Introduction 200


Xl


Sample collection 202

Result

Dynamic changes ofW or OlI' concentrations in influent. 204

Dynamic changes ofP and W or OlI' concentrations in eflluent 204

Spectral component ofoutput W or output OlI' 204

Relationship of amplitudes and phases between

output W or OlI' and output P 205

Discussion 207

Conclusion 208

References 208

Chapter 7: The Properties ofMinerals

X-ray diffraction analysis 239

Surface Area 239

Acetone adsorption isotherms 239

Phosphorus adsorption isotherms 240

xu

LIST OF TABLE

Table Page

1.1. Gradient table of input signal with period 17

1.2. Minimum time to run input and output signals 18

1.3. Standard deviation and CV ofamplitudes and phases of

input signals among repeated experiments 19

1.3 Averages and standard deviations of estimates ofwater velocity

and dispersion coefficient among repeated experiments 20

2.1 Frequencies and frequency responses used for example 1 47

2.2 Transfer function and its logarithmic equation in example 1 fitted by

modified Gauss-Newton method .48

2.3 Frequencies and frequency responses used for example 2 .49

2.4 Transfer function and its logarithmic equation in example 2 fitted by

modified Gauss-Newton method 50

3.1. Property of columns 81

3.2. Estimated dispersion coefficients (D) and ratio of sorption and

desorption rates for bauxite system at pH 10 81


desorption rates for goethite system at pH 10 82


desorption rates for hematite system at pH 10 83


desorption rates for kaolinite system at pH 10 84

Xlll

Table Page

3.6. Average and standard deviation of dispersion coefficients (D) and

ratio of adsorption and desorption rates for mineral systems at pH 10

among repeated experiments 85

3.7. Estimated dispersion coefficients and rates of adsorption and desorption

for bauxite system at pH 4 86


for goethite system at pH 4 87


for hematite system at pH 4 88


for kaolinite system at pH 4 89

3.11. Average and standard deviation of dispersion coefficients and rates of

adsorption and desorption for mineral systems at pH 4

among repeated experiments for mineral systems at pH 4 90

3.12. Equilibrium constants of reaction path and the overall equilibrium

constant of sodium nitrate adsorption/desorption at mineral/water

interface at pH 4 91

4.1. Sampling intervals of pH for different periods of signals 128

4.2. Regression coefficients oflinear relationship between adjusted W

or OIr amplitudes and those of nitrate '" .. , " .129

4.3. Regression coefficients of linear relationship between adjusted H+

or OIr phases and those of nitrate 130

XIV

Table Page

4.4. Ratios of concentration deviation from average 131

4.5. pRo from Sverjensky and Sahai (1996) 131

5.1. Physical properties of mineral columns " .169

5.2. Sampling intervals for input P and output P of a certain period '" .. 170

5.3. Maximum of relative amplitudes of subharmonics 171

5.4. Parameter estimates and 95% confidence interval ofbauxite system

at pH4 172

5.5. Parameter estimates and 95% confidence interval ofbauxite system

at pH 10 '" '" , 173

5.6. Parameter estimates and 95% confidence interval ofgoethite system

at pH 4 '" '" 174

5.7. Parameter estimates and 95% confidence interval of goethite system

at pH 10 , '" 175

5.8. Parameter estimates and 95% confidence interval of hematite system

at pH 4 '" , '" '" 176

5.9. Parameter estimates and 95% confidence interval of hematite system

at pH 10 '" '" '" 177

5.10. Parameter estimates and 95% confidence interval of kaolinite system

at pH 4 '" '" , '" 178

5.11. Parameter estimates and 95% confidence interval of kaolinite system

at pH 10 179

xv

5.12. Averages of parameter estimates among repeated experiments and

ratios ofadsorption and desorption rates 180

6.1. Sampling intervals of pH for different periods of signals 210

6.2. Maximum of relative subharmonic amplitudes 211

6.3. Regression coefficients and their 95% confidence intervals oflinear

relationships between adjusted amplitudes ofW or OIr and

amplitudes ofP 212

6.4. Regression coefficients and their 95% confidence intervals oflinear

relationships of phases between W or OIr and P 213

6.5. Ratios ofconcentration ofP and H+ or OIr in effiuent solution

for mineral systems at pH 4 and 10 214

7.1. Peak search report for gibbsite , , '" .243

7.2. Peak search report for goethite 244

7.3. Peak search report for hematite 245

7.4. Peak search report for kaolinite '" 246

7.5. Initial acetone concentrations and those after 18 hours sorption 247

7.6. Coefficients of models ofP sorption isotherms 248

XVI

LIST OF FIGURES

Figure Page

1.1. Schematic representation of experimental setup. .. . . . . . . . . . .. . . . . . . . .. . . . . . .. . .. . 20

1.2. Linear relationship between acetone concentration and absorbance 21

1.3. Three input signals with period of 12.8,2.2, and 1.2 minutes 22

1.4. Change of relative amplitude of subharmonics of input signals

with fundamental frequency 23

1.5. Changes of original and modified phases with fundamental frequency 24

1.6. Output signals for bauxite system 25

1.7 Output signals for goethite system 26

1.8 Output signals for hematite system 27

1.9 Output signals for kaolinite system 28

1.10 Change of relative amplitude of subharmonics of output signals

with fundamental frequency for bauxite system.... . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . ..... 29


with fundamental frequency for goethite system '" 30


with fundamental frequency for hematite system. .. .. . . .. . . . . . . . . . . . . . . . . . . . . ... . ..... 31


with fundamental frequency for kaolinite system... ... . 32

1.14 Linear relationship between phase and frequency '" 33

1.15 Linear relationship between logarithmic amplitude reduction

and squared frequency '" .,34

XVll

Figure Page

2.1 Fit of transfer function in example 1 51

2.2 Fit of logarithmic transfer function in example 1 52

2.3 Fit of transfer function in example 2 .,53

2.4 Fit of logarithmic transfer function in example 2 54

3.1. Diffuse Layer Model 92

3.2. Triple -Layer ModeL 93

3.3. Linear relationship between nitrate concentration and UV absorbance 94

3.4. Concentration of input nitrate varying with time .,95

3.5. Spectral components of input signals 96

3.6. Normalized concentration of nitrate varying with time in a bauxite

system at pH 4 97

3.7. Normalized concentration of nitrate varying with time in a bauxite

system at pH 10 98

3.8. Normalized concentration of nitrate varying with time in a goethite

system at pH 4 99

3.9. Normalized concentration of nitrate varying with time in a goethite

system at pH 10 100

3.10. Normalized concentration of nitrate varying with time in a hematite

system at pH 4 101

3.11. Normalized concentration of nitrate varying with time in a hematite

system at pH 10 102

XVlll

Figure

3.12. Normalized concentration of nitrate varying with time in

a kaolinite system at pH 4 103

3.13. Normalized concentration of nitrate varying with time in a kaolinite

system at pH 10 104

3.14. Relative amplitudes of subharmonics of output nitrate of a bauxite

system at pH's 4 and 10 ,. '" '" '" 105

3.15. Relative amplitudes of subharmonics of output nitrate of a goethite

system at pH's 4 and 10 '" ., '" '" , 106

3.16. Relative amplitudes of subharmonics of output nitrate ofa hematite

system at pH's 4 and 10 107

3.17. Relative amplitudes of subharmonics of output nitrate of a kaolinite

system at pH's 4 and 10 108

3.18. Frequency response of a bauxite system at pH 10 and fitting with

models derived from Two-Plane Model '" 109

3.19. Frequency response of a goethite system at pH 10 and fitting with

models derived from Two-Plane Model 110

3.20. Frequency response of a hematite system at pH 10 and fitting with

models derived from Two-Plane ModeL '" '" 111

3.21. Frequency response of a kaolinite system at pH 10 and fitting with

models derived from Two-Plane ModeL 112

3.22. Frequency response of a bauxite system at pH 4 and fitting with

models derived from Three-Plane Model. 113

XIX

Figure Page

3.23. Frequency response ofa goethite system at pH 4 and fitting with

models derived from Three -Plane Model 114

3.24. Frequency response ofa hematite system at pH 4 and fitting with

models derived from Three -Plane Model 115

3.25. Frequency response ofa kaolinite system at pH 4 and fitting with

models derived from Three -Plane Model. " .116

4.1. Schematic design of a pH detector 132

4.2. Typical dynamic concentrations ont" and N03- in effluent solution


4.3. Typical dynamic concentrations ofH+ and N03• in effluent solution


4.4. Typical dynamic concentrations ofW and N03- in effluent solution




4.6. Typical dynamic concentrations ofH+ and N03• in effluent solution


4.7. Typical dynamic concentrations ofH+ and N03- in effluent solution




xx

Figure Page

4.9. Typical dynamic concentrations ofIr and N03- in eflluent solution


4.10. Spectral components of output H+ ofbauxite system at pH 4 141

4.11. Spectral components of output H+ ofgoethite system at pH 4 '" 142

4.12. Spectral components of output H+ of hematite system at pH 4 143

4.13. Spectral components of output H+ of kaolinite system at pH 4 144

4.14. Spectral components of output H+ ofbauxite system at pH 10 145

4.15. Spectral components of output H+ ofgoethite system at pH 10 146

4.16. Spectral components of output H+ of hematite system at pH 10 147

4.17. Spectral components of output H+ of kaolinite system at pH 10 148

4.18. Linear relationship between the amplitudes of output N03- and

output It of bauxite system at pH 4 149

4.19. Linear relationship between amplitudes of output N03- and

output It ofgoethite system at pH 4 149


output It of hematite system at pH 4 150


output H+ of kaolinite system at pH 4 150


output H+ ofbauxite system at pH 10 '" 151


output It of goethite system at pH 10 151

XXi

Figure Page


output W of hematite system at pH 10 152



4.26. Linear relationship between phases of output N03- and

output W ofbauxite system at pH 4 153


output W ofgoethite system at pH 4 153






output W ofbauxite system at pH 10 155


output H+ ofgoethite system at pH 10 155




output W of kaolinite system at pH 10 156

5.1. Typical input and output signals ofbauxite system at pH 4 and 10 181

5.2. Typical input and output signals ofgoethite system at pH 4 and 10 182

XXll

Figure Page

5.3. Typical input and output signals of hematite system at pH 4 and 10 183

5.4. Typical input and output signals of kaolinite system at pH 4 and 10 184

5.5. Spectral components of input signals , 185

5.6. Spectral components for bauxite system at pH 4 and 10 186

5.7. Spectral components for goethite system at pH 4 and 10 187

5.8. Spectral components for hematite system at pH 4 and 10 188

5.9. Spectral components for kaolinite system at pH 4 and 10 189

5.10. Frequency response ofbauxite system at pH 10 and curve fitting

of transfer function derived from T Triple Layer Model. 190

5.11. Frequency response ofgoethite system at pH 10 and curve fitting

of transfer function derived from T Triple Layer Model 191

5.12. Frequency response ofhematite system at pH 10 and curve fitting

of transfer function derived from Two Plane Layer Model 192

5.13. Frequency response of kaolinite system at pH 10 and curve fitting

of transfer function derived from Three Layer Model. 193

5.14. Frequency response ofbauxite system at pH 4 and curve fitting

of transfer function derived from Triple Layer ModeL 194

5.15. Frequency response ofgoethite system at pH 4 and curve fitting


5.16. Frequency response of hematite system at pH 4 and curve fitting


XXlll

Figure Page

5.17. Frequency response of kaolinite system at pH 4 and curve fitting

of transfer function derived from Triple Layer Model. 197

6.1. Dynamic concentrations ofW and P in eflluent ofbauxite system

at pH 4 '" '" 215

6.2. Dynamic concentrations ofOIr and Pin eflluent ofbauxite system

at pH 10 '" 216

6.3. Dynamic concentrations ofW and Pin eflluent ofgoethite system

at pH 4 '" '" , 217

6.4. Dynamic concentrations ofOIr and Pin eflluent ofgoethite system

at pH 10 '" , 218

6.5. Dynamic concentrations ofW and Pin eflluent of hematite system

at pH4 219

6.6. Dynamic concentrations ofOIr and Pin eflluent of hematite system

at pH 10 '" '" '" 220

6.7. Dynamic concentrations ofW in eflluent of kaolinite system

at pH 4 '" '" , '" 221

6.8. Dynamic concentrations ofOIr in eflluent of kaolinite system

at pH 10 222

6.9. Spectral components of output W ofbauxite system at pH 4 '" .223

6.10. Spectral components of output OIr ofbauxite system at pH 10 224

6.11. Spectral components of output H+ ofgoethite system at pH 4 225

6.12. Spectral components of output OIr ofgoethite system at pH 10 226

xxiv

Figure Page

6.13. Spectral components of output W of hematite system at pH 4 227

6.14. Spectral components of output OK of hematite system at pH 10 228

6.15. Spectral components of output W of kaolinite system at pH 4 229

6.16. Spectral components of output OK of kaolinite system at pH 10 230

6.17. Linear relationship between adjusted amplitude ofW and amplitude

ofP ofbauxite system at pH 4 '" .231

6.18. Linear relationship between adjusted amplitude of OK and amplitude

ofP ofbauxite system at pH 10 231


ofP ofgoethite system at pH 4 232


ofP ofgoethite system at pH 10 '" '" 232


ofP of hematite system at pH 4 233

6.22. Linear relationship between adjusted amplitude of OH- and amplitude

ofP ofhematite system at pH 10 233


ofP of kaolinite system at pH 4 234


ofP of kaolinite system at pH 10 234

6.25. Linear relationship of phases between Wand P ofbauxite system

at pH 4 : 235

xxv

Figure Page

6.26. Linear relationship of phases between aIr and P ofbauxite system

at pH 10 235

6.27. Linear relationship of phases between Wand P ofgoethite system

at pH 4 '" '" '" 236

6.28. Linear relationship ofphases between aIr and P ofgoethite system

at pH 10 '" '" 236

6.29. Linear relationship of phases between H+ and P of hematite system

at pH 4 '" '" 237

6.30. Linear relationship of phases between aIr and P of hematite system

at pH 10 '" 237

6.31. Linear relationship of phases between H+ and P ofkaolinite system

at pH 4 '" '" '" 238

6.32. Linear relationship of phases between OH- and P ofkaolinite system

at pH 10 '" 238

7.1. X-ray diffraction pattern ofgibbsite '" '" 250

7.2. X-ray diffraction pattern ofgoethite 251

7.3. X-ray diffraction pattern ofhematite 252

7.4. X-ray diffraction pattern ofkaolinite 253

7.5. P sorption isotherm when minerals were previously adjusted to pH 4.0 254

7.6. pH changes after 36 hour P sorption at low pH condition 255

7.7. P sorption isotherm when minerals were previously adjusted to pH 9.6 256

7.8. pH changes after 12 hour P sorption at high pH condition 257

XXVi

Chapter 1

Parameter Estimation for the Convection-Dispersion Model for Non

reactive Transport Process via Transfer Function Approach

Abstract

Identification of mechanisms of solute transport in porous media, such as would

be implied by the Convection-Dispersion Equation (CDE), the mobile-immobile model,

and the Dual-porosity Model, are important for environmental chemistry and soil physics.

However, the widely used method, i.e. breakthrough curve with impulse input, may not

supply sufficient and accurate information for the identification procedure. A system

identification approach in frequency domain was proposed to study the transport process

for porous media. The sinusoidal input signals, i. e. the dynamic acetone concentration of

the influent to mineral columns, were generated by the design of an HPLC gradient

controller and attachment of an additional3-meter tube at the outlet of the pump. The

dominant spectral components of input signals were the designed fundamental

frequencies while all the subharmonics were negligible. The input signals were

repeatable among experiments. The output signals, i. e. the dynamic sequence of acetone

concentration in eflluent of mineral columns when excited with input signals, were also

sinusoids with dominant fundamental frequencies and the subharmonics were negligible,

and thus the transport processes ofacetone/water flowing at rate 4.00 ml min-t,

equivalent to 0.33 to 0.40 pore volumes per minute, through mineral columns were

viewed as linear system. Two simplified equations were derived from CDE to describe

the relationship between the amplitude reduction/phase shift and frequency. The

1

estimated water velocities ranged between 8.50 and 10.39 cm min-I, and the estimated

dispersion coefficients ranged between 0.39 and 1.01 cm2 min-I.

1. Introduction

Solute transport in porous media is important for studying the behavior and fate of

various chemicals in the subsurface environment. For homogeneous medias, the

convection-dispersion equation (CDE) for water flow can be reduced from the equation

ofBigger and Nielsen (1967) into

(1)

where c is the resident concentration of solute, t and x are respectively the time and space

coordinates, D is the dispersion coefficient, V is the solute velocity, and R, the retardation

factor, is 1 for nonsorbing solute. Parker and van Genuchten (1984) developed the

algorithm to determine the dispersion coefficient D from breakthrough data. For

heterogeneous media, different governing equations were proposed. Coats and Smith

(1964) and van Genuchten and Wierenga (1976) partitioned soil water into mobile and

immobile phases, and the mobile-immobile model was

(2)

where the subscripts m and im refer to the immobile and mobile phases, respectively, and

a is the mass transfer coefficient. Dykhuizen (1987) separated soil into two distinct

2

homogenous pore systems, and the dual-porosity model for structured soils was proposed

as

(3)

where f.J is a first order decay coefficient, Fs is a solute mass transfer term to which both

molecular diffusion and convective transport contribute, and WM is the ratio of the volume

of the interaggregate pores to that of total volume of all pores.

For the mobile-immobile model (2), the concentration of em cannot be measured.

For the dual-porosity model (3), the concentration Cm and CM cannot be separated in the

eftluent solution of the breakthrough curve of impulse input experiments. Thus, it is

difficult to select one from the three candidate models according to the structure of

porous media. The possible way of selection is based on a limited number of

measurements of inputs and usually noisy outputs, which is just the aim of system

identification theory. The breakthrough method, in which an impulse signal is used as

the input, may not be a good design because the distinguishability between models will

be reduced due to noise influences. An optimization of input signals will gather

measurements to minimize the uncertainty of the final result. Because the complexity of

the analytical solution of the models in the time domain, system identification in the

frequency domain will supply simple analytical solution and the least square approach

can be used to estimate model parameters. An additional advantage of exciting the

transport process by a sinusoidal input signal is that the linearity of the process can be

determined easily because the harmonic distortion is directly visible (Haber and

3

Keviczky, 1999~ Schoukens and Pintelon, 1991), and the linear models (1)-(3) are

excluded if the process is tested as nonlinear.

This chapter reports an attempt to use the system identification approach in the

frequency domain to design and set up the experiment, collect data and estimate both the

solvent velocity and dispersion coefficient. Although the sole objective of the study is

focused on the identification of the CDE ofwater flowing through columns packed with

homogenous fine minerals, the method can be extended to more complicated transport

mechanisms for heterogeneous media.

2. Materials and Methods

Chemicals: Acetone was certified A.C.S. reagent, and sodium nitrate (NaN03)

was analytical reagent. The water was of nanopure quality and degassed by boiling.

Minerals: gibbsite, goethite, hematite, and kaolinite are from Ward Science

Company. They were ground and wet sieved with deionized water, and the fraction of

325-500 mesh was collected and freeze-dried. X-ray diffraction analysis showed that

goethite, kaolinite, and hematite contained quartz. Each mineral was packed into an

empty stainless column with weight Woand of25 cm in length and 1 cm in diameter.

The weights ofcolumns packed with minerals were measured and recorded as Wj. The

weights ofgibbsite, goethite, hematite, and kaolinite packed, which were equal to Wj-Wo,

were 26.77,41.94,31.67,30.74 grams, respectively. Water was pumped into the dry

columns slowly with flow rate 0.5 ml min-I. After the columns were filled with water,

they were washed sequentially with 1 mMNaOH solution for 8 hours, 1 mMHCI solution

4

for 8 hours, and water for 1 hour, with flow rate 4 ml min-I. The weights of columns with

mineral and water were measured and recorded as W2. The mean water velocities passing

through columns with flow rate 4 ml min- l was calculated as 4 *25 in cm min-t, and~-W;

they were 8.68, 8.97, 8.22, 10.06 cm min- l for gibbsite, goethite, hematite, and kaolinite,

respectively.

Setup ofExperiment: The Waters Prep LC, a High Performance Liquid

Chromatography (HPLC), was used as the solvent delivery system. The schematic of the

setup of experiment is shown in Figure 1.1. The effective volume ofthe column was

19.635 cm3 (length: 25 cm, diameter: 1 cm). The solution A was 2.5 ml L-1

acetone/water, and solution B was pure water, and they were under a helium condition

with flow rate 13 ml min- l while the system was on. The ends of the column were

connected with the UVNis detector and the outlet of the pump ofHPLC. The detector

was used to measure acetone concentration at a wavelength of264 nm. The timer on the

detector was used for indexing data storage by computer. The detector was connected to

the computer with software Millennium® to store the absorbance data and the time sent

from the detector. The data collection interval was one recording per second.

Design ofinput signals: Seven input sinusoidal signals designed with periods

12.8,4.8, 3.0, 2.2, 1.8, 1,4, 1,2 minutes, numbered from 1 to 7, were carried out by

creating seven gradient tables and seven event tables on the gradient controller ofHPLC.

Acetone was used as a tracer because its polar molecular will not be adsorbed and

desorbed as ions by the charged surfaces of minerals at different pH conditions. A

gradient table included gradient segment time, flow rate, solvent composition, and rate

of change curve number. An example of a gradient table for an input signal with period

5

T, COS(271 t) +1, is shown in Table 1. If a gradient table was designed for signal # 1 withT

T = 1.2 minutes, the gradient table was stored by the number"1"; if a gradient table was

designed for signal #2 with T= 1.4 minutes, the gradient table was stored by the number

"2", and so on for other input signals. An event table included time of event, even type,

and event action/setting. The time of event was set as the period of a signal, event type

was always set as "8" which means "start running table #", and the event action/setting

was set as the identifying number of the gradient table for the input signal with the same

period as the number in the time of event. For example, to create the event table for

signal #1, the time of event is set as "12.8", event action/setting was set as "I", and event

table was stored as identifying number "I"; to create the event table for signal #2, the

time of event was set as "4.8", event action/setting was set as "2", and event table was

stored as identifying number "2", and so on for the other signals. The event tables had to

match the corresponding gradient tables to ensure one gradient table repeated again and

again when HPLC was running the gradient although a gradient table was designed for

only one cycle. The repeat running of a gradient and the corresponding event table could

be stopped and switched manually to another one without stopping the flowing condition.

The design of a gradient table was discrete in time, and the signal generated at the inlet of

the column was dominant of the designed single fundamental frequency to excite the

transport process inside the mineral columns.

Input signal experiments: Owing to the limitation of the experimental device, the

input and output signals were not monitored simultaneously. The experiments for

studying input signals were carried out separately from those for studying output signals

via disconnecting the column from the setup. The procedure was as follows.

6

Step 1. Turn on HPC, UV/Vis detector, and start computer and software

Millennium which communicated with detector and had functions of data storage and

retrieval. After the UVNis detector was ready, connect the software with the UV/Vis

detector.

Step 2. Run isocratically, which means no time actuated changes in flow or

solvent composition, or other time-dependent conditions to occur, with composition 0%

A, 100%B, O%C, O%D and flow rate 3.00 ml min-1 for 30 minutes or longer, and then

start the Millennium collect data. Continue running for 30 minutes. Check the online

curve in the window ofMillennium, and run longer time if the curve was not flat. Record

the time at UVNis detector noted as fBbefore continuing to step 3.

Step 3. Run isocratically with composition 100% A, O%B, O%C, O%D for 30

minutes. Check the online curve in the window ofMillennium, and run longer time if the

curve was not flat. Record the time at UVNis detector noted as fA before went to step 4.

Step 4. Run gradient of event table #1, record the start time f11 from the timer of

UVNis detector. Check the online curve in the window ofMillennium. The curve

usually became stationary after around 15 minutes. The minimum time to run the signals,

shown in Table 2, was empirically 15 minutes to make the system stable. After the

system was stable, ran 3-7 times of the period T of a signal. Record the time to get a

stationary signal, noted as f12 and the ending time f13 to switch to run another signal.

Step 5. Switch manually to run event table #2 without stopping flowing condition.

Repeat step 4. Record the start time f21, time f22tO get a stationary signal, and ending

time f23.

7

Step 6. Repeat step 5 to run the other signals. After all the signals have been run,

exit the software Millennium first, and then turn down the HPLC and detector.

Output signal experiments: Connect the column between 3-meter tubing and the

inlet ofUVNis detector. The procedure is the same as input signal experiments.

Data retrieval: The data collection by Millennium was continuous with one-

second interval before Millennium was stopped manually or a maximum of 650 minutes

was exceeded. The data ofboth input and output signals was stored in a row vector, and

the index of elements was time in minute. For example, the acetone concentration at x

minutes was the element with index 60x. Useful information was retrieved from the

dataset based on the starting times tA, tB, tIl, t2l, ... , t71, times t12, t22, ... , t27 to reach

stationary, and ending times t13, t23, ... , t73. The elements for baseline of solution Bare

from index 60(tB - 5) to 60tB' i.e., those elements in the last five minutes before ending

was retrieved. The elements were averaged and the mean value was noted as XB.

Similarly to solution B, the elements for baseline of solution A were from index

60(tA - 5) to 60tA' The elements were averaged and the mean value was noted as XA.

The elements for signal #1 in the input signal experiments were from index

tll +r~l *12.8 to l~J*12.8, where rxl and LxJare functions which round the112.8 12.8

elements ofX to the nearest integers towards infinity and minus infinity, respectively.

The average in terms of one cycle was noted as x]. Similarly to signal #1, the elements

for signal #k with period Tk in either input signal experiments or output signal

experiments, are from index t" +I;: 1*T, to l;:J*T,. The average in terms ofone

cycle is noted as Xk.

8

3. Mathematical model in frequency domain

3.1 Transfer function of the transport process

For the nonsorbing solute acetone transporting in the columns of minerals,

equation (1) was used to describe the process with R equal to one. The inlet boundary

condition is

c(O, t) =u(t) ,

and exit boundary condition is

c(oo,t) =0.

For linear partial differential equation (1), the analytical solution is in the product

of functions ofx and t (Bleecker and Csordas, 1992). Taking Fourier Transform with

respect to time to both sides of equation (1) gives

where C(x, JOJ) is the Fourier transform ofc(x, t),j is imaginary number, OJ is frequency

in radians min-I. The Fourier transform of the inlet boundary conditions is

C(O,jOJ) =U(jOJ)

and that of exit boundary condition is

C(oo,jOJ) =°The solution of equation (4) is in the form of

where PI and P2 are functions ofjOJ, rl and r2 are the roots of equation

Dr2

- Vr - JOJ =°

9

(4)

(5)

I.e.,

v( ~)1j =2D 1- V1+JJi2 '

V( ~)r2 =2D 1+V1+ JJi2

Since the outlet boundary is a bounded number,p2 is zero otherwise C(x,jm) goes to

infinity as x goes to infinity. Applying inlet boundary condition to equation (5), we have

PI =U(jOJ)

and hence

C(X,jOJ) =U(jOJ)exp(1jx).

Define

Y(jOJ) =C(L,jOJ)

where L is the length of the column. Equation (5) can be rewritten as

Y(jOJ) =U(jOJ)exp(1jL)

or

G(' )= Y(jOJ) =ex (VL(l_~l+ .4DOJ)]JW U(jOJ) p 2D J V 2

where G(jOJ) is the transfer function of the linear time-invariant system defined by

equation (1).

3.2 Formula for estimating solvent velocity and dispersion coefficient

When 4D2OJ «1, Taylor series expansion

V

10

(6)

Then, equation (6) can be approximated as

Furthermore, we have

In(1 G(jaJ) I) =- L~ aJ 2

V

and

LG(jaJ) =- L aJ.V

Equations (8) and (9) show the relationship of amplitude reduction and phase shift

between the solute concentrations ofinlet and exit solutions, and v can be estimated by

equation (9) and then D can be estimated by equation (8) if frequency aJ of u(t) varies

and the frequency response GOaJ) are determined.

4. Results

4.1 Linear relationship between acetone concentration and absorbance

(7)

(8)

(9)

The relationship between acetone concentration and absorbance is linear as shown

in Figure 1.2.

4.2 Input signals in time domain

11

In order to make comparison between experiments, the relative concentrations

2(xk - CB) were used to describe the input signals, where Xk is the absorbance of inputCA -CB

signal #k and CA and CB are averaged value of absorbance ofbaselines of solution A and

B in the input signal experiments. The time sequences of input signals with period 12.8,

2.2, and 1.2 minutes are shown in Figure 1.3.

4.3 Spectral component of input signals

For an input signal with frequencyf k = 1/Tk, k =1,2, ... ,7, its spectral components

with frequencies f k, 2/k, 3/k, ... were calculated with the fast Fourier Transform (FFT)

algorithm in MATLAB. Their amplitudes were noted as Aft' A 2ft , A 3ft , .... The

relative amplitude of the 1st subharmonic component with frequency lfk to its fundamental

frequency fic is

where I is a positive integer. The curves rift vs. fk' I = 1,2, ... ,6 were shown in Figure

(10)

1.4. From Figure 1.4, the change of subharmonics of input signals with their fundamental

frequency showed that all the relative amplitude of subharmonics were less than 1.4% of

those of the fundamental frequency and hence negligible. Thus, the input signals were

sinusoids with dominant fundamental frequency.

4.4 Variance of input signals among repeated experiments

12

The standard deviations and CV (%) of the amplitudes of input signals among

three repeated experiments were shown in Table 3. The low standard deviations and CVs

showed that the amplitudes of the generated input signals were repeatable.

The phases CPk of fundamental frequency fk of input signals, k = 1,2, ... 7, obtained

from FFT, were within -1t and 1t, and discontinuous, as shown in Figure 1.5. The phases

of input signals within -1t and 1t were modified to meet the requirements of actual phases

of a real system: continuous with frequency fk , and passing through origin. The

modification formula is

(11)

where n is a positive integer. The modified phases of input signals were also shown in

Figure 1.5. In this study, the "phase" refers to the modified values instead of those within

-1t and 1t. The standard deviations and CV (%) of phases of the input signals among three

repeated experiments were shown in Table 3. The low standard deviations and CVs

showed that the phases of input signals generated were repeatable.

Since the input signals were highly pure sinusoids and repeatable, the averages of

the amplitudes and phases among three repeats were used for data analysis.

4.5 Output signals and their spectral analysis

Similar to the input signals, the relative concentration 2(xk - CB) was used toCA -CB

describe the output signals, where Xk is the absorbance of output signals excited by input

signal #k, CA and CB were respectively the averaged values of absorbance ofbaselines of

solution A and B in the input signal experiments. The time sequences of output signals

13

with period 12.8,2.2, and 1.2 minutes are shown in Figure 1.6 to 1.9 for column packed

with gibbsite, goethite, hematite, and kaolinite. The spectral analysis is similar to that for

input signal in Section 4.2, and the changes of relative subharmonic amplitudes with

fundamental frequencies are shown in Figure 1.10 to 1.13. The figures showed that all

the relative amplitudes of subharmonics were less than 1.2% of those of the fundamental

frequencies ofoutput signals and hence negligible. Thus, the transport process is linear.

The closeness of relative amplitudes of output signals to those of input signals showed

that effect of environmental noise on the transport process was negligible and the main

noise was from measurement by UVNis detector.

The phases of output signals calculated by FFT were within -1t and 1t, and not

continuous with frequencies and passing through the origin. They are modified as those

of the input signals to meet the requirements of minimizing phases of a real system.

4.6 Estimate of solvent velocity and dispersion coefficients

The phase shift of output signals to input ones, noted as LG, is defined as the

modified phase of output signals minus those of the input ones. The model (9) is used to

describe the relationship between phase shift and frequency

where V is the water velocity to be estimated and OJ =2;if. The average of estimated

solvent velocity V and their standard deviations among repeated experiments for the

transport processes in different mineral columns are given in Table 4. The estimated

(12)

water velocities were compared with those calculated in section 2, and the result showed

14

that the estimated values were higher than calculated, and the reason may be due to the

presence of immobile phase of pore water.

A new variable, Yk, to represent amplitude reduction is defined as

(13)

where Aft ,input is the amplitude of input signal with fundamental frequencyJk determined in

section 4.2, and Aft is the amplitude of output signal excited by the input signal with

frequencY!k. The model (8) was used to describe the relationship between amplitude

reduction and frequency

where D is the dispersion coefficient to be estimated. The average of estimated

dispersion coefficients and their standard deviations among repeats for the transport

processes in different mineral columns are given in Table 4. The estimated dispersion

coefficients decrease with the increase of estimated water velocity.

5. Conclusion

(14)

The input signals generated by the setup and design were repeatable and sinusoids

with a pure designed frequency and negligible subharmonics. The transport of

acetone/water solution in mineral columns was linear since the subharmonics of the

output signals were negligible. Simplified models were derived from the Convection-

Dispersion Equation in frequency domain to describe the relationship between amplitude

reduction/phase shift and frequency. The estimated water velocities ranged between 8.50

15

and 10.39 cm min-I, and the estimated dispersion coefficients ranged between 0.39 and

1.01 cm2 min-I.

Reference

Biggar, J.W. and D. R. Nielsen, 1967. Miscible displacement and leaching phenomena.

Agronomy 11: 254-274.

Bleecker, D., and G. Csordas. 1992. Basic partial differential equations. Van Nostrand

Reinhold, New York.

Coats, KH., and B. D. Smith. 1964. Dead-end pore volume and dispersion in porous

media. Soc. Petrol. Engrs. J. 4:73-84.

Gerke, H.H. and M. Th. van Genuchten. 1993. A dual-porosity model for simulating the

preferential movement of water and solutes in structured porous media. Water

Resour. Res. 29: 305-319.

Schoukens, J. and R. Pintelon. 1991. Identification of linear systems: A practical

guideline to accurate modeling. Pergamon press, Oxford, U.K

van Genuchten, M. Th. and P. 1. Wierenga. 1976. Mass transfer studies in sorbing porous. I.

Analytical solutions. Soil Sci. Soc. Am. 1. 40: 473-480.

16

Table 1.1. Gradient table of input signal with period (1).

a 6 means %A, %B, %C and %D change lmear from mitIal condItIon to final condItIon.

Time Flow rate %A %B %C %D Curve profile

(min) (ml min-I) number of the

gradient segment

Initial 4.00 100 0 0 0

O.OST 4.00 98 2 0 0 6a

O.lOT 4.00 90 10 0 0 6

O.lST 4.00 79 21 0 0 6

0.3ST 4.00 21 79 0 0 6

0.40T 4.00 10 90 0 0 6

O.4ST 4.00 2 98 0 0 6

O.SOT 4.00 0 100 0 0 6

O.SST 4.00 2 98 0 0 6

0.60T 4.00 10 90 0 0 6

0.6ST 4.00 21 79 0 0 6

0.8ST 4.00 79 21 0 0 6

0.90T 4.00 90 10 0 0 6

0.9ST 4.00 98 2 0 0 6

LOOT 4.00 100 0 0 0 6

. .. . . . .

17

Table 1.2. The minimum time to run signals in the input and output signal experiments.

Period of A Signal Minimum time to run

(min) (min)

12.8 53

7.2 37

4.8 49

3.6 40

3 36

2.4 32

2.2 30

2 29

1.8 28

1.6 26

1.4 25

1.2 23

18

Table 1.3. The standard deviation and CV ofthe amplitudes and phases of input signals

among repeated experiments.

Period Amplitude Phase

(min) (radians)

Standard CV Standard CV

Deviation (%) Deviation

12.8 0.0040 0.4127 0.0029 0.3670

4.8 0.0036 0.4033 0.0015 0.0692

3.0 0.0026 0.3355 0.0143 0.4313

2.2 0.0029 0.4378 0.0182 0.4042

1.8 0.0044 0.7644 0.0410 0.7533

1.4 0.0052 1.1926 0.0067 0.0975

1.2 0.0049 1.4038 0.0348 0.4365

19

Table 1.4. The averages and standard deviations of estimates ofwater velocity and

dispersion coefficient among repeated experiments.

Mineral

Gibbsite

Goethite

Hematite

Kaolinite

Velocity Dispersion Coefficient

(cm min-I) (cm2 min-I)

Mean Std Mean Std

9.4232 0.03127 0.9472 0.02188

9.7727 0.005798 0.4840 0.008093

8.5034 0.01633 1.0100 0.09829

10.3902 0.007801 0.3879 0.003356

20

Reservoir:~ ISolutions Pump ~ Mineral

A and B . ColumnIIII

GradientController

Figure 1.1. Schematic representation of experimental setup. Solid arrow stands for mass

transfer and dash ones for data transfer.

21

0.5

0.4

Q>g~ 0.3

B«

0.2

0.1

0.5

y =0.20045 x + 0.0016492

1 1.5 2

Acetone concentration (ml L-1)

2.5 3

Figure 1.2. The linear relationship between the acetone concentration and its absorbance

at 264 run.

22

21=12.8 min

Q)CJ 1.5c:~...0

~IIIQ)

>+'III

0.5~

00 5 10 15 20 25

21=2.2 min

Q)CJ 1.5c:III-e0

~IIIQ)

~III

0.5~

00 2 4 6 8

21=1.2 min

Q)CJ 1.5c:III.0...0

~IIIQ)

~III

0.5~

00 2 3 4 5

time (min)

Figure 1.3. Three input signals with period of 12.8,2.2, and 1.2 minutes.

23

1.2

1.4 i --,--------r---,----,-------,-------,---,----o;:====:::;-]---.- 2f--e-3f--- 4f-+- Sf--+-- 6f

0.90.80.70.60.30.20.1O'--------'--------'-----__-'-----=--_...L-__...L-__---'----__---'----__---'----__--I

o

0.2

lrl'§E~ 0.8III

'0Gl

~ 0.61i.

~.~,.! 0.4

Figure 1.4. The change of relative amplitude of subharmonics of input signals with

fundamental frequency.

24

4

0

2 0

0

0

0

-20

QlIII

!Do

-4

-6

-8

0.2 0.3 0.4 0.5

Frequency f (min-1)

0.6 0.7 0.8 0.9

Figure 1.5. The changes of original and modified phases with fundamental frequency.

Circle stands for the original phases and cross stands for the modified phases. The solid

line stands the phase characteristics of continuity with frequency and passing through

origin.

25

2T=12.8 min

Q.)0 1.5c

~0

23 1IIIQ.)

.~

iil0.5Q)

0::

00 5 10 15 20 25

2T=2.2 min

Q.)0 1.5cIII-e0

23 1IIIQ.)

>~

0.5Q)0::

00 2 4 6 8

2T=1.2 min

Q.)0 1.5c

~0

23 1IIIQ.)

>~

0.5Q)0::

00 1 2 3 4 5

time (min)

Figure 1.6. Output signals with periods of 12.8,2.2, 1.2 minutes for gibbsite system.

26

2T=12.8 min

Q)(.) 1.5c:.e0

.B 1IIIQ)

.~

tii0.5a;

0::

00 5 10 15 20 25

2T=2.2 min

Q)(.) 1.5c:III-e0

.B 1IIIQ)

.~

tii0.5a;

0::

00 2 4 6 8

2T=1.2 min

Q)(.) 1.5c:

.e0

.B 1IIIQ)

>~

0.5a;0::

00 1 2 3 4 5

time (min)

Figure 1.7. Output signals with periods of 12.8,2.2, 1.2 minutes for goethite system.

27

2

CD0 1.5~

III-e0Ul.c 1IIICD>~

0.5Qja:::

00 5 10 15 20 25

2T=2.2 min

CD0 1.5~III-e0Ul.c 1III

CD.~-III 0.5Qja:::

00 2 4 6 8

2T=1.2min

CD0 1.5~III.c...0Ul.cIIICD>:;:;III

0.5Qja:::

00 2 3 4 5

time (min)

Figure 1.8. Output signals with periods of 12.8,2.2, 1.2 minutes for hematite system.

28

2T=12.8 min

Q)0 1.5c

~0

1i 1CIlQ)

.~

1ii0.5a;

a::

00 5 10 15 20 25

2T=2.2 min

Q)0 1.5c.fg...0

1iCIlQ)

>~

0.5a;a::

00 2 4 6 8

2T=1.2 min

Q)0 1.5c

~0

1i 1CIlQ)

>~

0.5a;a::

00 1 2 3 4 5

time (min)

Figure 1.9. Output signals with periods of 12.8,2.2, 1.2 minutes for kaolinite system.

29

1.4r-------,-------,----,----,----,----,...-------r---r;:=====:-

1.2

-.- 2f-e-3f-+t-- 4f-f- Sf-t- Sf

0.90.80.70.60.4 0.5

Frequency (min-1)

0.3OL-__.L-__.L-__-'------"-------_--'-----__--'-----~I"=_--L.--_-L-__---"---____l

o 0.1 0.2

0.2

~ 1..u'gE~ 0.8..'0

~.... 0.6Q.EIll'

.~,.~ 0.4

Figure 1.10. The change of relative amplitude of subharmonics of output signals with

fundamental frequency for gibbsite system.

30

1.2

1.4 i ---,---------,-----,----,---,-------.------,-------.;::===::;"l-.- 2f-e-3f-- 4f-+- Sf-t- Sf

~Ulu'isE..l! 0.8.gUl

'0

~"" 0.61S.E01

.~I 0.4

0.2

01-~__L__ ______=t~=:'I:::::~==~~=±::=:E=====2~==~~o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Frequency (min·1)


fundamental frequency for goethite system.

31

1.2

1.4 r -----,----,----,------,---,--------,,----r------O;::====:::;l

-- 2f-e-3f---- 4f-+- Sf-+- Sf

~rl.~

e~ 0.8III

'0

~.. 0.6Ii.EIII

.~1ii~ 0.4

0.2

0.1 0.2 0.3 0.4 0.5

Frequency (min")

0.6 0.7 0.8 0.9


fundamental frequency for hematite system.

32

1.2

1.4 r ---.---.------,---,----,------,---,--------r;::====::;l-.- 2f--e-3f~ 4f-+- Sf-+- Sf

~ 1

rl'co~i! 0.8"9III

'0

~~ 0.615.EasGI.~1;j~ 0.4

0.2

0.1 0.2 0.3 0.4 0.5

Frequency (min-1)

0.6 0.7 0.8 0.9


fundamental frequency for kaolinite system.

33

-10

-12

-14 o bauxite+- goethite+ hematite• kaolinite

0.90.80.70.60.30.20.1-16 L.':::===::::::r==='--L__---L-__--l.-__l--_-----l__--L__-L~_ ___l

o

Figure 1.14. Linear relationship between phase and frequency by equation (12).

34

O~~;::::~--""---"'---'-----'----'---I

-0.2

-0.4

>- -0.6

-0.8

-1,.- --,o bauxite+- goethite+ hematite• kaolinite

o 0.1 0.2 0.3 0.4

Square of frequency P(min-2)

0.5 0.6 0.7

Figure 1.15. The linear relationship between logarithmic amplitude reduction (y) and squared

frequency if) by equation (14).

35

Chapter 2

Parameter Estimation of a Transfer Function

Abstract

Parameter estimation of a transfer function is essentially a problem of complex

curve fitting. The existing methods are focused on transfer functions in typical form, i.e.

ratio of two complex polynomials. In this chapter, the modified Gauss-Newton method

was implemented to estimate parameters in non-typical transfer functions. The real and

imaginary parts of a transfer function can be regarded as multivariate nonlinear models

and they are transformed into univariate ones for implementation of the modified Gauss

Newton method by scaling. Two examples showed that (i) when a transfer function is

high damping, fitting of the original transfer function gives emphasis to low frequencies,

however, fitting of the logarithmic form overcame the problem; (ii) when a transfer

function is of light damping, the fitting of its original and logarithmic forms were

equivalent. Generally, therefore, fitting the logarithmic transfer function is a better

choice.

Introduction

In chapter 1, the transfer function for the Convection-Dispersion Equation was

derived and the parameters were estimated with frequency response data, i.e. the dynamic

change of the solute concentration ofthe effluent solution due to the solute concentration

36

dynamic changing as a sinusoidal function of time. When non-equilibrium adsorption

and desorption of nitrate or phosphate onto the variable charged mineral surfaces was

coupled with the transport, the simplification methods in Chapter 1 will not work, and an

algorithm for parameter estimation must be developed for the following Chapters 3 and 5.

The typical transfer function of a linear dynamic system is expressed as a ratio

of two frequency-dependent polynomials

n

'Lbr(jm)'G(jm) = r=O d-I

(jw)d + 'Lar(jm)'r=O

N(jm)=-=----.:....D(jm)

(1)

Non-typical transfer functions also exists in real system, for example, the transfer

function of heating dynamics

G( '01) = 2K} .Jlf(exp(L.Jlf) - exp(- L.Jlf»

where k, K, and L are constants (Ljung, 1999). The parameter estimation of a transfer

function from noisy measurements of frequency response data is called complex-curve

fitting (Levy, 1959) since G(jm) is a function of an imaginary variable}m. The noises

involved can be divided into two types: the first type is in the process and/or outputs,

and the second is in both inputs and outputs. When both input and output are signals

(2)

corrupted with noise, Schoukens and Pintelon (1991) proposed an algorithm to estimate

the parameter based on an error-in-variables (EV) method. This article is focused on the

first type of noise.

Various linear least squares methods for typical transfer function have been

studied by Levy(1959), Sanathanan and Koerner (1963), Lawrence and Rogers (1979),

Stahl (1984), Bayard et al. (1991), and Van den Enden et al. (1977).

37

By multiplying both sides ofEquation (1) with D(jOJ), Levy (1959) obtained

D(jOJ)Gm(jOJ) =N(jOJ) ,

where Gm(jOJ) is the measured frequency response. In this way, the model form can be

converted into parameter-in-linear equations, and thus, parameter estimation can be

carried out by least squares method. For example,

then the linear equations can be obtained as

2bo -RkaO + Ika\ =-RkOJk

b\OJk - Ikao - OJkRka\ =-IkOJk2

The disadvantage of this method is that the lower frequency terms have little influence

since in the cost function

K ~ID(')G N(' )1 2 = ~ID(j'OJ)12 Gmk

_ N(~OJk) 2= .t...J jOJk mk - jOJk .t...Jk=\ k=\ D(jOJk)

/D(jOJ )12actually becomes a weighting function increasing with OJ.

Sanathanan and Koerner (1963) modified the weighting function such that the

cost function is

(3)

(4)

(5)

(6)

K =±ID(jOJk)G~ -N~jOJk)12k=\ ID(jOJl-l

(7)

whereID(jOJ) IL-\2 is calculated with the parameter estimations in the last iteration step

L-l, and IDU'"i) I: will approach I as the parameter estimates converge to the trueID(jOJ) L-\

values. Other slightly different weighting functions were proposed by Lawrence and

38

Rogers (1979), and Stahl (1984). Whitefield (1986, 1987) compared the performance of

various methods and analyzed their asymptotic behavior.

Van den Enden et aI. (1977) derived the weighting function based on the cost

function in strict least squares sense as follows

F

K= IIGmk -Gk(jOJ)lz

k=1

F

=I {Rk - real[G(jOJk)DZ + {Ik -imag[G(jOJk)DZk=1

where Rk and Ik are the real and imaginary parts ofmeasured frequency response,

(8)

respectively. Non-linear equations will be obtained when minimizing the cost function

with respect to parameters, and they can be partitioned into nonlinear and linear terms.

By choosing initial values for the parameters, the nonlinear terms will be calculated as A

weighting function, and the linear terms will be estimated by the least squares method.

The resulting parameters are substituted back into the nonlinear terms. The iterative

process will give the converged parameter estimates. Compared with the Levy method

and the Sanathanan and Koerber method, this method has improvements in reducing

residuals (van den Enden et aI., 1977) and increasing estimation efficiency (Schoukens

and Pintelon, 1991).

Non-linear least squares method for both typical and non-typical transfer

functions was given in Schoukens and Pinte10n (1991) and Martin (1994). The cost

function (16) is rewritten as

(9)

39

where superscript H stands for the Hermittian transposition operator, and the Gauss-

Newton algorithm can be used for parameter optimization. The first order derivative of

G(jo;) with respect to its parameter vector Pis

J = aG(jo;).8P

At each iteration step the parameters are updated by formula

while its strict derivation from equation (9) has not been reported in literature. The

(10)

(11)

disadvantages of this method are (i) the total degrees of freedom is reduced to half that of

the other three methods, and (ii) the initial parameters must be set correctly in order to

ensure convergence (Martin, 1994).

This chapter reports on the implementation of the modified Gauss-Newton

method to estimate parameters of transfer functions in both typical and non-typical form.

2. Mathematical Methods

The real and imaginary parts ofa transfer function can be considered as

multivariate nonlinear models. The modified Gauss-Newton method can be used to

estimate the parameters after the multivariate nonlinear models are transformed into a

univariate model (Gallant, 1987). Two approaches were studied in terms ofthe forms of

a transfer function. The original form of a transfer function was fitted in the approach I,

while the logarithm of the original transfer function was employed in approach II.

Approach l The cost function in strict least squares sense is

40

K(P) = (y - f(ro, p)r (y - f(ro, P»y = [R1' ... , RF , 11' ... , IF J

real[G(jm) , P)]

real[G(jm F ,P)]f(co,P) =

imag[ G(jm) , P)]

imag[ G(jm F' P)]

(12)

where Rk and Ik are the measured real and imaginary parts at frequencies 0Jk, subscript k

= 1,2, ... , F with F is the number of frequencies tested. The iteration algorithm is

Step 0: Set starting values ofP and A at 100.

Step 1: Calculate the Jacobian matrix

oreal[G(jaJ), P)]

oP

J =af(ro,P) =oP

o real[G(jaJF' P)]

OPoimag[G(jaJ) ,P)]

oP

oimag[G(jaJF , P)]

oP

(13)

which is 2F by m, where F and m are the numbers frequencies and parameters,

respectively. Since it is difficult to get the explicit expressions of the first derivatives,

numerical solutions were used.

Step 2: Update parameters by equation

Step 3: Calculate K(Pk +) at iteration step k+1 as equation (12).

41

(14)

Step 4: Update Awith l if K(Pk+I) < K(Pk) , and go to step 5; otherwise update A2

with 10l and go to step 2.

Step 5: if the stop criterion K(Pk) -K(Pk;l) <10-8 is not met, go to step 1.K(Pk ) +10-

Approach ll. If a transfer function is highly damping, Approach I will

emphasize low frequencies. To deal with the problem, the magnitude/phase expression

was used for fitting instead of real/imaginary expression. A logarithmic transfer function

is defined as

H(jaJ) =log[G(jaJ)] = real([H(jaJ)] + jimag([H(jaJ)] =logIG(jaJ)1 + jLG(jaJ) (15)

where IG(jaJ)1 and LG(jaJ) are the amplitude and phase frequency characteristics,

respectively. The cost function in strict least squares sense becomes

K(P) =(y - (00, P»t (y - (00, P»

y=[A1wp '" AFwp Cl'IW2' ... , Cl'FWJt

real[H(jm l , P)]w1

real[H(jm F , P)]w(oo,P)=

imag[H( jm l , P)]w2

imag[H( jmF , P)]w2

where Ak is the measured logarithmic magnitude at frequencies mk, i.e. logarithmic

amplitude ratio of output to input, QJk is the measured phase shift of a minimum-phase

(16)

system at frequencies 0JJc, i.e., phase of output minus that of input, the subscript k =1,2,

... , F with F is the number of frequencies tested, Wi and W2 are the weights. Since the

logarithm of magnitude and phase have different units, weights should be given for

42

normalization. The weights can be chosen such WI = 11 I and W 2 = 11 I that themax( Ak ) max( rpk )

logarithmic magnitude and phase are ranged within [-1, 1]. The iteration algorithm is the

same as Approach I.

3. Results

Two examples were used to demonstrate the mathematical approaches for

parameter estimation of transfer functions in typical and non-typical forms.

Example 1. Consider a high damping system with the following non-typical

transfer function

[LV( 4D . K )JG(jOJ)=exp - 1- 1+-

2JOJ(I+. I +K3 )

2D V JOJ +K2

where the constants V, and L are 10.3902, and 25, respectively; D, Kl, K 2, and K3 are

(17)

parameters to be estimated. The transfer function is non-typical and cannot be converted

to a typical one. The original data are shown in Table 2.1. The fitted curves by

Approaches I and II are shown in Figures 2.1 and 2.2, and the statistical results are shown

in Table 2.2. Figure 2.1 shows that the subset of low frequencies were fitted well in

contrast with that of high frequency as the real and imaginary vales approached zero.

Figure 2.2 shows that the whole range of frequency was fitted well, and thus, the problem

oflow frequency emphasis in Figure 2.1 was improved in Approach II. From Table 2.2,

the asymptotic standard errors of parameters by Approach II are less than those by

Approach I. The asymptotic 95% confidence intervals of estimates by Approach II are

within those by Approach I. Thus, Approach II is better than I.

43

Example 2. Consider a light damping system with the following non-typical

transfer function

[LV( 4D K K)]G(j01)=exp - 1- 1+-2

}01(I+. 1 +. 3 )2D V j01 + K 2 j01 + K 4

(18)

where the constants V, and L are 9.4232, and 25, respectively; D, K 1, K2, K3 and K4 are

parameters to be estimated. The transfer function is non-typical and cannot be converted

to a typical one. The original data are shown in Table 2.3. The logarithmic magnitudes

at high frequencies in Table 2.3 were greater than those in Table 2.1, thus, transfer

function (18) had relatively lighter damping than transfer function (17). The fitted curve

by Approach I and II are shown in Figures 2.3 and 2.4, respectively. The statistical

results by Approach I and II are shown in Table 2.4. Both Figures 2.3 and 2.4 showed

that all frequencies were fitted well, and the phenomena of no low frequency emphasis

did not occur. Table 2.4 showed that the asymptotic standard errors of parameter K 1, K3,

K4, and D by Approach II were less than those by Approach I, while that ofK 2 by

Approach II was more than that by Approach I. The asymptotic 95% confidence

intervals of estimates ofK3, K4 and D by approach II were within those by Approach I;

the asymptotic 95% confidence intervals of estimates ofK 1by approach II has overlap

with that by Approach I; the asymptotic 95% confidence intervals of estimates ofK2 by

approach II has no overlap with that by Approach I. Thus, Approach I and II are

equivalent in terms of asymptotic standard error and asymptotic 95% confidence interval.

5. Conclusion

44

The real and imaginary parts of a transfer function or its logarithmic form can be

regarded as multivariate nonlinear models, and they can be transformed into a univariate

nonlinear model by the scaling method suggested in this article. The modified Gauss

Newton method can be used to estimate the parameters of the univariate nonlinear

models. Examples showed that fitting the logarithmic form of a transfer function is a

better choice than fitting the original transfer function when the system is high damping

because the low frequency emphasis will be overcome; however, when the system is light

damping, the two fittings are equivalent.

References

Bayard, D. S., F. Y. Hadaegh, Y. Yam, R. E. Scheid, E. Mettler, and M. H. Milman.

1991. Automated on-orbit frequency domain identification for large space

structures. Automatica, 27: 931-946.

Gallant, A. R. 1975. Seemingly unrelated nonlinear regressions. Journal of

Econometrics, 3: 35-50.

Gallant, A.R. 1987. Nonlinear Statistical Models. Wiley, John & Sons, Inc.

Lawrence, P. 1., and G. 1. Rogers. 1979. Sequential transfer function synthesis from

measured data. Proceedings. IEE, 136: 104-106.

Levy, E. C. (1959). Complex curve fitting, IRE Trans. Automat. Contr., Vol. AC-4, 37

43.

Ljung, L. 1999. System identification theory for the user. Prentice Hall PTR, Upper

Saddle River, New Jersey.

45

Martin L. 1994. A Global Approach to Accurate and Automatic Quantitative Analysis of

NMR Spectra by Complex Least-Squares Curve Fitting, Journal ofMagnetic

Resonance, SeriesA, Volume 111, Issue 1, November 1994, Pages 1-10.

Sanathanan, C.K. and J. Koerner (1963). Transfer function synthesis as a ratio of two

complex polynomials, IEEE Trans. Automat. Contr., Vol. AC-8, 56-58.

Schoukens, J. R. Pintelon, and J. Renneboog (1988). A maximum likelihood estimator

for linear and nonlinear systems - A practical application of estimation techniques

in measurement problems, IEEE Trans. Instrum. Meas., Vol IM-37, no.1, pp. 10

17.

Schoukens, J., and R. Pintelon. 1991. Identification of linear system: A practical

guideline to accurate modeling. Pergamon Press, Oxford, England.

Stahl, H. 1984. Transfer function synthesis using frequency response data. International

Journal of Control, 39: 541-550.

Van den Enden A W. M., G.C.Groendaal and E.vn Zee (1977). An improved complex

curve-fitting method, Proceedings of the conference on computer aided design of

electronic, microwave circuits and systems, Hull, United Kingdom, pp. 53-58.

Whitefield, AH. 1986. Transfer function synthesis using frequency response data.

International Journal of Control. 43: 1413-1426.

Whitefield, AH. 1987. Asymptotic behavior of transfer function synthesis methods.

International Journal of Control. 45: 1083-1092.

46

Table 2.1. The frequencies and the frequency responses used for example 1.

Frequency Real Imaginary Magnitude Phase

(min-I) (radians)

0.0781 0.7317 0.2600 -0.2529 -5.9417

0.1389 -0.3450 0.3840 -0.6613 -10.2636

0.2083 -0.2812 -0.1539 -1.1377 -15.2072

0.2778 0.2191 -0.0509 -1.4918 -19.0780

0.3333 -0.0037 0.1596 -1.8348 -23.5387

0.4167 0.0029 0.1097 -2.2099 -29.8713

0.4545 0.0112 -0.0943 -2.3547 -32.8688

0.5000 0.0056 0.0765 -2.5677 -36.2015

0.5556 -0.0459 -0.0297 -2.9071 -40.2661

0.6250 0.0082 -0.0297 -3.4795 -45.2827

47

Table 2.2. Transfer function (17) and its logarithmic equation in example 1 fitted by the

modified Gauss-Newton method.

TransferSource OF Sum of F value

FunctionSquares Mean Square

Regression 4 10.4237 2.6059 840

Residual 16 0.0496 0.0031Original

Uncorrected Total 20 10.4733Form

(Corrected Total) 19 6.3639

Regression 4 7.7316 1.9329 8880

Residual 16 0.0035 0.00022

Logarithmic Uncorrected Total 20 7.7351

Form (Corrected Total) 19 1.5300

Approach Parameter Estimate Asymptotic Asymptotic 95%

Standard confidence Interval

error Lower Lower

K1 0.4520 0.2108 0.0052 0.8988

K2 0.9722 0.3457 0.2393 1.7050

K3 3.6441 0.0953 3.4421 3.8462

D 0.4966 0.1727 0.1305 0.8628

K1 0.6115 0.0521 0.5009 0.7220

K2 1.0883 0.1083 0.8588 1.3178

II K3 3.7181 0.0369 3.6399 3.7963

D 0.2676 0.0171 0.2314 0.3038

48

Table 2.3. The frequencies and the frequency responses used for example 2.

Frequency Real Imaginary Logarithmic Phase

(min-I) Magnitude (radians)

0.0781 -0.0334 -0.5792 -0.2364 -1.6284

0.1389 -0.5170 -0.1605 -0.2665 -2.8406

0.2083 -0.2278 0.4467 -0.2998 -4.2408

0.2778 0.3456 0.2865 -0.3479 -5.5910

0.3333 0.3781 -0.1586 -0.3872 -6.6804

0.4167 -0.1443 -0.3196 -0.4551 -8.2780

0.4545 -0.2964 -0.1324 -0.4886 -9.0046

0.5000 -0.2677 0.1203 -0.5324 -9.8470

0.5556 -0.0216 0.2578 -0.5871 -10.9121

0.6250 0.2080 0.0716 -0.6577 -12.2350

0.7143 0.0281 -0.1707 -0.7619 -13.9742

49

Table 2.4. Transfer function (18) and its logarithmic form in example 2 fitted by the

modified Gauss-Newton method.

TransferSource OF Sum of Mean Square F value

FunctionSquares

Regression5 13.8060 2.7612 36159

ResidualOriginal 17 0.0013 0.000076

Uncorrected TotalForm 22 13.8073

(Corrected Total)21 7.3848

Regression5 8.6099 1.7220 97661

Residual17 0.0003 0.000017

Logarithmic Uncorrected Total22 8.6102

Form (Corrected Total)21 1.2916

Approach Parameter Estimate Asymptotic Asymptotic 95%Standard confidence Interval

error

Lower Lower

K1 0.2101 0.0037 0.2022 0.2180

K2 0.0541 0.0147 0.0231 0.0851

K3 1.7814 0.1129 1.5433 2.0196

K4 8.3195 0.6266 6.9976 9.6414

D 0.1937 0.0937 -0.0040 0.3914

K1 0.2187 0.0032 0.2120 0.2254

K2 0.1531 0.0186 0.1139 0.1922

II K3 1.7459 0.0664 1.6057 1.8861

K4 8.2217 0.3884 7.4023 9.0412

D 0.1347 0.0492 0.0308 0.2387

50

0.5

a::: 0

-0.5

+

+ +

0.70.60.50.40.30.20.1-1 L--__---L.-__--'- -"-----__---l.-__--.1 --i.-__----l

o

0.5

o

-0.5

+

++

+

0.70.60.50.3 0.4

Frequency f (min-1)

0.20.1-1 L--__---L.-__--'- -'--__-'-__-----L --'-----__-----'

o

Figure 2.1. Fit of transfer function (17) in example 1 by the modified Gauss-Newton

method.

51

Or--<=o----_____,_----,--------,----,------,-------,-------,

-0.5

~ -1:;,-'E~ -1.5

:::E(,)

'E -2J::t&.3 -2.5

-3

0.70.60.50.40.30.20.1-3.5 L-__---'-- '----__---'--__----.J'----__---'-- L....=j~_ __'

o

O~--_____,_---_,_--_____,---___.__---~--____,_------,

-10

-III +I: -20«I'6l!-Q)

l(l -30J::a.

-40

0.70.60.50.20.1-50 '---------'-----'--------'-------'------'---------'------'

o

Figure 2.2. Fit oflogarithmic transfer function (17) in example 1 by the modified Gauss-

Newton method.

52

1

0.8

0.6

0.4

0::: 0.2

0

-0.2

-0.4

-0.60 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.6 ,---------,,--------,-----,-----,-------,-------,-------,------,

0.80.70.60.3 0.4 0.5

Frequency f (min-1)

0.20.1

0.2

0.4

o

-0.8'--------1--------'---------'---------'------'------'------'--------'o

-0.2

-0.6

-0.4

Figure 2.3. Fit of transfer function (18) in example 2 by the modified Gauss-Newton

method.

53

0..-----,---,-----,------,---------,-------,------,------,

0.80.70.60.50.40.30.20.1-1.8 "--__L-__-'-----__-'--__--'----__--'----__-'--__---'---'__---l

o

-1.6

-1.4

-0.4

-0.2

-8:E -0.6C0)

;; -0.8

"'E -1.s'C~ -1.2.3

0

-2

-4-VIClU -6'tie!-Q)

-8VIlU

..c:a.

-10

-12

-140 0.1 0.2 0.3 0.4 0.5

Frequency f (min-1)

0.6 0.7 0.8

Figure 2.4. Fit oflogarithmic transfer function (18) in example 2 by the modified Gauss-

Newton method.

54

Chapter 3

Kinetics of Nitrate AdsorptionlDesorption at the MinerallWater


Abstract

The kinetic study of adsorption/desorption of electrolyte sodium nitrate at variable

charge mineraVwater interfaces is important for environmental chemistry and surface

chemistry. Although the widely used Triple-Layer Model (TLM) is successful in

accounting for the effect of electrolyte specific adsorption in the observations of

equilibrium experiments, direct evidence from kinetic study is needed to validate the

surface complexation models. The disadvantages in the methods available for kinetic

study include that they cannot study adsorption and desorption simultaneously, the useful

sampling interval is short, and/or direct measurement of each ions in bulk solution may

not be available. The objectives of this chapter are (i) to design and conduct novel

column experiments with constant pH 4 or 10 and sinusoidal varying sodium nitrate

concentration with time in the influent solution, and (ii) to identify the proper surface

complexation models according to responses of nitrate concentrations in the effluent

solutions. Linear results were obtained for all four systems when input signals were at

pH 10, and for the goethite system when input signals were at pH 4, the gibbsite,

hematite, and kaolinite systems were approximately linear when input signals are pH 4.

The Three-Plane Model was the proper mechanism of electrolyte specific adsorption at a

mineraVwater interface for the mineral system at pH 4. The Two-Plane Model was the

55

proper mechanism of negative adsorption of nitrate at a mineraVwater interface for the

mineral system at pH 10 due to the net negatively charged surface of minerals.

Introduction

The fate and transport of nitrate in surface and ground waters are ofconsiderable

concern. In order to accurately assess the potential hazard of nitrate to mankind and the

environment, it is necessary to determine the rate and mechanisms ofchemical

transformations, and mass transfer rates in environmental profiles. The study of

mechanism of nitrate adsorption!desorption onto the surface of minerals will first help to

understand and solve the problems of nitrate contamination. Surface Complexation

models (SCMs) have been developed to model the adsorption of ions at the

mineraVaqueous interface, and they assume that minerals have surface functional groups

similar to ligands in aqueous chemistry to form complexes with adsorption ions (Hayes,

1987). The surface complexation models coupled with hydrological models can be used

to predict the reactive transport nitrate.

Surface complexation models like the Constant Capacitance Model (HoW and

Stumm, 1976; Schindler and Stumm, 1987), Diffuse Layer Model (Stumm et aI., 1970;

Huang and Stumm, 1973; Dzombak and Morel, 1990), Triple -Layer Model (Yates,

1975; Yates, 1974; Davis et aI., 1978), and Four-Layer Model (Bowden et aI., 1980),

which combine electrical double-layer theory with aqueous complexation theory have

been used to describe inorganic ion adsorption. All surface complexation models

describe the mineral-water interface in terms of electric double layer theory, where the

interface is divided into a compact layer and a diffuse layer. The diffuse layer consists of

56

counterions that approach the surface to balance the charge distribution in solution.

Surface complexation models assume that (1) adsorption on oxides takes place at specific

surface sites, (2) adsorption reactions on oxides can be described quantitatively by mass

reaction expressions, (3) surface charge results from the adsorption reactions themselves,

and (4) the effect of surface charge on adsorption can be taken into account by applying

an electrostatic term derived from electrical double layer theory to the mass action

expressions for surface reactions.

The General Two-Plane models (Westall and Hohl, 1980; Dzombak and Morel,

1986a) include the Constant Capacitance SCM and the Diffuse Layer SCM (Healy et ai.

1977; Huang, 1981; James and Parks, 1982). The schematic representation ofpotential as

a function of distance from the surface according to Diffuse Layer SCM is shown in

Figure 3.1. The models divide the mineral-water interface into two layers, a planar

compact layer for specific (i.e., chemical) adsorption onto one type of surface site, and a

diffuse layer ofcounterions to balance the surface charge. Surface protonation,

deprotonation, and the adsorption of strongly adsorbed cations and anions (surface

coordinate) are responsible for the surface charge. The electrolyte ions are considered to

be counter ions in the diffuse layer, and they affect ion adsorption through the explicit

dependence of the Gouy-Chapman diffuse layer charge on ionic strength.

Compared with Two-Layer Models, the Triple Layer Model (TLM), which is also

referred to as Triple-Layer Model, adds another adsorption plane between the solid

surface and diffuse layer to account for the counter ion's forming of ion-pair complex

(Yates, et aI., 1974; Davis et ai. 1978; James and Parks, 1982; Sposito, 1984; Sverjensky,

1993; Sverjensky and Sahai, 1996; Sahai and Sverjensky, 1997a, b; Sahai and

57

Sverjensky, 1998; Criscenti and Sverjensky, 2002). The schematic representation of

potential as a function ofdistance from the surface according to Davis et ai. (1978) is

shown in Figure 3.2. By accounting specifically for counter ion binding via the

formation of an ion-pair surface complex, the increase of surface charge as a function of

ionic strength and pH (Uehara and Gillman, 1981) can now be recognized as a simple

consequence ofan increase in counter ion binding with increasing counter ion

concentration. In the TLM ofDavis and Lekie (1987b), there is an inner surface plane,

the 'a-plane', populated by potential determining ions (pdi's) (e.g., Ir and aIr); an

outer surface plane, the' J3-plane', populated by specifically adsorbed electrolyte cations

and anions, and an outermost 'diffuse layer', the 'D-plane', populated by a diffuse ion

swarm to balance the charge at the surface.

According to the Triple-Layer Model (Yates et aI., 1974; Yates, 1975; Davis et

aI., 1978; Sahai and Sverjensky, 1996, 1997, 1998), the schematic representation of

potential as a function ofdistance from the surface is shown in Figure 2 ofChapter 3.

The calculation of surface speciation by Sahai and Sverjensky (1997a, 1997b, 1998) is

described as follows.

Protons are adsorbed at the surface a-plane closest to solid surface

> SOH +H~ <=> > SOH;

where the symbol ">" stands for that the surface-site is bonded to the mineral, and

(1)

subscript aq stands for the aqueous solution. The "intrinsic" (Le., the effect of electrical

double layer is taken into account) equilibrium constant of the reaction (1) is

!"Po !"Po

e RT = KaPPe RT8,1

58

(2)

where the superscript "int" and "app" refer to the "intrinsic" and "apparent" (i.e., the

effect ofelectrical double layer is not taken into account) equilibrium constants.

Protons are also desorbed at the surface O-plane

and the "intrinsic" equilibrium constant is

a a -F'¥o -F'¥o>80- H a+qK int = e RT = KaPPe RT

s,2 s,2.a>80H

(3)

(4)

The (1:1) electrolyte M~- is adsorbed by ion-pairs >SOH2+-L-, >SO·_~ at the (3-

plane which is a small distance outside the O-plane, and the reactions are

and

(5)

(6)

the "intrinsic" equilibrium constants are, respectively,

F('Po-'Pp )

= KaPP e RTs,L- (7)

and

F(-'Po+'Pp )

= Kapp e RTs,"M+ (8)

where aj represents the activity of the jth species. The total concentration of sites, NT

(moles kg-I), is calculated from the site-density, Ns (sites nm-2) according to

59

where F is Faraday's constant (C morl), A is the specific surface area (m2gol

) of the

mineral, Cs is the amount (g LOI)of solid mineral dispersed in the solution, p is the

solution density (kg LOI), C>i is the molar concentration of the ith surface site (moles kg-I),

andNA is Avogadro's number. The net charge at O-plane and f3-plane are, respectively,

(10)

and

(11)

There is a diffuse swarm ofcounter ions, and the closest distance of approach of the

diffuse swarm defines the d-plane. The charge balance requires that sum of the charges

at the O-plane (0'0), f3-plane (O'~), and the net electrical double layer (O'd) be equal to zero,

(12)

The three planes are treated as a pair of parallel-plate capacitors connected in series and

the following relationship are obtained:

(13)

and

(14)

where CI is the capacitances (Farads m-2) of the media between the O-plane and f3-plane,

and C2 is the capacitances (Farads m-2) of the media between the f3-plane and d-plane

(i.e., the inner edge of electrical double layer). For a 1: 1 electrolyte, the relationship

between potential ('¥d) and net charge in electrical double layer (O'd) follows the Gouy-

Chapman Theory

60

where 80 is the permittivity offree space (equal to 8.854xlO-I2 c2r Im-I), 8 w is the

(15)

dielectric constant of the aqueous medium, and the I is the "true" ionic strength (molar)

of the system given by

where ci,aqis the concentration (moles kg-I) and Zi,aq is the valency of the ith aqueous

species (Sposito, 1984).

(16)

Overall, the equations (1) to (16) and the seven parameters (Ks,l' Ks,2, Ks,L-, Ks,M+'

Ns, Cl, and C2) are used to calculate the surface species by non-linear algorithms.

The surface complexation models so far are based on the study of equilibrium

status of electrolyte adsorption and desorption. Because thermodynamic study cannot

supply the information ofdynamic process, the study of mechanisms of surface

complexes and the formation needs information from kinetic study. There are two

disadvantages in the traditional experimental methodology including batch experiment

and pressure jump method (Ashida et aI., 1980; Astumian et aI., 1981; Sasaki et aI., 1983;

Mikami et aI., 1983a,b; Hachiya et aI., 1984a, b; Chang et aI., 1994; GrossI and Sparks,

1995; Lin et aI., 1997; Wu et aI., 1998; Liu and Huang, 2001). The first disadvantage is

that the electrolyte adsorption and desorption cannot be studied simultaneously. The

objective of the traditional methodology is focused on either adsorption or desorption.

The second disadvantage is that the traditional methodology may not supply accurate

information. The quality of sampling is constrained by the narrow interval of peak or

jump of solute concentration in the breakthrough curve ofbatch or flow method. In the

61

pressure jump method, the ion concentration cannot be measured directly and the

measurement of conductivity is mix information.

The objectives of this chapter are (i) to develop an experimental method to study

the kinetics ofadsorption and desorption of electrolyte NaN03 at the mineral/water

interface different pH condition, (ii) to identify the proper surface complexation models.


Chemicals: Sodium nitrate (NaN03) was analytical reagent grade. Saturated

NaOH solution was 120g in 100 ml water, and kept for one month to allow the Na2C03

and NaHC03to precipitate. Diluted NaOH is fresh made with boiled water and the

supernatant of the saturated NaOH. 2% RN03 was certified AC.S. reagent grade. The

water was of nanopure quality and boiled to degass.

Minerals: gibbsite, goethite, hematite, and kaolinite were from Ward Science

Company. They were ground and wet sieved with deionized water, and the fraction of

325-500 mesh was collected and freeze-dried. X-ray diffraction analysis showed that

goethite, kaolinite, and hematite contained quartz. The weights of minerals packed in

columns, the water contents, and the water velocity estimated in Chapter 1 are listed in

Table 3.1. The columns were sequentially washed with 1 mMNaOH for around 8 hours,

1 mMRN03 for around 8 hours, 1.25 ml Lo l acetone solution for around 30 hours. Then,

the columns were washed with pH 4.0 RN03 until the eflluent pH was within 4.0±0.1 for

the nitrate sorption/desorption experiments at pH 4, or washed with pH 10.0 NaOH until

62

the eflluent pH was within 10.0±0.1 for the nitrate sorption!desorption experiments at pH

10.

Setup ofExperiment: The instrument setup was the same as that in the transport

experiment with acetone as inert tracer in Chapter 1. Four liters ofwater was adjusted to

pH 4.0 with 2% RN03 or to pH 10.0 with diluted NaOH solution. Solution A, 0.2 mM

NaN03 solution, was made with two liters of the solution, and the remaining two liter

solution was solution B. The change ofpH of solution A after adding NaN03 was

negligible. Solutions A and B were used to generate input signals for experiments.

Solution C was 4 liters ofNaN03with accurate concentration 0.300 mM without

adjusting pH, and solution D was water. The solutions C and D were used to calibrate

the UV absorbance method to absolute nitrate concentration. The wavelength to measure

nitrate concentration was 210 nm.

Input signal Design: Twelve input sinusoidal signals with periods 12.8, 7.2, 4.8,

3.6,3.0,2.4,2.2,2.0, 1.8, 1.6, 1.4, 1.2 minutes, were designed by using combinations of

gradient tables and event tables as described in Chapter 1. The composition of solution A

and B designed at gradient controller ofHPLC with period Twas

A% =[tcOS(2; t) + t] *100 (17)

B%=100-A,

where t is the time ofeach step, and thus, the nitrate concentration of the mixed solution

is changing with time. The nitrate concentration varied with time at the inlet of column,

called input signal, is dominant of single one frequency.

Input and output signal Experiments: Columns were packed with various

minerals. Input and output signal Experiments were carried out via

63

disconnecting/connecting columns to the experimental setup as Chapter 1. The effects of

tubing and other device on solute transport were cancelled out when the ratio of output

and input in frequency domain. Experiments include four minerals: gibbsite, goethite,

hematite and kaolinite, and two pH levels: pH 4 and pHI0.

3. Mathematical models and algorithms for parameter estimation

In the frequency response experiments, input signals were termed as the

sinusoidal dynamically changing concentration of sodium nitrate with constant solution

pH 4 or pH lOin the influent solution. The systems were termed the convection

dispersion process of aqueous solution and the adsorption/desorption process at the

mineral/water interface occurring inside of columns packed with minerals. The output

signals were termed the dynamically changing concentration ofnitrate in the effluent

solution.

3.1. The transfer functions for systems at different pH conditions when the Two-Plane

Model is assumed to be the mechanism of adsorption/desorption process at the

mineral/water interface.

The reaction will be the adsorption and desorption of nitrate between diffuse layer

(Ld) and aqueous solution (Laq) as follows

(

and the transfer function is

64

(ZV 4D Jexp -(1- 1+-

2G(s))

H(s) = 2D V

exp( - ~G(S»)

D>O

D=O

(18)

to represent the convection-dispersion process coupled with electrolyte adsorption-

desorption process

KG(s) =s(1 + a)

s+Kd

(19)

where Z is 25 (cm) and Vis a known constant listed in Table 3.1. When Kd» s = jm,

G(s) can be approximated as

KG(s) =s(1+_a) =s(1+KJ,

K d

where K1 is a constant Ka l. The derivation of equation (18) to (20) is shown in

Kdl

Appendix 1.

3.2. The transfer functions for systems at different pH conditions when the Triple-

Layer model is assumed to represent the mechanism of adsorption/desorption at the

mineral/water interface.

(20)

The reaction path will be the sequential exchange of nitrate among >SOH2+-L- (L~),

diffuse layer (Ld), and aqueous phase (Laq") as follows

Kal )

( Kdl

The transfer function is equation (9) with

Ka2 )

Laq .

65

or when Kal and Kd1 are much greater than s, Ka2, and Kd2,

where K1 is a constant Ka} . The derivation of the transfer function (21) and (22) isKd}

shown in Appendix 2.

3.3 Algorithm for model selection and parameter estimation.

(21)

(22)

The algorithm developed in Chapter 2 was used to estimate the parameters in the

above transfer functions. The models (19) and (20) were chosen first to fit the

experimental data. If the models could not describe the experimental data, then models

(21) and (22) were chosen.

4. Results

4.1 The linear relationship between nitrate concentration and absorbance.

The relationship between nitrate concentration and absorbance was linear as

shown in Figure 3.3.

4.2 The input signals in time

The dynamic change in nitrate concentration of the input signals is shown in

Figure 3.4 with three periods as examples.

66

4.3 Spectral component of the input signals

For an input signal with fundamental frequency/k = l/Tk, k =1,2, ... ,12, its

spectral components with frequencies / k, 2/k, 3/k, ... 6/k, were calculated with algorithm

fast Fourier Transform (FFT) in MATLAB®. Their amplitudes were noted as All' A2/1

,

A3/1

, ••• A6It. The relative amplitude ofa spectrum component with frequency lfk, I =

2,3, ... , 6, compared with its fundamental frequency fk is

(23)

where I is a positive integer. The five curves rIfl vs. he, I = 2, 3, ... ,6 are shown in Figure

3.5. The relative amplitudes of subharmonics were less than 3% of those of the

fundamental frequency and they were negligible. Thus, the input signals were sinusoids

dominant with only fundamental frequency.

4.4 The variance of input signals among repeated experiment.

The CVs of the input signals among four repeated experiments were less than

0.76%, thus, amplitudes of input signals generated were repeatable.

The phases {fJk of fundamental frequency fk of input signals, k =1,2, ... 12,

calculated by FFT, were within -1t and 1t, and not continuous. To convert the phase

values into continuous data passing through the origin, the modification formula is

where n is a positive integer. In this chapter, the "phase" refers to the modified values

(24)

instead of those within -1t and 1t. The CVs of phases of the twelve input signals among

repeated experiments were less than 0.6%, thus, the phases of input signals generated

67

were repeatable. The averages of the amplitudes and phases among repeated experiments

were used for data analysis.

4.5 The output signals in time

The dynamic change of nitrate concentration of four systems at pH 4 and pH 10

are shown in Figures 3.6 to 3.13 with selected signals with period 12.8,2.4, 1.6, and 1.2

minutes as examples. Except for the kaolinite system at pH 4, the output signals showed

a regular sinusoidal pattern. The amplitudes of the outputs at pH 4 were highly reduced

relative to those of pH 10.

4.6 Spectral component of output signals

The amplitudes offundamental frequency and subharmonics were calculated by

the FFT algorithm. The relative subharmonic amplitudes with fundamental frequencies

were calculated by equation (23) and their changes with fundamental frequency are

shown in Figures 3.14 to 3.17 for gibbsite, goethite, hematite, and kaolinite systems at pH

4 and 10. The relative amplitudes of subharmonics were less than 5% ofthose of the

fundamental frequencies of output signals at pH 10, and they were negligible, thus, the

sorption/desorption process was linear. The relative amplitudes of subharmonics were

less than 5.2% ofthe fundamental frequencies of output signals for goethite system at pH

4, thus, the sorption/desorption process of nitrate was linear. The relative amplitudes of

subharmonics were less than 11.4%, 14.0%,37.3% ofthe fundamental frequencies of

output signals for systems ofgibbsite hematite, and kaolinite at pH 4, and thus, the

68

sorption/desorption processes were approximately linear and they were treated as linear

systems.

The phases ofoutput signals calculated by FFT were within -1t and 1t, and not

continuous with frequencies. Similar to input signals, they were modified into

continuous data passing through the origin.

4.7 Estimates of parameters in the transfer function models for systems at pH 10.

The processes ofdispersion and negative sorption/desorption ofnitrate were

dominant for the experiments at pH 10. Assuming that the desorption rate is much higher

than the frequencies of input signals, equations (18) to (20) were used to simulate the

frequency response ofgibbsite, goethite, hematite and kaolinite at pH 10. The original

data and the fitted curves are shown in Figures 3.18 to 3.21. The estimated dispersion

coefficients and the ratio ofadsorption and desorption rates are listed in Tables 3.2 to 3.5.

The averages and standard deviations among repeated experiments are shown in Table

3.6. The dispersion coefficients were 12%-28% less than those of acetone experiments.

The ratio ofadsorption and desorption coefficients varied between -0.0802 and -0.1646.

The negative ratio implied that the adsorption rate was negative, corresponding to

negative adsorption ofnitrate at pH 10 when the surface charge of the minerals was

negative. Because the models derived from Two-Plane Model fitted the experimental

data for mineral systems at pH 10, a tentative interpretation is that the ion-pair SOH2+

N03- is negligible in the f3-plane due to repulsion of net negative surface charge while

ion-pair SO--Na+ is dominant in the f3-plane.

69

4.8 Estimate ofparameters in transfer function models for systems at pH 4.

Equations (18) and (21) were used to describe the transfer function ofthe nitrate

sorption/desorption process in gibbsite and goethite systems at pH 4.0. Equations (18)

and (22) were used to describe the transfer function in nitrate adsorption/desorption

process at hematite and kaolinite systems at pH 4.0 since the desorption rates were so

much higher than the frequency in radians that an accurate estimate is impossible if

equation (21) was used. The original data and the fitted curves are shown in Figures

3.22 to 3.25. The estimate ofdiffusion coefficient D is set to zero if it was not

significantly different from zero or negative. The parameters with respect to adsorption

and desorption rates and the diffusion coefficients are listed in Tables 3.7 to 3.10. The

averages and standard deviations among repeated experiments are shown in Table 3.11.

Because the models derived from the Triple-Layer Model fitted the experimental data for

mineral systems at pH 4, the Triple Layer Model is a tentative predictive model.

4.9 The equilibrium constants ofelectrolyte adsorption/desorption at mineral/water

interface at pH 4.

The equilibrium constants of the adsorption and desorption rates to partition

between aqueous solution and diffuse layer for gibbsite and goethite systems at pH 4 are

calculated as follows

K =Ka)) Kd

)

The equilibrium constants of the adsorption and desorption rates to partition between

diffuse layer and fl-Iayer at pH 4 ofinput signals for all four systems are calculated as

follows

70

(25)

K =Ka2

2 Kd2

(26)

The overall equilibrium constants of electrolyt~adsorption and desorption rates to

partition between aqueous solution and J3-layer at pH 4 are calculated as follows

K=K*K1 2

associated with the reaction

SOH +H;q +L~ => SOH; -D.

(27)

The values Kl, K2, and K are shown in Table3.12. The sequence ofK1 values is

gibbsite < goethite < hematite < kaolinite; the sequence ofK2 values is gibbsite> goethite

> hematite> kaolinite. The sequence ofK values is gibbsite> goethite> hematite>

kaolinite.

5. Conclusion

The spectral analysis showed that generated input signals were sinusoids with

only fundamental frequencies. The spectral analysis of output signals when input signals

were pH 10 showed that the subharmonic components were negligible and thus, the four

systems were linear at pH 10. The spectral analysis of output at pH 4 showed that the

goethite system was linear, and gibbsite, hematite, and kaolinite systems were

approximately linear. Kinetics models, transfer functions, according to Two-Plane

Model and Triple-Layer Model were derived. For mineral systems at pH 10, the

identified transfer functions were from Two-Plane Model, and the possible reason is that

nitrate was negatively adsorbed due to the repulsion from the net negative charge at

mineral surface. For mineral systems at pH 4, the identified transfer functions were from

71

Triple-Layer Model, and the possible reason is that nitrate was adsorbed at p-plane and

diffuse layer from aqueous solution.

Appendix 1

Derivation of the transfer function of Two-Plane Model coupled with

convection-dispersion process.

In Two-Plane Model, two layers of diffuse layer and bulk solution are allowed.

The reaction path for the three cases is assumed as follows

L"4Ka )

( Kd

Assume the reactions are linear, and the following kinetic model can be given

8L"4 82Laq 8L"4-=D---V--KaL +KdL

8t 82z 8z "4 d

8Ld =KaL - KdL8t "4 d

(28)

where t and z are time and distance from the point where influent solution is injected, D,

V are constants of dispersion coefficient and solution velocity. The initial conditions to

the pde above are:

Laq (O,z) =O,Ld (O,z) =0,

and the boundary condition is :

L"4 (t,O) =u(t) .

The length of column is Z, and define the nitrate concentration of the effluent is

yet) =L"4 (t, Z) .

72

(29)

(30)

(31)

Do Laplace transform to the pde's with respective to t,

2 A A

A A a Laq aLaq(s+Ka)Laq -KdLd =D-

2--V-

a z az

where s is Laplacian, iaq andid are the Laplace transforms ofLaq and Ld, respectively.

The equations can be simplified as

a2i a1 A

D--aq-V~=G(s)La2z az aq

G(s) =S(l + Ka )s+Kd

Similar to Appendix 1, the general solution to the equations above is

When Ki»s =}O1, G(j01) can be approximated as

where the constant K1 is equal to K a .

K d

Appendix 2

Derivation of the transfer function of Triple Layer Model for nitrate

adsorption/desorption coupled with convection-dispersion process.

The elementary reaction path is assumed as follows

(32)

(33)

(34)

(35)

( KdJ

Ka2) L

~

73

where L3f!' Ld and L~ are nitrate in aqueous solution, diffuse layer, and J3-plane,

respectively, and the constants Kat and Kdl are the adsorption and desorption rates of

nitrate from aqueous solution to diffuse layer, and Ka2 and Kd2 are the adsorption and

desorption rates ofnitrate from diffuse layer to J3-plane. The nitrate in aqueous solution

is mobile and governed by both convection-dispersion process and reactions, while

nitrate in diffuse layer and J3-plane is immobile and governed by reactions only. Assume

the reactions are linear, and the following kinetic model can be given

(36)

where t and z are time and distance from the point where influent solution is injected, D,

V are constants of dispersion coefficient and solution velocity. The initial conditions to

the pde above are:

Laq (0, z) =O,Ld (0, z) =O,L~ (0, z) =0,

and the boundary condition is:

Laq (t,O) =u(t).

The length ofcolumn is Z, and define the nitrate concentration of the effluent

yet) =Laq (t, Z) .

Do Laplace transform to the pde's with respective to t,

74

(37)

(38)

(39)

2 A A

A A a Laq aLaq(s+Ka)Laq -Kd)Ld =D-

2--V--az az

(s+Kd) +Ka2 )id -Kd2i p =Ka)iaq

Ka2i d - (s +Kd2 )ip =0

where s is Laplacian, iaq, i d , and 4are the Laplace transforms ofLaq, Ld and L~,

respectively. The equations can be simplified as

The general solution to the equations above is

(40)

(41)

where Cl and C2 are constants to be determined. Since the second term of the solution is

not bounded, it is deleted from the solution and the new general solution is

A (ZV [ 4DG(S)]]Laq =c) exp 2D 1- 1+ V 2 • (43)

Do Laplace transforms to u(t) and y(t) with respect to t, and their transforms are denoted

as U(s) and Y(s). Inserting U(s) and Y(s) to the equation above, we have

c) =U(s)

and

Y(s) =U(S)exp(:r; [1~ 1+ 4D:'(S) ]}

The transfer function of the system can be defined as

75

(44)

(45)

H(s) = Y(s) = eXP(ZV[I_ 1+ 4DG(S)]].U(s) 2D V Z

(46)

• SZ Ka KdWhen Kal' Kd1 are much greater than Ka2, Kd2 and s, the ratlos--,-_z , and __z are

Ka l Ka} Ka1

approximately zeros, and thus, the transfer function can be approximated by

G(s) =S(1 + s +Kaz +Kdz Jr:::J S[1 + Kl(s +Kaz +Kdz)] (47)L + S(Kdt + K02 + Kd2 ) + Kd1Kd2 S +KdKa1 Kat Kat Ka1 Kat Z

Kawhere K =__11 Kd

1

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80

Table 3.1. Property ofcolumns.

Mineral Weight packed (g) Water content (%) Water velocity (cm min-i)

Gibbsite 26.770 54.074 9.423

Goethite 41.940 51.488 9.896

Hematite 31.670 59.924 8.503

Kaolinite 30.740 49.042 10.390

Table 3.2. Estimated dispersion coefficients (D) and the ratio of adsorption and

desorption rates for gibbsite system at pH 10.

Asymptotic 95%

Repeated Parameter Estimate Confidence Interval

Experiment Lower Upper

#1 K1 -0.1187 -0.1246 -0.1128

D 0.4130 0.4065 0.4196

#2

#3

D

D

-0.1280

0.8350

-0.1248

0.8239

81

-0.1355

0.8179

-0.1293

0.8137

-0.1205

0.8522

-0.1203

0.8340

Table 3.3. Estimated dispersion coefficients (D) and the ratio of rates of sorption and desorption

for goethite system at pH 10.

Asymptotic 95%



#1 K1 -0.0840 -0.0890 -0.0790

D 0.4464 0.4406 0.4523

#2

#3

D

D

-0.0820

0.4521

-0.0745

0.4338

82

-0.0868

0.4464

-0.0806

0.4269

-0.0771

0.4578

-0.0683

0.4407


for hematite system at pH 10.

Asymptotic 95%



#1 K1 -0.1639 -0.1671 -0.1606

D 0.8266 0.8189 0.8343

#2

#3

D

D

-0.1652

0.8123

-0.1644

0.8070

83

-0.1691

0.8032

-0.1684

0.7978

-0.1614

0.8214

-0.1604

0.8162


for kaolinite system at pH 10.

Asymptotic 95%



#1 K1 -0.1542 -0.1704 -0.1381

D 0.3224 0.3078 0.3370

#2

#3

D

D

-0.1654

0.2966

-0.1555

0.2950

84

-0.1826

0.2820

-0.1630

0.2888

-0.1483

0.3112

-0.1480

0.3012

Table 3.6. The average and standard deviation of parameters of equation (20) among repeatedexperiments for mineral systems at pH 10.

Mineral Parameter

Gibbsite K1

D

Goethite

D

Hematite

D

Kaolinite

D

Average

-0.1238

0.6906

-0.0802

0.4441

-0.1645

0.8153

-0.1584

0.3047

85

Std

0.0047

0.2405

0.0050

0.0094

0.0007

0.0101

0.0061

0.0154

Table 3.7. The estimated dispersion coefficients and rates of adsorption and desorption forgibbsite system at pH 4.

Asymptotic 95%Confidence Interval

RepeatedExperiments Parameter Estimate Lower Bound Upper Bound

Ka1 2.2325 2.0306 2.4344

Kd1 9.1069 7.9796 10.2341

#1 Ka2 0.9775 0.9152 1.0399

Kd2 0.1918 0.1464 0.2372

0 0.3101 0.2119 0.4083

Ka1 1.9646 1.8197 2.1095

Kd1 7.3235 6.5511 8.0959

#2 Ka2 0.8795 0.8273 0.9317

Kd2 0.1718 0.1282 0.2154

0 0.1347 0.0308 0.2387

Ka1 2.2981 1.9946 2.6017

Kd1 8.7379 7.1439 10.3318

#3 Ka2 0.9358 0.8438 1.0279

Kd2 0.1567 0.0700 0.2434

0 0.3321 0.1752 0.4890

86

Table 3.8. The estimated dispersion coefficients and rates ofadsorption and desorption forgoethite system at pH 4.

Asymptotic 95%Repeated Confidence Interval

Experiments Parameter Estimate Lower Bound Upper Bound

Ka1 1. 9624 1. 8578 2.0670

Kd1 6.7871 6.3040 7.2702

Ka2 0.8301 0.7809 0.8793

Kd2 0.1849 0.1323 0.2375

#1 D -0.1658 -0.2457 -0.0858

Ka1 2.0980 1. 9464 2.2496

Kd1 7.7788 7.2696 8.2881

Ka2 0.9251 0.8901 0.9601

Kd2 0.2334 0.1827 0.2841

D 0

Ka1 2.2602 2.1127 2.4077

Kd1 7.5167 6.8508 8.1825

Ka2 0.8057 0.7507 0.8606

Kd2 0.2542 0.2064 0.3020

#2 D -0.1417 -0.2392 -0.0442

Ka1 2.4275 2.2679 2.5870

Kd1 8.5061 8.0211 8.9911

Ka2 0.8779 0.8456 0.9102

Kd2 0.2864 0.2423 0.3306

D 0

Ka1 2.2601 2.1005 2.4198

Kd1 7.4510 6.7364 8.1657

Ka2 0.8044 0.7442 0.8645

Kd2 0.2555 0.2031 0.3079

#3 D -0.1637 -0.2701 -0.0572

Ka1 2.4564 2.2758 2.6370

Kd1 8.6009 8.0533 9.1486

Ka2 0.8882 0.8518 0.9245

Kd2 0.2925 0.2441 0.3410D 0

87

Table 3.9. The estimated dispersion coefficients and rates ofadsorption and desorption forhematite system at pH 4.



K1 0.5494 -0.0250 1.1238

Ka2 5.1630 -6.5273 16.8534

Kd2 3.6582 2.3048 5.0116

#1 0 -0.7715 -2.4844 0.9415

K1 0.7971 0.6936 0.9006

Ka2 1.7219 1.3304 2.1134

Kd2 2.8303 2.3511 3.3095

0 0

K1 0.8536 0.6518 1.0554

Ka2 1.6871 -0.0037 3.3780

Kd2 2.9970 1.7607 4.2334

#2 0 -0.0002 -0.4669 0.4665

K1 0.8537 0.7613 0.9460

Ka2 1.6861 1.3627 2.0095

Kd2 2.9960 2.5414 3.4505

0 0

K1 0.8293 0.6519 1.0067

Ka2 1.7325 0.2386 3.2264

Kd2 2.8696 1.7901 3.9491

#3 0 -0.0197 -0.4459 0.4065

K1 0.8362 0.7513 0.9211

Ka2 1.6678 1.3806 1.9549

Kd2 2.8262 2.4336 3.2188

0 0

88

Table 3.10. The estimated dispersion coefficients and rates of adsorption and desorption for

kaolinite system at pH 4.



K1 3.6427 3.5412 3.7443

#1 Ka2 0.2363 0.1512 0.3214

kd2 1.6701 1.2226 2.1176

0 0.2811 0.2094 0.3528

K1 3.7181 3.6399 3.7963

#2 Ka2 0.1645 0.1334 0.1955

kd2 1.0883 0.8588 1.3178

0 0.2676 0.2314 0.3038

K1 3.3904 3.3119 3.4690

#3 Ka2 0.2638 0.2029 0.3247

kd2 1.4927 1.2447 1.7407

0 0.1745 0.1101 0.2388

89

Table 3.11. Average and standard deviation ofparameters in equation (21) among repeated

experiments for mineral systems at pH 4.

Mineral Parameter Average Std

Gibbsite Ka1 2.1651 0.1767Kd1 8.3894 0.9414Ka2 0.9309 0.0492Kd2 0.1734 0.0176

0 0.2590 0.1082

Ka1 2.1609 0.1719Kd1 7.2516 0.4036Ka2 0.8134 0.0145Kd2 0.2315 0.0404

Goethite 0 -0.1571 0.0133Ka1 2.3273 0.1991Kd1 8.2953 0.4498Ka2 0.8971 0.0248Kd2 0.2708 0.03250 0

K1 0.7441 0.1691Ka2 2.8609 1.9938Kd2 3.1749 0.4233

Hematite 0 -0.2638 0.4398

K1 0.8290 0.0290Ka2 1.6919 0.0275Kd2 2.8842 0.0969

0 0

Kaolinite K1 3.5837 0.1716Ka2 0.2215 0.0513Kd2 1.4170 0.29820 0.2411 0.0580

90

Table 3.12. The equilibrium constants of reaction path and the overall equilibrium constant of

electrolyte sodium nitrate adsorption/desorption at mineral/water interface at pH 4.

Mineral

Gibbsite

Goethite

Hematite

Kaolinite

KallKdl

0.258

0.281

0.829

3.584

91

Ka2IKd2

5.369

3.313

0.587

0.156

Kal *Ka2lKdllKd2

1. 386

0.929

0.486

0.560

Potential

\Po

I111! Diffuse layer: of counter ionIIIIIIIIIII1

.................................................................................... 1

Bulk solution

O-plane(mineral Surface)

Figure 3.1. Diffuse Layer Model.

d-plane

92

Distance x

Potential

'Po

~-plane d-plane(electrolyteadsorption)

Distance x

Bulk solutionDiffuse layerof counter ion

IIIIIIIIII

............................................1 .

O-plane(MineralSurface)

Figure 3.2. Triple -Layer Model.

93

1.6,..-----------.-------,-----,-------,-----,-----,

1.4

1.2

y= 6.8924x ·0.0188

R2 = 0.9988, P < 0.0001

0.8

>-

0.6

0.4

0.2

0 0

-0.20 0.05 0.1

x0.15 0.2

Figure 3.3. The linear relationship between concentration of nitrate (y) and UV absorbance at

210 nm (x).

94

1l:

T=12.8 min0

~ 0.5CQ)Ul:

8 0

~.~

~ -0.5~

0z-1

0 2 4 6 8 10 12 14

1l: T=2.4 min0

~ 0.5CQ)Ul:

8 0

~.~

~ -0.5~

0z-1

0 0.5 1 1.5 2 2.5

0.4l: T=1.2 min0

~ 0.2CQ)Ul:

8 0

~N

1-0·20z

-0.40 0.2 0.4 0.6 0.8 1 1.2 1.4

lime (min)

Figure 3.4. The concentration of input nitrate varying with time.

95

3

2.5

l1: 2

tuf!i 1.5..CD

j!

0.5

-- 2f--e-3f--- 4f-+- Sf--'i<- Sf

o~~~~~~~~~o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Frequency (min-1)

Figure 3.5. Spectral components of input signals.

96

1c: T=12.8 min0

~ 0.51:Q)

g8 0

i.~

l -{).50z

-10 2 4 6 8 10 12 14

0.4c: T=2.4 min0

~ 0.21:Q)0c:0 00

i.~

l-o·20z

-0.40 0.5 1 1.5 2 2.5

0.04c:0

~ 0.02+' T=1.2 minc:Q)0c:0 00

i.!::!l-o·020z

-{).040 0.2 0.4 0.6 0.8 1 1.2 1.4

lime (min)

Figure 3.6. Normalized concentration of nitrate, deviated from the averages, varying with time

in a gibbsite system at pH 4. The legends stand for the period of input signals.

97

1c: T=12.8 min0

:0::;

~ 0.51:4l

"c:8 0

"i.!::!

/~ .{l.5...0z

-10 2 4 6 8 10 12 14

1c: T=2.4 min0

~ 0.5-c:4l

"c:8 0

"iN

~ .{l.5...0z

-10 0.5 1.5 2 2.5

0.4c: T=1.2 min0

~ 0.21:4l

"c:8 0

"i.!::!~ .{l.2

z-0.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4Time (min)


in a gibbsite system at pH 10. The legends stand for the period of input signals.

98

1c: T=1208 min0~

l! 005CQ)(,)c:8 0

i.!:::!

l-o·50z

-10 2 4 6 8 10 12 14

0.4c:0

~ 002-c:Q) T=2.4 min(,)c:

8 0

i.!:::!l-0020z

-0.40 0.5 1 1.5 2 205

0.06c: T=1.2 min0 0.04~C 0.02Q)(,)c:0 00

io!:::! -0.02iiiE5 -0.04z

-0.060 0.2 0.4 0.6 0.8 1 1.2 1.4

lime (min)

Figure 3.8. Normalized concentration ofnitrate, deviated from the averages, varying with time

in a goethite system at pH 4. The legends stand for the period of input signals.

99

1c:

T=12.8 min0

~ 0.5'EQ)

g8 0i.~

1-0·5

~-1

0 2 4 6 8 10 12 14

1c:

T=2.4 min0

~ 0.5-55(,Jc:0 0uiN

l-O·55z

-10 0.5 1 1.5 2 2.5

0.4c: T=1.2 min0;;e! 0.2~(,Jc:8 0

i.~

~ -0.2...0z

-0.40 0.2 0.4 0.6 0.8 1 1.2 1.4

lime (min)


in a goethite system at pH 10. The legends stand for the period of input signals.

100

1c: T=12.8 min0

~ 0.51:Q)

g8 0

i.~

~ ~.5

~-1

0 2 4 6 8 10 12 14

0.15c:0 T=2.4 min~ 0.1-c:Q)(.) 0.05c:00

i 0.~(ijE ~.050z

~.10 0.5 1 1.5 2 2.5

0.015c: T=1.2 min0 0.01:;:;l!:!-c: 0.005Q)(.)c:8 0

i.~ ~.005(ij

§ ~.01

z~.015

0 0.2 0.4 0.6 0.8 1 1.2 1.4lime (min)


in a system ofhematite at pH 4. The legends stand for the period of input signals.

101

1c: T=12.8 min0

:;:l

l!! 0.5-c:Q)0c:8 0

"i.~

~ -0.5

~-1

0 2 4 6 8 10 12 14

1c: T=2.4 min0

~ 0.5i:Q)0c:8 0

"i.~

~ -0.5...0z

-10 0.5 1 1.5 2 2.5

0.2c: T=1.2 min0

~ 0.1i:Q)0c:8 0

"i.~

~ -0.10z

-0.20 0.2 0.4 0.6 0.8 1 1.2 1.4

lime (min)


in a system of hematite at pH 10. The legends stand for the period of input signals.

102

1c:0

:;::;T=12.8 mil!! 0.5

CQ)

~

8 0

i.~

~ -{l.50z

-10 2 4 6 8 10 12 14

0.1c:

T=2.4 min0

~ 0.05-c:Q)CJc:0 0(,)

i.~

~ -{l.050z

-{l.10 0.5 1 1.5 2 2.5

0.02c: T=1.6 min0

~ 0.01-c:Q)CJc:8 0

i.~

~ -{l.01

z-{l.02

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6lime (min)


in a system ofkaolinite at pH 4. The legends stand for the period ofinput signals.

103

1c T=12.8 min0

~ 0.5t:CD

~

8 0i.!::!~ ~.5...~

-10 2 4 6 8 10 12 14

1c

T=2.4 min0:;::l!! 0.5-cCDto)c8 0i.!::!~ ~.5

z-1

0 0.5 1 1.5 2 2.5

0.4c0

~ 0.2- T=1.2 mincCD

~

8 0i.!::!~ ~.20z

~.40 0.2 0.4 0.6 0.8 1 1.2 1.4

lime (min)


in a system of kaolinite at pH 10. The legends stand for the period of input signals.

104

12 ____ 2fpH4 --e- 3f

~ 10 ----><- 4f--+- 5f

C -+- 6fQ)c:::: 88.c::::0u!l! 6"08-II)

4Q)

.~

1ii~ 2

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

4.5 pH 10____ 2f

--e- 3f- 4 ----><- 4f~~ --+- 5f~ 3.5 -+- 6fc::::

8. 3c::::0

~ 2.5"0 28-II)

~ 1.5i 1~

0.5

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Frequency (min-1)

Figure 3.14. The relative amplitudes of subharmonics of output ofa gibbsite system at pH's 4

and 10.

105

pH 4___ 2f

5 -e- 3f~ ---- 4f~ -+- 5fe 4 -+- 6fCDc:0Q.c:0 30

!!!0CDQ.

2IIICD

.O!:1ii~ 1

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

3pH 10

___ 2f

-e- 3f

~2.5 ---- 4f

-+- 5f- -+- 6fc:CD 2c:0Q.c:00

!!! 1.5-0CDQ.IIICD

.O!:1ii~ 0.5

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Frequency (min-1)

Figure 3. 15. The relative amplitudes of subharmonics of output of a goethite system at pH's 4

and 10.

106

15pH 4 - 2f

--e- 3f~ --- 4f~ -+- Sf- ----+- 6fc:Q)c: 100c.c:00

I!!-0Q)c.f/)

5Q)

.~

1il~

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

2.5pH 10

_ 2f

--e- 3f~ --- 4f~ 2 -+- Sf- ----+- 6fc:Q)c:0c.c: 1.500

I!!-0Q)

1c.f/)

Q)

.~-l'II~ 0.5

00 0.1 0.2 0.3 0.4 0.5 0.6 O. 0.9

Frequency (min-1)

Figure 3.16. The relative amplitudes of subharmonics of output ofa hematite system at pH's 4

and 10.

107

pH 4_ 2f

40 -B- 3f~ --- 4f~ 35 -+- Sf... ---+-- 6fc:(I) 30c:00-c: 250u!!! 20"0(I)0-en 15(I)

.~

1ii 10~

5

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1.5 _ 2f

-B- 3f

~--- 4f-+- Sf

1: ---+-- 6f(I)c:8.c:0u!!!...u(I)0-en

0.5(I)

.~...ltJ

~

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Frequency (min-1)

Figure 3.17. The relative amplitudes of subharmonics of output of a kaolinite system at pH's 4

and 10.

108

O,----------,ep;:::::::r:;-----,----,----,-----,-----,------,---,------,

-0.05Q)

-g:g -0.1~~

~ -0.15..l::tC1lCl.3 -0.2

-0.25

o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

-2

-4

Q) -6UlC1l

..l::a..

-8

-10

-12

0 0.1 0.2 0.3 0.4 0.5 0.6

Frequency f (min-1)

0.7 0.8

Figure 3.18. The frequency response of a gibbsite system at pH 10. The fitted curves were fitted

using model (18) and (20) derived from the Two-Plane Model.

109

O'--~I""'i'=h=:---.----,----,-------,-------,-------.----,-------,

-0.05

Q)

-g~ -0.1fi:E'0E -0.15.r;-.t:nl.3 -0.2

-0.25

o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

-2

-4

Q) -6IIInl.r;a..

-8

-10

-12

0 0.1 0.2 0.3 0.4 0.5 0.6

Frequency f (min-1)

0.7 0.8

Figure 3.19. The frequency response ofa goethite system at pH 10. The fitted curves were

fitted using model (18) and (20) derived from the Two-Plane Model.

110

O.--"""""':=------,-----,---.-------.----r------.---,----,

-0.1

Q)

-g -0.2:!:::c~~ -0.3'0E~ -0.4'EIIICl.3 -0.5

-0.6x

o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

-2

-4

Q) -6IIIIII~

a..-8

-10

-12

0 0.1 0.2 0.3 0.4 0.5 0.6

Frequency f (min-1)

0.7 0.8

Figure 3.20. The frequency response ofa hematite system at pH 10. The fitted curves were


111

0x x

x

-0.02 xx

~ -0.04 x:::l~

r::: x~ -0.06~-0E -0.08..c.-'r:::ca -0.1 x

.9-0.12

-0.14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

O~---,----,----,--------,,-------,----,------,------.,-----,

-2

-4

CIl

lG..c. -6a.

-8

-10

o 0.1 0.2 0.3 0.4 0.5 0.6Frequency f (min-1)

0.7 0.8

Figure 3.21. The frequency response ofa kaolinite system at pH 10. The fitted curves were


112

0"..-----,----,-------,----,---,-----.--,-------,--------,

0.90.80.70.60.50.40.30.20.1-2.5 '--_------'---__-'--_----'-__---'----__"'--_-----L__--L-__'--_----'

o

-2

-0.5CI)

-g-'f:i -1~

'0E£ -1.5'CCtl

~

0

-2

-4

..... -6II)CCtl

-8l~ -10~

a.. -12

-14

-16

-180 0.1 0.2 0.3 0.4 0.5 0.6

Frequency f (min-1)

0.7 0.8 0.9

Figure 3.22. The frequency response of a gibbsite system at pH 4. The fitted curves were fitted

using model (18) and (21) derived from the Three-Plane Model.

113

O,.,..----.----.---------.------.----.--------r---.---..------,

0.90.80.70.60.50.40.30.20.1-2 '---_-----'---__-'--_-----'__-'-__-'--_-----'--__--'-__L--_---'

o

CIl -0.5'B-'cfi~

'0 -1E

.z='Cl'Cl

.3 -1.5

0

-2

-4

-lh -6c:l'Cl'Bl'Cl -8-=-CIl

lG.z= -10a.

-12

-14

-160 0.1 0.2 0.3 0.4 0.5 0.6

Frequency f (min-1)

0.7 0.8 0.9

Figure 3.23 The frequency response of a goethite system at pH 4. The fitted curves were fitted

using model (18) and (21) derived from the Three -Plane Model.

114

0x

-0.5 x

CI)'0 x:::l -1-'c~ x~(,) -1.5'E.&:.-'CCll -2 x

.9 x

-2.5 x xx x

-30 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

-5 ++

_ -10 +I/Jc:: +Cll'CJg. -15 :1-CI)I/JCll

.&:.D.. -20

-25+

-300 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Frequency f (min-1)

Figure 3.24. The frequency response ofa hematite system at pH 4. The fitted curves were fitted


115

Or---=---___._-------,----r-------.-------..----,-----

-0.5

Q) -1-g~ -15i .~o -2'E.&:.~ -2.5C1l

.s -3

+

0.1 0.2 0.3 0.4 0.5 0.6 0.7

+

O~--___._----,.----.,------.--------,------,------___,

-5

-10

_ -15~C1l~ -20-Q) -251(l

.&:.a. -30

-35

-40

0.70.60.50.3 0.4

Frequency f (min-1)

0.20.1-45 L-__-'--__--'- -'--__-"-__----.L ~*'-_ __.J

o

Figure 3.25. The frequency response of a kaolinite system at pH 4. The fitted curves were fitted


116

Chapter 4

Dynamics of Aqueous nitrate and W/OH- Concentrations in Effluent

Solutions from Columns of Variable Charged Minerals

Abstract

The dynamics of ion concentrations in aqueous phase can supply information

about mechanisms and modeling of ion adsorption/desorption reactions. The

concentrations of nitrate and sodium in the influent solutions of column experiments were

designed to change as sinusoidal functions of time while the concentration ofWIaIr was

constant. The concentrations ofW laIr and nitrate in the effluent were monitored by a

pH detector simultaneously with nitrate by an UVIVis detector. Results indicated that for

all the four mineral systems at both pH 4 and pH 10, the dynamic concentration ofW

laIr in effluent were approximately sinusoidal functions of time. The spectral analysis

of dynamic concentration ofW laIr in the effluent solutions indicated that the W IOIr

adsorption/desorption processes were approximately linear. The relationships of

amplitudes and phases between W IOIr and nitrate were linear, and the relationship

between the dynamic concentrations ofH+ IOIr and nitrate in the effluent solutions

deviated from their averages were linear, thus only dynamics of nitrate in the effluent

solution could sufficiently describe the mineral system under the experimental condition.

The possible mechanisms ofW laIr adsorption/desorption under the experimental

condition is the charge balance of the bulk solution while sodium and nitrate are adsorbed

and desorbed by the charged mineral surface.

117

1. Introduction

In the study of adsorption!desorption at the interface ofvariable charge minerals

and water, the charge of the solid phase and ion adsorption at equilibrium is affected by

both the concentration of electrolyte and pH (Uehara and Gillman, 1981; Sverjensky and

Sahai, 1996; Sahai and Sverjensky, 1997). The calculation of surface speciation at

equilibrium was studied by Westall (1979, 1982), Uehara and Gillman (1981), Papelis et

al., (1988), Hayes et aI.(1991), Allison et aI., (1992), Schecher and McAvoy (1992), and

Sahai and Sverjensky (1998). The methods by Uehara and Gillman (1981) and Triple

Layer Models (Sahai and Sverjensky, 1998) are discussed in detail.

Uehara and Gillman (1981) derived an equation to describe relationship ofsurface

charge density 00, pH and counter ion concentration in the equilirium solution n, which

can be simplified as

(1)

where K is a constant related to dielectric constant, absolute temperature, counter ion

valence.

According to the Triple-Layer Model (Yates et aI., 1974; Yates, 1975; Davis et

aI., 1978; Sahai and Sverjensky, 1996, 1997, 1998), the schematic representation of

potential as a function ofdistance from the surface is shown in Figure 2 of Chapter 3.

The surface reactions include adsorption and desorption ofW/OK at the surface O-plane

and~ and L- at the ~-plane by ion-pairs.

> SOH +H;q ~ > SOH;

>SOH ~ > SO- +H;q

118

(2)

(3)

> SOH; +L~q <:::> > SOH; - r

> SO- +M;q <:::> > SO- - M+

(4)

(5)

where the symbol ">" implies that the surface-site is bonded to the underlying bulk

mineral, subscript aq represents a species in aqueous solution, and >SOH2+-L-, >SO--~

stand for ion-pair at the p-plane.

The Uehara and Gillman Equation (1) and the reactions (2) to (6) implied that the

adsorption/desorption ofWIOH', M+, and L- were coupled together. The study of the

relationships among their dynamics will supply information of mechanisms involved and

facilitate the modeling of the systems. In this chapter, the column study was focus on the

effect of dynamic change ofaqueous concentration of sodium nitrate on ion adsorption

while the pH of influent solution was constant.

The objectives of the study included (1) measuring the dynamic concentrations of

N03- and WIOH' in the effluent solutions after the influent solutions with varying

NaN03 concentration and constant pH passing through a column packed with variable

charged mineral, (2) the linearity ofH+IOH' adsorption/desorption process excited by

change ofaqueous NaND3 concentration, and (3) the relationship between the

concentrations ofNaN03 and WIOH' in effluent solutions to propose the possible

mechanisms ofN03- and H+IOH' adsorption/desorption and to validate the model

derivation in Chapter 3.


119

The experimental setup. The HPLC and UVNis detector apparatus was the same

as described in Chapter 3. A pH detector was connected to the outlet ofUVNis detector

to monitor the pH dynamics of the eflluent solution while the nitrate concentration was

simultaneously monitored by the UVNis detector. The pH detector, shown in Figure 4.1,

was used to continuously monitor the pH of low volume flows of eflluent from stainless

steel columns with the experimental minerals packed inside. The pH detector was

composed ofa pH electrode and a flow cell. The flow cell was made ofKynar (PVDF)

with 50 micro liter internal volume. The pH detector was supplied with inlet and outlet

fittings sized for standard 1/16" OD tubing which was connected to the outlet of the

UVNis detector. The Flat Surface Electrode, Sensorex Model 450C, was a combination

pHfReference electrode with a double reference junction design. The reference electrode

was a sealed, gel-filled design and includes a peripheral, semi-porous polyethylene

junction. The small internal volume, 50 micro liter, of the pH detector was achieved by

locating the flat surface pH electrode at the top of the flow cavity. The resulting

rectangular cross-section flow path had no protruding parts which could interfere with a

clean, sweeping flow. The electrode was supplied with a 76 cm (30 inch) cable and with

BNC connectors connected to an Accumet Research pH meter, which was programmed

to collect pH data automatically and continuously in equal time intervals. The RS232 port

of the pH meter was connected to the serial port of a computer by cable, and the pH data

collected by the pH meter was transferred to the computer by a HyperTerminal program.

In order to obtain an appropriate number of sampling points of pH, the sampling intervals

of pH, shown in Table 1, were varied as the period of input signals of sodium nitrate

designed in Chapter 3.

120

Other experimental materials and methods were the same as Chapter 3.

3. Results

In the column experiments, the concentrations of nitrate, sodium and W IOlI' in

the influent solution were termed input nitrate, input sodium, and input WIOlI',

respectively. Similarly, the concentrations of nitrate, sodium and WIOlI' in the eflluent

solution were termed output nitrate, output sodium, and output WIOlI', respectively. The

adsorption!desorption of nitrate at the gibbsite Iwater interface when the input W IOlI'

was at pH 4 were termed nitrate adsorption!desorption process ofgibbsite-nitrate system

at pH 4, and similar for other ions, minerals and input WIOlI' conditions.

3.1 Input H+ or input alI': the dynamical changes ofW or alI' concentrations in the

influent solution.

Input W or input OIr concentrations were constant since the pH of solutions A

and B are equal.

3.2 Output W or output OlI': The dynamical changes ofH+ or Olf concentrations in the

eflluent solution.

The output W or Olf deviated from its average and output N03" deviated from its

average of four mineral systems at pH 4 are shown in Figures 4.2 to 4.5. The output W

or alI' deviated from its average and output N03"deviated from its average offour

systems at pH 10 are shown in Figures 4.6 to 4.9. The output W ofkaolinite system at

pH 4 showed obvious distortion.

121

3.3 The spectral component of output W or aIr.

A FFT transform was applied to the output W or OIr. The relative amplitudes of

subharmonics ofoutput W or aIr were shown in Figures 4.10 to 4.17. For the four

systems at pH 4, the relative amplitudes of subharmonics were less than 13% of

fundamental frequencies except kaolinite as 35%. The relative amplitudes of

subharmonics of systems at pH 10 were less than 10% of fundamental frequencies. Thus,

the system, adsorption/desorption ofW or aIr at mineral/water interface at both pH 4

and pH 10 were approximately linear.

3.4 The relationship of amplitudes and phases of fundamental frequencies between output

H+ or OIr and output nitrate

In order to eliminate the slight difference of pH of solutions A and B among

repeated experiments, the following method was used to adjust the amplitude ofoutput

W and aIr

or

A =A 0.1OHk OHk [On-]

where AHk was the amplitude of the kth fundamental frequency of output W at pH 4,

AOHk was the amplitude of the kth fundamental frequency of output OK at pH 10, while

[W] and [OIr] are the measured concentrations ofW or OIr in solutions A and B

desired to be pH 4 and pH 10, respectively. The relationships between the adjusted

122

amplitudes ofoutput W or OK and the amplitudes of nitrate are shown in Figures 4.18 to

4.21 for systems at pH 4, and in Figures 4.22 to 4.25 for systems at pH 10. The

amplitudes of nitrate (ANo3) and adjusted amplitudes ofW or OK (An Ion) were

regressed with model

A

Anion =bo +b\AN03 '

and the coefficients are shown in Table 2.

The relationships ofphases between output W or OK and output nitrate are

shown in Figures 4.26 to 4.29 for systems at pH 4, and in Figures 4.30 to 3.33 for

systems at pH 10.

The phases of output nitrate (f/JN03) and those of output W or OK ((jJH/OH) were regressed

with model

and the coefficients are shown in Table 3.

The output W or OK (y) was predicted from

A

Y =An Ion cOS(aJt + (jJn Ion) +Yo

=(bo + b\ *AN03)COS(aJt + Po + P\(jJN03) +Yo

and the output nitrate (x) was predicted from

x =AN03 cos(aJt + (jJN03) + xo

where aJ is the frequency in radians min-·, yo and Xo are the average concentrations of

solutions A and B respectively for H+ or OK and nitrate, and t is time. Since the values

of bo, P. and flo were respectively not significantly different from or very close to 0, 1,0

or -7t , bowas set as 0, P. as 1, and flo as 0 or -7t. Consequently the aqueous

concentrations deviated from average status by

123

where

The aqueous sodium concentration (z) deviated from average status (zo) was calculated

according to charge balance:

{

z -Zo = (1- y)(x- xo) for pH 4

z-zo =(1+y)(x-xo) jorpH10

The ratios of output nitrate deviated from its average concentration, output sodium

deviated from its average concentration, and output W or OIr deviated from its average

concentration, are shown in Table 4. These linear relationship between ion concentration

deviated from their average concentrations indicated that only one ion concentration was

necessary to describe the dynamic of the ion adsorption/desorption processes at the

mineral/water interfaces.

4. Discussion

The net zero charge points of pH (plIo) of the four minerals from Sverjensky and

Sahai (1996) are shown in Table 5. A possible interpretation of the negative signs of the

ratios (Y -Yo) in Table 4 is as follows. For systems at pH 10, the surface charge isx-xo

negative, and the adsorption of sodium and negative adsorption of nitrate from bulk

solution into the diffuse layer and the p-Iayer results in the decrease of OIr to balance

charge ofbulk solution; on the contrary, desorption of sodium and negative desorption of

124

nitrate from bulk solution into the diffuse layer and the f3-layer results in the increase of

OIr to balance charge of the bulk solution. For gibbsite and goethite systems at pH 4,

the surface charge is positive, and the adsorption of nitrate and negative adsorption of

sodium from bulk solution into the diffuse layer and the f3-layer results in the decrease of

W to balance charge of the bulk solution; on the contrary, the desorption of nitrate and

negative desorption of sodium from bulk solution into the diffuse layer and the f3-layer

results in the increase ofW to balance charge ofbulk solution. For hematite and

kaolinite systems at pH 4, the reason why the sign of the ratio (Y -Yo JiS positive is notx-xo

clear.

5. Conclusion

When the sodium nitrate concentration in influent solution was varying as

sinusoidal functions of time and the pH was constant of 4 or 10, the output H+ or OH-

were also sinusoidal functions of time with obvious distortion for the kaolinite system at

pH 4. The spectral analysis of output W or OIr showed that the mineral systems at both

pH 4 and 10 were approximately linear. The relationships ofamplitudes and phase

between output W or OIr and output nitrate were linear, and the relationship between the

ion concentrations deviated from their averages were linear. The dynamics of nitrate was

sufficient to describe the mineral systems, and it supplied the reason of modeling the

mineral system without taking sodium, W IOIr into account in Chapter 3. The possible

mechanism of the dynamics ofWIOIr concentration in the effluent solution may be due

125

to adsorption and desorption of sodium and nitrate and charge balance of the bulk

solution.

References

Allison, J.D., D.S. Brown, and KJ. Novo-Gradac. 1991. MINTEQA2IPRODEFA2, a

geochemical assessment model for environmental systems: version 30 user's

manual, Environmental Research Laboratory, Office ofResearch and

Development, U.S. Environmental Protection Agency, Athens.

Hayes, KF., G. Redden, W. Ela, and J. Leckie. 1991. Surface complexation models: An

evaluation of model parameter estimation using FITEQL and oxide mineral

titration data. J. Colloid. Interface Sci. 142:448-469.

Papelis, C., H.K F., and L.J. O. 1988. HYDRAQL: A program for the computation of

chemical equilibrium composition of aqueous batch systems including surface

complexation modeling of ion adsorption at the oxide/solution interface.

Technical Report No. 306, Stanford Univ.

Sahai, N., and D.A. Sverjensky. 1997. Solvation and electrostatic model for specific

electrolyte adsorption. Geochimica et Cosmochimica Acta 61 :2827-2848.

Sahai, N., and D.A. Sverjensky. 1997a. Evaluation ofintemally consistent parameters for

the triple-layer model by the systematic analysis ofoxide surface titration data.

Geochim. Cosmochim. Acta 61:2801-2826.

Sahai, N., and D.A. Sverjensky. 1998. Geosurf: A computer program for predictive

modeling ofadsorption on surfaces from solution. Computers Geosci. 24:853

873.

126

Schecher, W.D., and D.C. McAvoy. 1992. MINEQL+: a software environment for

chemical equilibrium modeling. Computers, Environment and Water Systems

11:65-76.

Sverjensky, D.A., and N. Sahai. 1996. Theoretical prediction of single-site surface

protonation equilibrium constants for oxides and silicates in water. Geochim.

Cosmochim. Acta 60:3773-3797.

Uehara, G., and G. Gillman. 1981. The mineralogy, chemistry, and physics of tropical

soils with variable charge clays. Westview Press, Boulder, Colorado.

Westall, I.C. 1982. FITEQL. A computer program for determination of chemical

equilibrium constants, Version 2.0. Report 82-01 Chemistry Department, Oregon

State University, Corvallis,OR, USA.

127

Table 1. Sampling intervals ofpH for different periods of signals.

Period ofInput Signals of Sodium Nitrate Interval of pH sampling

(min) (sec)

1.2 2or3

1.4 20r3

1.6 20r3

1.8 20r6

2 2 or 6

2.2 20r6

2.4 6

3.0 6

4.8 6 or 9

7.2 6 or 12

12.8 12

128

Table 2. The regression coefficients of the linear relationship between adjusted W or

OK amplitudes and those of nitrate. 1

Slope b l Intercept bo

Estimates 95% Confidence Interval Estimates 95% Confidence Interval

pH Mineral Upper Lower Upper Lower

Bound Bound Bound Bound

Gibbsite 0.4081 0.3831 0.4332 -0.0001 -0.0010 0.0008

Goethite 0.4335 0.4200 0.4471 -0.0012 -0.0017 -0.0007

4 Hematite 0.3284 0.3100 0.3467 -0.0033 -0.0109 0.0043

Kaolinite 0.2664 0.2406 0.2922 -0.0011 -0.0023 -0.0000

Gibbsite 0.2611 0.2551 0.2672 -0.0008 -0.0012 -0.0004

Goethite 0.2592 0.2478 0.2706 -0.0012 -0.0020 -0.0005

10 Hematite 0.2610 0.2532 0.2688 -0.0016 -0.0021 -0.0011

Kaolinite 0.2551 0.2428 0.2674 -0.0023 -0.0032 -0.0015

129

Table 3. The regression coefficients of the linear relationship between W or OIr phases

and those ofnitrate.2

Slope PI Intercept Po

Estimates 95% Confidence Interval Estimates 95% Confidence Interval

pH Mineral Upper Lower Upper Lower


Gibbsite 1.0141 1.0123 1.0159 -3.1582 -3.1812 -3.1352

Goethite 1.0059 0.9915 1.0203 -3.0860 -3.2728 -2.8992

4 Hematite 1.0046 0.9914 1.0178 0.1947 -0.0905 0.4799

Kaolinite 0.9905 0.9603 1.0207 -0.0897 -0.6831 0.5038

Gibbsite 1.0122 1.0022 1.0222 -3.0070 -3.1366 -2.8773

Goethite 1.0057 0.9913 1.0201 -3.0872 -3.2740 -2.9004

10 Hematite 1.0115 0.9983 1.0247 -3.1903 -3.3676 -3.0130

Kaolinite 1.0137 1.0044 1.0230 -3.1735 -3.2815 -3.0655

130

Table 4. The ratios ofconcentration deviation from average

pH4 pH 10

Mineral H+ Na+ H+ OH- Na+ OH--- -- -- -- -- --NO; NO- Na+ NO; NO; Na+

3

(Y- Yo J (~J (Y- Yo J (Y- Yo J (z-zo J (Y- Yo Jx-xo x-xo z-zo x-xo x-xo Z-Zo

Gibbsite -0.408 1.408 -0.290 -0.261 0.739 -0.353

Goethite -0.434 1.434 -0.303 -0.259 0.741 -0.350

Hematite 0.329 0.671 0.490 -0.261 0.739 -0.353

Kaolinite 0.266 0.734 0.362 -0.255 0.745 -0.342

Table 5: The pHo from Sverjensky and Sahai (1996)

Mineral pRo

Gibbsite 8.1

Goethite 8.8

Hematite 8.3

Kaolinite 5.1

131

970277 electrode +970282 flow cell

flat pH glass&HOPE referencejunction

1/4-28

flangless tube fittingfor 1/16" 00 tubing

Figure 4.1. The schematic design of a pH detector

132

Compression nutto hold electrodeinto flow cell

CPVC body

Black epoxybody

PVOF Flow cellinternal volumeapprox 40uL

0.06 r------,------,------,---------,,-------,------,------,

0.04

-::E 0.02Sc:o~ 0cQ)

g8 -0.02

T=12.8 min

FH+l~

-0.04

14012010080604020-0.06 '--------'----'--------'--------'-------'--------''--------'

o

0.03 r---------r--------,--------,------,------,

" T=2.4 min

0.02

~ 0.01c:o~ 0cQ)(,)8-0.01

-0.02

252010 15Sampling sequence

5-0.03 '-- ~ -'---- -"----- _____L _____'

o

Figure 4.2. Typical dynamic concentrations ont" and N03- in the effluent solution fromthe gibbsite system at pH 4 with designed highest period 12.8 minutes and lowest period2.4 minutes

133

~~

0.06 ,------.--------r-----,~....,_____,_____--_._--____,_---___,__--__,

T=12.8 min

0.04

I 0.02

co~ 01:Q)

g8 -0.02

-0.04

-0.06 '--__-'--__---l... L--__---'----__--"- --l.--__---'

o 20 40 60 80 100 120 140

0.01 ,-----,----,-----r----.----,------,-----.--------,,.------,

T=1.6 min./

0.005

~Sco~ 01:Q)(,)

§u

-0.005

45403515 20 25 30Sampling sequence

105-0.01 '--_--'-__-'--_----..l__-'-__L--_-----'--__--'--__'--__

o

Figure 4.3. Typical dynamic concentrations ofH+ and N03- in the efiluent solution fromthe goethite system at pH 4 with designed highest period 12.8 minutes and lowest period1.6 minutes

134

0.1 ~---.-------,r-------,-------,------,----~---,

T=12.8 min

0.05

i'S§~ 01:CDg8

-0.05

70605040302010-0.1 '-----__-L-__-----' ---"---__-----' ---"---__-----''-__--'

o

X 10-3

4.-------,--------.-------.------.-------.------,

3T=1.4 min

2

i'S1c::::o~ 01:CD

~ -1

-2

-3

302510 15 20Sampling sequence

5-4'----- .l.....- .l.....- ...L.- --L- ---'-- ----'

o

Figure 4.4. Typical dynamic concentrations ofH+ and N03- in the eflluent solution fromthe hematite system at pH 4 with designed highest period 12.8 minutes and lowest period1.4 minutes

135

0.1

T= 12.8 min

0.05......~

Sc0

~ 0c:CI)(,)c8

~.05 :

~.10 10 20 30 40 50 60 70

X 10-3

8,--------,------::-::-:-0--------,---------.----------,

6T= 2.4 min

4

oi 2co

:;::;~c: -2CI)(,)

<3 -4

-6

-8


5-10 '----------'------'-------'---------'------'

o

Figure 4.5. Typical dynamic concentrations ofH+ and N03- in the eflluent solution fromthe kaolinite system at pH 4 with designed highest period 12.8 minutes and lowest period2.4 minutes

136

0.1 ,-------,------:-~____=_--,__--_,_--__r---,_--_____,

T=12.8 min /

0.05

~S 0co~t:~ -0.05c8

-0.1

70605040302010-0.15 '-------'-----'-----'-----------"----------'------'--------'

o

0.015 .------,-------,---,------,-----,---,-----,----,----,-------,

T=1.0 min

0.01

~ 0.005""""co~ 0t:Q)o§ -0.005u

-0.01


642-0.015 '---------'-------'----'-----'-------'----'---------'-----'-----'------'

o

Figure 4.6. Typical dynamic concentrations ofIr and N03- in the eflluent solution fromthe gibbsite system at pH 10 with designed highest period 12.8 minutes and lowest period1.0 minutes

137

0.1 ,-----.------:-="'"""1--=,----.---------,----,-------,,-------,

T=12.8 min,/

0.05

oi'Sc:o~C~ -0.05c:8

-0.1

70605040302010-0.15 L-__---'-----__--L L-__-L-__--L -L-__-----I

o

0.02 ,__---,-----r---.--~--.,_-_____._--~-~--,__-___,

0.015T=1.0 min

0.01

~S 0.005c:o~ 0cQ)

g -0.005

8-0.01

-0.015.. '

-0.02 L-_-'-_--.-.J'--_---'-----_----l.__--'-----_--'-__L-_--L.__L-__

o 2 4 6 8 10 12 14 16 18 20Sampling sequence

Figure 4.7. Typical dynamic concentrations ofW and N03- in the effluent solution fromthe goethite system at pH 10 with designed highest period 12.8 minutes and lowest period1.0 minutes

138

0.15 r------.------,---,------,-------r---.,.------,

0.1

_ O.OS~

Sc 0o~~ -o.OSg8 -0.1

T=12.8 min ,/FOHl~

-0.15

70605040302010-o.2'-----'---------'------'-----'---------'-----l..-...-----'

o

0.015

T=1.0 min

0.01

-~ O.OOSSc0

~ 0...1:Q)(,,)

<3 -0.005

-0.01

-0.0150 2 4 6

"


16 18 20

Figure 4.8. Typical dynamic concentrations ofIr and N03" in the eflluent solution fromthe hematite system at pH 10 with designed highest period 12.8 minutes and lowestperiod 1.0 minutes

139

0.1 ,------,------:-=""--r-.,-----,---------,---------,.---...,.----------,.'..

T=12.8 mi.n···

0.05

70605040302010-0.15 '--__---'--__-----'-- -'--__-'-__--'- --L-__-----'

o

0.04 ,-------,--------,------,-------,-------,T=1.2 min

0.03

0.02

~..s 0.01t:o~ 0cQ)

§ -0.01

o-0.02

-0.03 "


5-0.04 '----------'--------'------'----------'--------'

o

Figure 4.9. Typical dynamic concentrations ont" and N03- in the eflluent solution fromthe kaolinite system at pH 10 with designed highest period 12.8 minutes and lowestperiod 1.2 minutes

140

oLB~::::±:J==~±::s~~~±::::=I=======L~0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Frequency f (min·1)

Figure 10. Spectral components of output W of the gibbsite system at pH 4

141

0.70.60.50.20.1

6

71---,---=--.----.------,------,-------,---;:=~:::;l

-- 2f---e-3f--- 4f-+- Sf--+- Sf

O'--------'-----'-'-------"-----"------'---------'------'-------.J

o

Figure 4.11. Spectral components of output W ofthe goethite system at pH 4

142

14-.- 2f-B-3f

--- 4f12 -+- Sf

--+- Sf

~ 10UI

~'0§III

~ 8If)

~

~6:t:!

is.E0<CD

j 4

2

oL~~~~=====::===2:':JL--L-_-L-_-----l.-_~o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Frequency f (min-1)

Figure 4.12. Spectral components of output Olf of the hematite system at pH 4

143

30

351----r---,---r-----r----,------.-------r-;::=~:::;_]

-.- 2f-e-3f-..- 4f-+- Sf-+- at

~25...~o

~~20II)

'0

~:!!! 15CoE<Q)

110

5

0.450.40.350.2 0.25 0.3

Frequency f (min-1)

0.150.10"--__---'----__------' --'--__-----'-- -'---__--'- -'--__-'0.05

Figure 4.13. Spectral components ofoutput OIr of the kaolinite system at pH 4

144

0.90.80.70.4 0.5 0.6

Frequency f (min-1)

0.30.20.10'-----'----'----'----'-----'---'-----'-------'----'--------'-----'o

9-.- 2f

-e-3f8 --- 4f

-+- Sf-+- Sf

7

~..l:! 6'0

i~ 5II)

'0

.g 4""Q.E«CD 3~.!!l.

2

Figure 4.14. Spectral components of output OH- of the gibbsite system at pH 10

145

0.1 0.2 0.4 0.5 0.6

Frequency f (min-1)

0.7 0.8 0.9

Figure 4.15. Spectral components of output Off of the goethite system at pH 10

146

OL-_~__-::-'-::-_------:c'-----''''-----:-''---=----'--~_---L-__-'--_---:-'------:_---'--__-'o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Frequency f (min-1)

Figure 4.16. Spectral components of output Olf of the hematite system at pH 10

147

8r---,---,---,----,----,---------,---,------,----.-------,

0.90.80.70.60.4 0.5

Frequency f (min-1)

0.30.20.1

----- 2f-e-3f----- 4f-t- Sf-4-- Sf

OL-__.L-__.L-__-'--__--'-----__--'-----__--l..-__---l-__---l.-__--I

o

7

Figure 4.17. Spectral components of output OIr of the

148

0.025.--------,------,------r-----r-----.------,

+

0.02 +

~

~ 0.015

'0Ql-g~ 0.01E«

+ +

+

o.oos y =0.4081 * x - 0.0001, R2 =0.9802

OL...:..------'-----'------'---------'----------"-----------.Jo 0.01 0.02 0.03 0.04 0.05 0.06

Amplitude of NO; (mM)

Figure 4.18. The linear relationship between the amplitudes of output N03- and output Irof the gibbsite system at pH 4

0.03

0.025

0.02-~-Z 0.015'0Ql

"'C 0.01:::l-=a.E« O.OOS

+0 +

+

+

-f

y = 0.4335 * x - 0.0012, R2 = 0.9932

~.005 '--__---' --'- -----'-- -----"--- --'- --l

o 0.01 0.02 0.03 0.04 0.05 0.06


Figure 4.19. The linear relationship between the amplitudes ofoutput N03- and output Wof the goethite system at pH 4

149

0.4 r-----.---,------,---,----------,----,----,---,------,----,

+0.35

0.3

:E' 0.25.§.

":z:'0 0.2

~:t:e

f 0.15

0.1

0.05

y =0.3284 • x - 0.0033, R2 =0.9773

+ +

+

+

+

+

+

0.2 0.3 0.4 0.5 0.6

Amplitude of NO; (mAtI

0.7 0.8 0.9

Figure 4.20. The linear relationship between the amplitudes of output N03- and output Wofthe hematite system at pH 4

0.025 ,..-----,-------,----,----,----,--------,-------,--------,------,

+0.02

i'~ 0.015

'5~::I~ 0.01

~

0.005+

+

++

y=0.2664*x -0.0011, R2 =0.9722

+

O'--------'--------"------'----'-----'-----------'--------"---------'---------lo 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09


Figure 4.21. The linear relationship between the amplitudes of output N03- and output Wofthe kaolinite system at pH 4

150

0.03 ,-----.-------r--,---,-------,--,-----,------,---,-------,

+0.025 +

-t. +

! 0.02

Io'0 0.015

CD

~iE-

0.01«

0.005

++

+

y = 0.2611 * x - 0.0008, ~ = 0.9951

O'----'-------'---L------'------L__-"-----_-L-_-----l__-'--_-----'0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11


Figure 4.22. The linear relationship between the amplitudes ofoutput N03" and outputOIr of the gibbsite system at pH 10

0.03 ,----,------,----,------,-------r--,-~____._-___,--__._-___,

y =0.2592 * x - 0.0012, R2 =0.9830

0.025

! 0.02

Io'0 0.015

CD

~:t::Q.E 0.01«

0.005

++

+ +

++

+

++

0'----'-------'---L------'------L---"-----_-L-_--l.__...L-_-----'

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11


Figure 4.23. The linear relationship between the amplitudes of output N03- and outputOIr of the goethite system at pH 10

151

0.03 ,------.-----,--,----.------,---,------,-----,---,------------,

0.025

! 0.02

J:o'0 0.015Q)

~-~E 0.01«

++ ++

++

+

+

o.oosy = 0.2610 * x - 0.0016, R2 = 0.9920

0'------'--------'---'----'--------'-----'----'-------'----'-----'0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11


Figure 4.24. The linear relationship between the amplitudes ofoutput N03- and outputOIr of the of hematite system at pH 10

0.03 .-----,---,----------r--....,---~------.--_._--,_-_____,

0.025 +++

+y =0.2551 * x - 0.0023, R2 =0.9799

~ 0.02

J:o'0 0.015Q)

~:t::c..E 0.01«

+

+

++ +

+

+++

o.oos -t

0.110.10'----'----'--------'----'----'-----'-----'--------'-------'0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09


Figure 4.25. The linear relationship between the amplitudes of output N03- and outputOIr of the kaolinite system at pH 10

152

-5,-----,-------,--------.---------,,------.------.----,

-10

'W -15l'Cl

i.;.."'i:: -20'0Q)

~ -25D.

-30

y = 1.0141·x -3.1582, ~= 1.0000

-35 L- ...l.- -'- -----"-- --' -'----- -'

-30 -25 -20 -15 -10 -5 0

Phase of NO; (radians)

Figure 4.26. The linear relationship between the phases of output N03- and output W ofthe gibbsite system at pH 4

oc------.-----~----_,__----~-----,

y =1.0059' x - 3.0860, R2 =0.9982

-5

-10

~..15!.5 -15

+'a

ja.

-20 +

-25

+

-30 L- -'-- -'----- -'-- -'- ----'

-25 -20 -15 -10 -5 0


Figure 4.27. The linear relationship between the phases of output N03- and output Wofthe goethite system at pH 4

153

y =1.0046 * x + 0.1947, R2 =0.9987

0

-5

-10

-~.!! -15

I:c -20'0Q)

HI -25.ca.

-30

+-35

-40-40 -35 -30 -25 -20 -15


-10 -5 o

Figure 4.28. The linear relationship between the phases ofoutput N03- and output W ofthe of hematite system at pH 4

-5 r-----.-------,--------,-------,-----.,-----,

-10

I -15

l":I:: -20

+

'0Q)

IIIf -25y = 0.9905 * x - 0.0897, R2 = 0.9972

-30

-5-10-15-20-25-30-35 '-- '-- L-- -L- ...L- --'----- ---'

-35


Figure 4.29. The linear relationship between the phases of output N03- and output W ofthe kaolinite system at pH 4

154

O,--------r-----,-------,----------,-------,

-5

'W -10ctl

I5-15'0Q)

~ -20a..

-25

y = 1.0122 * x - 3.0070, ~ = 0.9991

-30 "-- -----'-- ---'--~ l__ __'_ __'

-25 -20 -15 -10 -5 0


Figure 4.30. The linear relationship between the phases of output N03- and output OH- ofthe gibbsite system at pH 10

0.----------,-----,-------,---------,----------.

-5

-~ -10ctl

I5-15'0Q)

lQ.s: -20a..

-25

y = 1.0057 * x - 3.0872, R2 = 0.9982

-30 "-- -----'-- ---'-- l__ __'_ --'

-25 -20 -15 -10 -5 0


Figure 4.31. The linear relationship between the phases of output N03- and output OK ofthe goethite system at pH 10

155

-5.--------,------,-----,---------,------,.1-----,

-10

'Wca

1-15

~o'0CD -20

~a..

-25

y=1.0115*x -3.1903, R2 =0.9985

-30 '--- --L ----'-- -'----- ---'- ----'

-25 -20 -15 -10 -5 0


Figure 4.32. The linear relationship between the phases of output N03- and output OlI" ofthe of hematite system at pH 10

O.--------,------,-----,--~-___,----_,

-5

'W -10ca

1a-15

'0CDl/I

1! -20a..

-25

y = 1.0137 * x - 3.1735, R2 = 0.9993

-30 '--- --'- --'---- L-- --L --'

-25 -20 -15 -10 -5 0


Figure 4.33. The linear relationship between the phases of output N03- and output OlI" ofthe kaolinite system at pH 10

156

Chapter 5

Kinetics of Phosphorus AdsorptionlDesorption at the MineralfWater


Abstract

Surface complexation models, such as the Two-Plane Model and the Three-Plane

Model, for describing the mechanisms of phosphate (P) adsorption at mineral/water

interface are based on the results from equilibrium status, and need validation from

kinetic study. However, the currently available methodology for kinetic study is not able

to describe the phosphate adsorption and desorption occurring simultaneously as a

reversible reaction. The objective of this chapter is to propose and test a novel technique

to study kinetics ofP adsorption/desorption at the mineral/water interface and identify the

possible mechanisms using a system identification approach. In the column experiments,

a set of input signals, sinusoidal dynamically changing concentration ofP in influent

solution, were delivered through a system, a column within which the solute transport

and adsorption/desorption occurred at the mineral/water interface, and the output

signals, dynamically changing concentration ofP in eflluent solution, were obtained.

The spectral analysis of the input signals indicated that the input signals were dominated

by only one single frequency. The spectral analysis of the output signals demonstrated

that all four systems at both pH 4 and 10 were approximately linear. The Three-Plane

Model was the proper model for P adsorption/desorption for all four mineral systems at

pH 4 and for gibbsite and goethite at pH 10. The Two-Plane Model was the proper

model for P adsorption/desorption for hematite and kaolinite at pH 10. The reason why

157

mechanisms differed from minerals and pH levels may be due to the competition between

electrostatic repelling ofP from solid phase and specific adsorption ofP onto solid phase

at pH 10. The specific adsorption ofP onto hematite was negligible, and the

adsorption/desorption ofP had the same behavior as nitrate at pH 10 except that the rates

for P were less than those for nitrate. The specific adsorption ofP onto kaolinite was not

negligible, however, electrostatic repelling may dominate the specific adsorption. The

specific adsorption ofP onto gibbsite and goethite were significant and thus, P existed in

the p-Iayer at pH 10. The P sorption isotherms at different pH conditions were reported

in Chapter 7.

Introduction

The kinetic study of adsorption/desorption ofP at a variable charge mineraVwater

interface is important for environmental chemistry, surface chemistry, and plant nutrition.

The adsorption ofP on metal oxides has been of interest in soil science due to the

concern with a mechanism ofP retention in soils. The mechanisms ofP adsorption onto

mineraVwater interface are focused on the number of planes and the structure of the

planes in the surface complex.

Davis and Lackie (1980) have proposed the Triple-Layer Model (TLM): a surface

plane for potential-determining ions and a specific adsorption plane (p-plane) for counter

ions that bond weakly by the forming of an ion-pair surface complex. The schematic

representation of potential as a function of distance from the surface according to Davis

et at. (1978) is shown in Figure 2 of Chapter 3. This model is suitable for weakly bonded

158

ions, and it is used for modeling the kinetics of nitrate adsorption/desorption at

mineraVwater interfaces. The version ofTLM proposed by Hayes and Leckie (1986a, b)

allows both surface coordination and ion-pair complex in to the model structure. Bowden

et at (1980) introduced a four-plane model in which the additional plane was added to

allow stronger bonding oxyanions and metals to form charged complexes at a plane

slightly farther away from the surface than the proton or hydroxide ions but closer than

counter ions. The model location ofan adsorbed ion and its net charge depend on both

the ion size and its relative binging affinity, instead ofjust the relative binding affinity.

The model is used to study P adsorption onto goethite and the effect of ionic strength and

adsorption density (Barrow et at, 1980; Bolan and Barrow, 1984).

Ifthe P adsorption mechanism is electrostatic attraction onto the net positively

charged mineral surface, then P is adsorbed at f3-plane in the TLM ofDavis and Lackie

(1980). This mechanism is supported by the kinetics of the adsorption-desorption ofP on

the y-Alz03 surface using the pressure-jump technique (Mikami et at, 1983). If the P

adsorption mechanism is displacement of an aquo (Breeuwsma et at, 1973; Huang, 1975)

or hydroxyl group (Rajan et aI., 1974; Rajan, 1975) from a metal oxide surface by P, then

P is adsorbed at the f3-Plane in the TLM ofHayes and Lackie (1986a, b). The mechanism

ofdisplacement of the aquo from a metal surface by P and the location of adsorbed P in

the f3-plane was excluded by the kinetics of the adsorption-desorption ofP on the y-Alz03

surface using the pressure-jump technique (Mikami et aI., 1983). The disadvantages of

the pressure jump method are (1) it cannot study the adsorption and desorption

simultaneously (2) it cannot measure the ions in the aqueous phase separately, instead of

159

conductivity, mix information of all the ions. Thus, alternative method for kinetic study

is still needed to validate the proposed mechanisms.

The experimental design of system identification approach is to excite the system

with input signal, varying concentration ofP in bulk solution as sinusoidal functions of

time, so that the P adsorption and desorption will repeat in turn. The corresponding

dynamic ofP in effluent solution to the input signal is called output signal. The

relationship between the input and output signals is called frequency response. The

frequency responses with a line offrequencies of input signals can be used to identify the

models derived from different surface complexation models, and models with best fit and

reasonable parameter estimation are identified as the proper model and the corresponding

surface complexation model is assumed to be proper mechanism.

The objectives of this Chapter are (1) to design the input signals, and measure the

output signals, (2) to analyze the linearity of the system, (3) to derive mathematical

models, transfer function, from different surface complexation models, (4) to identify the

proper transfer function and estimate the parameters including the adsorption and

desorption coefficients.


Chemicals: Sodium phosphate monobase (NaH2P04) and sodium phosphate

dibase (Na2HP04) were analytical reagent grade. A saturated NaOH solution of 120g in

100 ml water was prepared, and stored for one month. Diluted NaOH was fresh made

160

before use. 2% RN03was certified A.C.S. reagent. The water was ofnanopure quality

and degassed by boiling.

Minerals: Gibbsite, goethite, hematite, and kaolinite were from Ward Science®

Inc. They were ground and wet sieved with deionized water, and a fraction of325 to 500

mesh was collected and freeze-dried. X-ray diffraction analysis showed that the goethite,

kaolinite, and hematite contained quartz, and that the gibbsite was mostly gibbsite. The

weights ofminerals, the water contents, and the dispersion coefficients and water velocity

estimated by a system identification approach using acetone as an inert tracer are listed in

Table 1. The columns were sequentially washed with 1 mMNaOH for around 8 hours, 1

mA1RN03 for around 8 hours, and water for around 1 hour at flow rate of4 m1 min-I.

The columns were washed with 1.25 ml L-1 acetone solution for around 30 hours at flow

rate of4 m1 min-I. The columns were washed with pH 4.00 RN03 0.1 mMNaN03 for

around 30 hours, and washed with pH 10.00 NaOH 0.1 mMNaN03 for around 30 hours.

Experimental setup: The column was connected to the outlet ofHPLC with an 3

meter 1/16" ill tubing. An pH detector was attached to the outlet of column, and an

fraction collector was connected to the outlet of the pH detector. Four liters ofwater

were adjusted to pH 4.0 with 2% RN03 or to pH 10.0 with diluted NaOH solution.

Solution A, 4 mg kg- l P solution was made with 2 liters of the solution, and the rest 2

liters was solution B. The change of pH of solution A after adding NaH2P04 or Na2HP04

was negligible.

Design of input signals. Twelve input sinusoidal signals with periods 60, 40,

25.6,20, 12.8, 10, 7.2, 4.8, 3.6, 3.0, 2.4, 2.2, 2.0, 1.8, 1.6, 1,4, 1,2 minutes, were designed

by using combinations ofgradient tables and event tables as those in the transport

161

experiment with acetone as inert tracer. The sampling intervals for input and output

signals of a certain period were listed in Table 2. The method for P measurement is by

Murphy and Riley (1962).

Experiments for studying P input. The change ofP concentration in the influent

solution with time is termed P input. The input signal design at the gradient controller of

HPLC with period Tis

2:rcos(-t) +1

A% = T *100.2

(1)

The input signals for exciting the transport and sorption/desorption process in the column

were measured without column connected to the setup.

Experiments for studying P output. The change ofP concentration in the eflluent

solution with time is termed P output. The column was connected between HPLC outlet

and the fraction collector. The experiments included four minerals, two pH levels (pH 4

and pH 10), with three replicates.

3. Mathematical models and algorithm for parameter estimation

3.1. The transfer functions for systems oftransport and different reactions.

Transfer functions were used to describe the system adsorption/desorption process

at the mineraVwater interface based on various surface complexation models similar to

those derived in Chapter 3.

1. The elementary reaction path is assumed as follows

162

L~( Kd

Ka) L

p (2)

where L8Jl and Lp are P in aqueous solution and J3-plane, respectively, and the constants Ka

and Kd are the adsorption and desorption rates ofP from aqueous solution to the J3-plane.

The overall transfer function is

in which the core transfer function to describe the reactions is

G(s) =S(1 + Ka Js+Kd

where s is the Laplacian operator, L is the length of the column (25 cm), D is the

dispersion coefficient, and V is the velocity of the water listed in Table 1. When the

system is under sinusoidal excitation, s =JOJ , where OJ is the frequency of input P in

radians/minute.

2. The elementary reaction path is assumed as follows

(3)

(4)

L~( Kdl

Ka2 )

L p (5)

where Ld is diffuse layer, and the constants Kal and Kdl are respectively the adsorption

and desorption rates ofP from aqueous solution to diffuse layer, and Ka2 and Kd2 are

respectively the adsorption and desorption rates ofP from diffuse layer to J3-plane. The

core transfer function is

163

(6)

When Ka) »OJ and Kd) »OJ, the core transfer function can be reduced to

(KKd JG(s)=s l+K) + ) zs+Kdz

with

K =Ka)) Kd

)

3.2 Parameter estimation ofthe transfer function

The algorithm developed in Chapter 2 is used to estimate the adsorption and

desorption coefficients of the transfer function while the dispersion coefficient D and

water velocity V are constants.

4. Results

4.1 The input signals in time-domain

For the comparison among experiments, define the relative concentration as

(7)

(8)

2(x - CB) , where x is the input P and CA and CBare respectively the concentrations ofPCA -CB

in solution A and B in an experiment. The time sequence of input P with period of25.6

minutes is shown as the curves labeled as "No column" in Figures. 2-5.

4.2 The Spectral component of the input signals

164

For an input P with frequencyI = liT, its spectral components with frequencies j,

2/, 3j, ...were calculated with fast Fourier Transform (FFT) algorithm in MATLAB®.

Their amplitudes were noted as Af, A2f, ... , A6f. The relative amplitude of a sub-

harmonic spectrum component with frequency /fto with its fundamental frequencylis

(9)

where 1=2,3, ... ,6. The changes of curves Rlfwithlof input P are shown in Figure 6. All

the relative amplitudes of sub-harmonics were less than 5%, and hence they were

considered negligible. Thus, the input signals were viewed as sinusoids with a single

dominant fundamental frequency.

4.3 Variance analysis of input signals among repeated experiments

The CVs ofthe input signals among four repeated experiments were less than

5.2%, thus, the amplitudes of input P generated were repeatable.

The phase f/J of fundamental frequencyI of input P, originally obtained from

MATLAB®, was within the interval [-x, x] and hence not continuous. The phase f/Jwas

modified as ((J to meet the requirements of continuity with frequencyf and passing

through the origin. The modification formula is

((J =f/J - 2m,

where n is a positive integer. In this chapter, the "phase" is referred to the modified

values ((J instead of those f/J within [-x, x]. The CVs of phases of three repeated

experiments were less than 2.4% when -5.05<qJ<-0.54, and less than 8.8% when-0.54 <((J

<0.14. The input signals generated were repeatable.

165

4.4 The output P and their spectral analysis.

Similar to input P, the relative concentration defined by 2(x - CB) , where x isCA -CB

output P excited by input P, and CA and CB were respectively the concentrations ofP in

solutions A and B, respectively, was used to describe the output P. The time sequences

of output P with frequency of25.6 minutes are shown in Figure 2 to 5 when the column

was packed by gibbsite, goethite, hematite, and kaolinite and the pH of solutions A and B

were 4 or 10. The spectral analysis is similar to the input signal in Section 4.2, and the

changes of relative subharmonic amplitudes with fundamental frequencies are shown in

Figures 6-13. The maximum relative amplitudes of subharmonics are given in Table 3.

The maximum relative amplitudes of subharmonics of systems at pH 4 were greater than

those at pH 10. The four systems at both pH 4 and pH 10 were all treated as linear

system, i.e., only the spectral component of fundamental frequency is taken into account.

The phases of output P calculated by FFT were within [-1t, 1t], and not continuous

with frequencies and passing through origin. They were modified as those of input P and

the modified phases were used in this chapter and Chapter 6.

4.5 Model selection and parameter estimation.

The model selection procedure was first equation (4), then equation (7), finally

the equation (6). If any parameter estimate was not significantly different from zero, then

the model selection may be stopped. The model selected for gibbsite and goethite

systems at both pH 4 and pH 10 was equation (6), for hematite and kaolinite systems at

pH 4 was equation (7), and for hematite and kaolinite systems at pH 10 was equation (4).

166

The original data and curve fitting of one of the three repeated experiments are shown in

Figures 5.10 to 5.17 as example. The parameter estimate for the three repeated

experiments are shown in Tables 5.4 to 5.11, and the averages and the standard

deviations are shown in Table 5.12. The ratios of adsorption and desorption rates Kaj,

Kdj , Ka2, Kd2 were also shown in Table 5.12.

5. Conclusions

The designed input signals were dominant of only one single frequency. The

output signals were approximately sinusoidal functions of time. Spectral analysis

indicated that the P adsorption/desorption process at the variable charge mineraVwater

interface at pH 4 and pH 10 was approximately linear. The Three-Plane Model appears

to be a better model than the Two-Plane model for P adsorption/desorption at

mineraVwater interface for all the four minerals at pH 4 and gibbsite and goethite system

pH 10, while Two-Plane Model for hematite and kaolinite system at pH 10.

Reference

Barrow, N.J., J.W. Bowden, A.M. Posner, and J.P. Quirk. 1980. An objective method for

fitting models of ion adsorption on variable charge surfaces. Australian Journal of

Soil Research 18:37-43.

Bolan, N.S., and N.J. Barrow. 1985. Modeling the effect of adsorption of phosphate and

other anions on the surface charge ofvariable charge oxide. Journal of soil

Science 35:273-281.

167

Bowden, lW., S. Nagarajah, N.J. Barrow, AM. Posner, and lP. Quirk. 1980. describing

the adsorption ofphosphate, citrate, and selenite on a variable charge surface.

Australian Journal of Soil Research 18:49-60.

Breeuwsma, A, and l Lyklema. 1973. physical and chemical adsorption of ions in the

electrical double layer on hematite (alpha-Fe203). Journal of Colloid Interface

Sciences 43:437-448.

Davis, lA, and lO. Leckie. 1980. Surface ionization and complexation at the

oxide/water interface. 3. Adsorption ofanions. Journal of Colloid and Interface

Science 74:32-43.

Hayes, KF., and l Leckie. 1986a. Modeling ionic strength effects on cation adsorption at

hydrous oxide/solution interfaces. Journal of Colloid and Interface Science

115:564-572.

Hayes, KF., and l Leckie. 1986b. Mechanism oflead ion adsorption at the

goethite/water interface. in Geochemical Processes at Mineral Surfaces, lA

Davis and K F. Hayes, Eds. ACS Symposium Series No. 323, Chapter 7,

American Chemical Society, Washington, D.C.

Huang, C.P. 1975. Journal ofColloid Interface Sciences 53:178.

Mikami, N., M. Sasaki, K Hachiya, RD. Astumain, T. Ikeda, and T. Yasunaga. 1983a.

kinetics of the adsorption-desorption of phosphate 0 the r-Al203 surface using the

pressure-jump technique. l ofPhysical Chemistry 87: 1454-1458.

Murphy, l, and H.P. Riley. 1962. A modified single solution method for the

determination ofphosphate in natural waters. Anal. Chim. Acta 27:31-36.

168

Rajan, S.S.S. 1975. Adsorption of divalent phosphate on hydrous aluminum oxide.

Nature 262:45-46.

Rajan, S.S.S., K.W. Perrott, and W.M.H. Saunders. 1974. Identification of phosphate

reactive sites of hydrous alumina from proton consumption during phosphate

adsorption at constant pH values. Journal of soil Science 25:438-447.

169

Table 5.1. Weights ofminerals packed, water contents, dispersion coefficients and water

velocities estimated from the experiments of acetone transport.

Mineral

Gibbsite

Goethite

Hematite

Kaolinite

Weight packed

(g)

26.7700

41.9400

31.6700

30.7400

Water content

(%)

54.0744

51.4883

59.9236

49.0418

170

Water velocity

9.4232

9.8965

8.5034

10.3902

Dispersion

coefficients

D (cm2 min-1)

0.9472

0.5021

1.0100

0.3879

Table 5.2. Sampling intervals for input P and output P of a certain period.

Period ofInput and Output Sampling Interval Number of Sample per

P (second) Period

(min)

60 120 30

40 96 25

25.6 48 32

20 48 25

12.8 24 32

10 24 25

7.2 24 18

4.8 12 24

3.6 12 18

3.0 6 30

2.4 6 24

2.2 6 22

2.0 6 20

1.8 6 18

171

Table 5.3. Maximum of relative amplitudes of subharmonics for mineral systems and pH

levels

Mineral Maximum Relative amplitude of subharmonics

(%)

pH4 pH 10

Gibbsite 25 13

Goethite 18 10

Hematite 27 15

Kaolinite 41 10

172

Table 5.4: Parameter estimates and 95% confidence interval ofgibbsite system at pH 4

Repeated Parameter Estimate 95% Confidence Interval

Experiment Lower Bound Upper Bound

Kal23.3356 20.5862 26.0851

#1 Kd1

4.2551 3.7755 4.7346Ka2

0.0597 0.0531 0.0664Kd2

0.0449 0.0206 0.0693Kal

22.6942 20.2937 25.0947#2 Kd1

5.004 4.4892 5.5187Ka2

0.0796 0.0742 0.085Kd2

0.0317 0.0146 0.0488Kal

28.8383 25.054 32.6227#3 Kd1

4.9492 4.3231 5.5754Ka2

0.056 0.0492 0.0629Kd2

0.0587 0.0354 0.082

173

Table 5.5: Parameter estimates and 95% confidence interval ofgibbsite system at pH 10



Kaj9.6658 8.6149 10.7167

#1 Kdj2.2604 1.9281 2.5927

Ka20.1455 0.114 0.1769

Kd20.1894 0.1427 0.236

Kaj9.5868 7.4649 11.7087

#2 Kdj2.167 1.6174 2.7166

Ka20.1137 0.0774 0.15

Kd20.1209 0.0708 0.171

Kaj12.5076 10.4677 14.5476

#3 Kdj2.6807 2.1954 3.1659

Ka20.1072 0.0853 0.1291

Kd20.1181 0.0859 0.1503

174

Table 5.6: Parameter estimates and 95% confidence interval ofgoethite system at pH 4



Kaj27.9688 25.2016 30.736

#1 Kdj5.3724 4.8828 5.8619

Ka20.0787 0.0708 0.0866

Kd20.064 0.0437 0.0842

Kaj27.2585 23.4324 31.0847

#2 Kdj5.5731 4.8342 6.3119

Ka20.0844 0.0723 0.0966

Kd20.0671 0.0415 0.0927

Kaj28.0109 24.5635 31.4582

#3 Kdj5.6667 5.0235 6.3099

Ka20.0804 0.0709 0.0899

Kd20.0659 0.0424 0.0894

175

1 Table 5.7: Parameter estimates and 95% confidence interval ofgoethite system at pH 10



Kaj4.4997 3.452 5.5473

#1 Kdj5.5745 3.8708 7.2782

Ka20.8277 0.5351 1.1204

Kd20.6697 0.4796 0.8599

Ka]3.7008 3.421 3.9807

#2 Kd]3.4068 3.1186 3.6951

Ka20.3794 0.3401 0.4187

Kd20.2848 0.2464 0.3232

Ka]3.6803 3.3502 4.0104

#3 Kd]3.5406 3.2038 3.8775

Ka20.3511 0.3104 0.3918

Kd20.2587 0.2149 0.3024

2

176

1 Table 5.8: Parameter estimates and 95% confidence interval of hematite system at pH 4



Ka20.2821 0.1615 0.4027

#1 Kd20.2009 0.1421 0.2597

Kr4.6235 3.6803 5.5668

Ka20.2609 0.1855 0.3362

#2 Kd20.2095 0.1676 0.2513

Kr4.8318 4.1996 5.464

Ka20.4925 0.3831 0.6018

#3 Kd20.2463 0.2092 0.2833

Kr3.0957 2.7135 3.4778

2

177

1 Table 5.9: Parameter estimates and 95% confidence interval of hematite system at pH 10

2

3

4



Ka]0.2882 0.2588 0.3175

#1 Kd]0.8097 0.6398 0.9796

Ka]0.4367 0.4009 0.4726

#2 Kd]1.1458 0.9511 1.3405

Ka]0.4225 0.3813 0.4636

#3 Kd]1.1625 0.9302 1.3949

178

1 Table 5.10: Parameter estimates and 95% confidence interval of kaolinite system at pH 4



Ka20.1892 0.1436 0.2348

#1 Kd20.1798 0.1186 0.2411

Kr8.1852 7.0721 9.2982

Ka20.2022 0.1455 0.2588

#2 Kd20.1524 0.0986 0.2061

Kr7.2835 5.975 8.5919

Ka20.248 0.1998 0.2962

#3 Kd20.2121 0.1757 0.2485

Kr7.3874 6.6456 8.1292

2

179

1 Table 5.11: Parameter estimates and 95% confidence interval of kaolinite system at pH

2 10



Ka]0.1907 0.1652 0.2161

#1 Kd]0.551 0.4048 0.6973

Ka]0.2492 0.2159 0.2826

#2 Kd]0.752 0.5066 0.9973

Ka]0.2137 0.1826 0.2447

#3 Kd]0.7877 0.517 1.0584

3

4

180

1 Table 5.12. Average parameter estimates among repeated experiments and the ratios of

2 adsorption and desorption rates.

Mineral pH Ka] Kd] K] Ka2 Kd2 K 2

(Ka]/Kd]J

Gibbsite 424.9560 4.7361 5.2693 0.0651 0.0451 1.4435

Goethite 427.7461 5.5374 5.0107 0.0812 0.0657 1.2360

Hematite 44.1837 0.3452 0.2189 1.5768

Kaolinite 47.6187 0.2131 0.1814 1.1747

Gibbsite 1010.5867 2.3694 4.4682 0.1221 0.1428 0.8553

Goethite 103.9603 4.1740 0.9488 0.5194 0.4044 1.2844

Hematite 100.3825 1.0393 0.3680

Kaolinite 100.2179 0.6969 0.3126

3

181

0.08....-----r-------,---,-------.------,---,-------,

-- No column-+- pH4-e- pH 10

-0.04

0.06

0.04

-0.06

_ 0.02.....:...."5Egg 0~t:Ql(,)c:o(,)

a. -0.02 L +-I-.l--l--"l"'t:::'-0

35302515 20Sampling Sequence

105-0.08 '--__--'---__----'-- L--__--'--__-----L -'--__--'

o

1

2 Figure 5.1. Typical input signal with period of25.6 minutes and output signals of the3 gibbsite system at pH 4 and pH 104

182

0.08 ,--------.-------,----,-------.--------,-------.---------,

0.06

0.04

_ 0.02....~

'0ES§ 0~1:Q)to)c:oto)

l:L -0.02

-0.04

-0.06

-- No column-+- pH4-e- pH 10


105-0.08 L-__--'-__---'- L--__--'---__----'- ...L..-__-----'

o

123 Figure 5.2. Typical input signal with period of25.6 minutes and output signals of the4 goethite system at pH 4 and pH 105

183

0.08 ,-------,-------,---,------,--------,------,--------,

o

0.06

0.04

0.02-....~

~S5

:;:::::;~cQ)u5u

a. -0.02

-0.04

-0.06

--- No column-+-- pH 4-e- pH 10

353025105-0.08 '---__--'--__---'-- L--__---'-__--'- ..L-__---'

o

12 Figure 5.3. Typical input signal with period of25.6 minutes and output signals of the3 hematite system at pH 4 and pH 10

184

0.08 ,-----,------,---.,-----,---------,----,------------,

0.06

0.04

_ 0.02.....~

Ic:o 0~1:(J)()c:o()

a.. -0.02

-0.04

-0.06

--- No column-+- pH4---e- pH 10

353025105-0.08 '--__---'-----__---'- "---__--L-__--L ...L-__---l

o

12 Figure 5.4. Typical input signal with period of25.6 minutes and output signals of the3 kaolinite system at pH 4 and pH 104

185

4.5

5r-----..,-----.----,---------,---------.-----;:======:;_l--- 2f--e-3f-- 4f-+- Sf-t- at

4

1

~rJ 3.5.~

E 3

~III

~ 2.5

~:I:!

~ 2...~! 1.5

0.5

0.1 ~2 ~3 ~4

Fundamental Frequency f (min-I)

0.5 0.6

2 Figure 5.5. Spectral components of input signals

186

30

- --- 2f~ -e- 3fe::..In 25 --- 4f(,)

'c -+- Sf~ ----+- 6f~ 20ell~..c:::lIn- 150Q)'0:::l~

Q. 10EellQ)>i 5

~

00.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11

15

- --- 2f~ -e- 3fe::..In --- 4f(,)

'c -+- Sf§

10----+- 6f

~:::lIn-0Q)'0:::l-'a. 5EellQ)>i~

00.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11

Fundamental Frequency f (min-1)

12 Figure 5.6. Spectral components vs. fundamental frequency jfor gibbsite system at pH 43 and pH 104

187

20

- -- 2f~ -B- 3fIII ---;<- 4f0'2 15 -+- 5f0

E ---+- 6flU.=.~~

III

'0 10CI)

-g~

Q.ElU

5CI)

.::!:«l~

00.02 0.04 0.06 0.08 0.1 0.12 0.14

0.30.250.05O'-----L----.L----_-L- --'-- ---'--- ----'o

10

- -- 2f~ -B- 3fIII 8 --.<- 4f0'2 -+- 5f0

E -+- 6flU

:5 6~

III

'0CI)

-g 4iElUCI)

> 2ii~

12 Figure 5.7. Spectral components vs. fundamental frequency jfor goethite system at pH 43 and pH 104

188

0.0550.050.0450.040.0350.030.0250.02o~~~~~

0.015

30

- - 2f~ --e- 3f~ 25 --- 4f(,)

'2 -+- Sf

~ 20-+- 6f

a:I:§:::lIII"- 150Q)

-g:t:::Q. 10Ea:IQ)

~ 5a:I

~

0.60.50.1

20

- - 2f~ --e- 3fIII --- 4f(,)

'2 15 -+- Sf

~ -+- 6fa:I:§:::lIII"- 100Q)

1:':::l

:t:::Q.Ea:I

5Q)

>i~

12 Figure 5.8. Spectral components vs. fundamental frequency jfor hematite system at pH 43 and pH 10

189

50

- -- 2f~ ~ 3fIII 40 -..- 4f(,)

'2 -t- Sf0

E -+- 6fCll:a 30~

III

'0CD

~ 20:a.ECllCD

i 10

~

00.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

12

-~III 10(,)

'20

E 8Cll:a~III

'0 6CD-g....:a. 4ECllCD.~ca 2~

00 0.1 0.2 0.3 0.4

Fundamental Frequency f (min-1)

-- 2f~ 3f-..- 4f-t- Sf-+- 6f

0.5

12 Figure 5.9. Spectral components vs. fundamental frequencyjfor kaolinite system at pH 43 and pH 104

190

0

-0.5

(1)-1-g-'c

C)

~ -1.5

'0E -2

oJ:-'Ccu~ -2.5

...J +-3

-3.5 l--_---'--_----'--__l--_-'-_----'--__"-----_---L-_----'-__--'--_--l

o 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

O~-,_-__,_--,__-_,_-___,--_,___-____._-__r--_,_____-__,

-1

-2+

'W -3cu'6~-4(1)

l(loJ: -5D..

-7

-8'---------'-----------'-----'--------'---------'----'------"-------'---..L--------'o 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Frequency f (min-1)

12 Figure 5.10. Frequency response of gibbsite system at pH 10 and curve fitting of transfer3 function derived from Triple Layer Model4

191

O~---,-------r---,--------,------r----,------,

-0.5CD-g-'c~ -1

::l!:'0E<5 -1.5'Ccu

.9-2

0.350.30.250.20.150.10.05-2.5 '--__-'-----__--'- -'-----__-L-__----'-_--=_-'--__---.J

o

O.------.---------r---,-------.-------r-----,-------,

-1

-2

'W -3cu'6,g-4

CD

~-5D..

-6

-7

0.350.30.250.15 0.2

Frequency f (min-1)

0.10.05-8'-------'---------'-----"--------'-------'------'------.J

o

12 Figure 5.11. Frequency response of goethite system at pH 10 and curve fitting of transfer3 function derived from Triple Layer Model

192

0

-0.2 xx

x

-0.4 x(I)

"'C.a -0.6·cfi x~ -0.8(,,) x·e -1.c:- x.1::j -1.2

-1.4x

-1.6 xx

-1.80 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

-2

- -4IIIc:al

~ ~-(I)IIIal.c:a. -8

-10

-120

+

0.1 0.2 0.3 0.4

Frequency f (min-1)

0.5 0.6 0.7

12 Figure 5.12. Frequency response ofhematite system at pH 10 and curve fitting of3 transfer function derived from Two Plane Model

193

0

-0.1x

G) -0.2x-g

..-·2 -0.3 xri~to) -0.4

x·e..c:1:: -0.5 x

t'll

~ x

....J -0.6

-0.7

-0.80 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0

-1+

-2 +

-~ -3 +t'll'CJg,-4 +G)Ul +J!! -5a..

.{) +

-7

-80 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Frequency f (min-1)

12 Figure 5.13. Frequency response of kaolinite system at pH 10 and curve fitting of3 transfer function derived from Triple Layer Model

194

Orc---..---,--,.....---,-------,---,-------,-----,--,--------,

-0.5Q)

-g-'c~ -1~

'0E;; -1.5'CtV

.9-2

-2.5 '----_--'--_----'-__'----_---'-----_-----'--__.1.--_---'-_--'-__--'--_-----'

o 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0..------,----------,---,----,-----,--,--------.-----,---.,-----,

-1

-2

_-3~tV:0-4~Q) -5!ll

J::.c..-6

-7

-8

-9 l--_--'--_--'-__'----_---'-----_-----'--__-'----_---'-_--'-__...L.-_-----'

o 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Frequency f (min-1)

12 Figure 5.14. Frequency response of gibbsite system at pH 4 and curve fitting of transfer3 function derived from Triple Layer Model

195

O~---,------,----,---------r----.------,-----,

-0.5

Q)

-g- -1'c~

::E'0 -1.5Es:.-'lij -2

~ +

-2.5

0.140.120.10.080.060.040.02-3L-------'-----------'------'--------'----'--------'----'

o

0

-2

-4-IIIc:lU -0'ClU..;.Q)

-8IIIlUs:.D..

-10

-12

-140 0.02 0.04 0.06 0.08

Frequency f (min-1)

0.1 0.12 0.14

12 Figure 5.15. Frequency response of goethite system at pH 4 and curve fitting of transfer3 function derived from Triple Layer Model

196

0.------=:::;:--,----,--,---.----------,---,-----,-----,---,------,

-0.5

CD

] -1'2fi~o -1.5'E.J::'Cj -2

-2.5

x

-3 L.-_---'-----_--'-__L.-_-'--_-----'-__.l--_----'---_--'-__--'----_--'

o 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

O.,------,------r--.-----,-------,---,------,---------,-----,--------,

-1

-2

+

-6

-7

-8'------'--------'---'----'--------'----'--------'------'-----'--------'o 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Frequency f (min-1)

12 Figure 5.16. Frequency response of hematite system at pH 4 and curve fitting of transfer3 function derived from Triple Layer Model

197

Or-"":"""--.-----,--,...-----,-----,---.,--------,-----.----.------,

-0.5

CD -1-g~ -1.5i~ -2'E~

t -2.5as~~ -3

-3.5

x

x

x

-4'----'--------'----.l------'--------'----L------'------!-----'---_--'o 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Or--,-------,---,-----,--------,--,...-----,-------,---.-------,

-2

-4

++

-8

-10

-12

-140 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Frequency f (min-1)

12 Figure 5.17. Frequency response of kaolinite system at pH 4 and curve fitting of transfer3 function derived from Triple Layer Model

198

Chapter 6

Dynamics of Phosphate and W/OH- Concentrations in Effluent

Solutions from Columns of Variable Charged Minerals

Abstract

The relationship between dynamics ofPhosphate (P) and WIOlI" concentrations

in eflluent solutions in a column experiment will supply information about mechanism

and modeling ofion adsorption/desorption reactions at mineral/water interfaces. This

relationship was studied by the experimental design aiming at system identification, i.e.

the concentration ofP in influent solutions varied as sinusoidal functions of time while

their pH remained constant as 4 or 10. The results showed that concentrations ofP and

W or OlI" in eflluent solutions for the gibbsite, goethite, hematite, and kaolinite systems

at both pH 4 and 10 were approximately sinusoidal functions of time; the spectral

analysis of the dynamic concentration ofW or OlI" in eflluent solutions indicated the

four systems at both pH 4 and 10 were approximately linear. The relationships of

amplitudes and phase between dynamic concentrations ofW or OlI" and P in eflluent

solutions were linear. The relationships between the dynamic concentrations ofW or

Olf and P in eflluent solutions were linear, and this linear relationship indicated that

dynamics ofP could be sufficient to describe the whole system. Based on the linear

relationships, the possible mechanisms ofadsorption/desorption ofP and H+ or OH- may

be the specific adsorption ofP at ~-plane with ligand exchange, the electrolyte adsorption

due to net surface charge, and charge balance of the aqueous solution.

199

1. Introduction

Phosphorus can be adsorbed onto an oxide surface when the surface charge is zero or

even negative. Besides the electrostatic attraction, the specific anion coordination with

the surface metal by a ligand exchange mechanism also occurs (Parfitt et aI., 1975; Rajan,

1976). The generalized ligand exchange reaction for phosphate ions can be written as

follows:

aSOH(s) + H bP04b-3 (aq) +cH+ (aq) <=>

SaHcP04(s) + (a - b)OH- (aq) +bHzO(I)

where S refers to a metal ion in a hydroxylated mineral, OH to a reactive surface

hydroxyl, and b::;3 is the degree of protonation of the phosphate ion (Goldberg and

Sposito, 1985). Surface complexes resulting from ligand exchange contain no water

molecules between the surface Lewis acid site (S) and the adsorbed ion and, therefore,

are referred to as inner-sphere. Studies of the molar ratio of hydroxyl released per P

adsorbed provided evidence of the ligand exchange mechanism (Breeuwsma and

Lyklema, 1973; Rajan et aI., 1974; Rajan, 1975; Rajan, 1976). The hydroxyl release is

not equivalent to anion adsorption due to negative charging process (Hingston et aI.,

1972; Rajan et aI., 1974; White, 1981). Bowden et aI. (1980) introduced a four-plane

model to model the P adsorption and the surface complex. In the four-plane model, the

(1)

additional plane was added to allow stronger bonding ions to reside closer to the surface

but not at the surface.

The surface complexation models so far are based on the study of equilibrium

status ofP adsorption and desorption. However, thermodynamic study cannot supply the

200

information of dynamic process, and thus, the study of mechanisms of surface P

complexes and the formation needs information from kinetic study. There are two

disadvantages in the traditional experimental methodology including batch experiment

and pressure jump method. The first disadvantage is that the P adsorption and desorption

cannot be studied simultaneously. The objective of the traditional methodology is focus

on either adsorption or desorption. The second disadvantage is that the traditional

methodology may not supply accurate information. The quality of sampling is

constrained by the narrow interval of peak or jump of solute concentration in the

breakthrough curve ofbatch or flow method. In the pressure jump method, the ion

concentration cannot be measured directly because the measurement of conductivity is

mix information.

In this Chapter, a novel experimental design was used to study P adsorption and

desorption simultaneously at mineral surfaces. The study included (1) measuring the

dynamic concentrations ofP and H+IOK in the eflluent solutions after applying influent

solutions with varying P concentration and constant pH passing through a column packed

with variable charged mineral, (2) confirming the linearity ofWIOK

adsorption/desorption process excited by change of aqueous P concentration, and (3)

establishing the relationship between the concentrations ofP and H+IOK in eflluent

solutions to proposed possible mechanisms ofP and WIOK adsorption/desorption to

validate the model derivation in Chapter 5.


201

The experimental setup: The setup ofHPLC and UVNis detector were the same

as Chapter 5. The pH detector discussed in Chapter 4 was connected to the outlet of the

column. The effluent flow may be disturbed when the solution passes through the flow

cell, i.e., the solution sampled by the fraction collector may be different from that

collected when the pH detector was connected within the outlet of column and the

fraction collector. In order to eliminate the possible sampling disturbance, the pH

detector was detached from the outlet ofthe column, and the fraction collector was

directly connected to the outlet of the column.

Sample collection: In order to get the appropriate number of sampling points of

pH, the sampling intervals ofpH are shown in Table 6.1.

For the study ofgibbsite, goethite, and hematite systems at both pH 4 and pH 10,

the pH sampling and solution sampling were carried out in the same experiment. The

procedure ofpH and solution sampling were as follows.

Step 1: connect the pH detector to the outlet of the column whiling running an

input signal.

Step 2: monitor the pH changes by graphing the pH dynamics.

Step 3: after the pH curve varied periodically and the peaks and valleys did not

changed with time, start pH sampling at the beginning of the input signal at controller.

The sampling intervals were the same as in Chapter 4. Three periods were sampled, and

the average was used for data analysis.

Step 4. Detach the pH detector from the outlet of the column, and connect the

fraction collector to the outlet ofcolumn.

202

Step 5. Sample the solution with vials at different intervals as Table 5.2 in

Chapter 5. One period was sampled, and the solutions were stored in 5°C for Plater

analysis.

Step 6. Switch to another input signal, and repeat steps 1-5.

For the study of the kaolinite system at both pH 4 and pH 10, the pH sampling

and solution sampling were carried out as separate experiments. P sampling was started

after running an input signal for 2 hours. The procedure for pH sampling was as follows.

Step 1: Connect the pH detector to the outlet of the column.

Step 2: Run an input signal, and monitor the pH changes by graphing the pH

changes.

Step 3: After the pH curve varied periodically and the peaks and valleys did not

change with time, the pH sampling began at the beginning of the input signal. The

sampling intervals are shown in Table 5.2 in Chapter 5. Three periods were sampled, and

the average was used for data analysis.

Step 4. Switch to another input signal, and repeat steps 1-3.

Three repeated experiments were carried out in time, and the average was used for

data analysis.

The procedure for pH sampling was as follows.

Step 1: Connect the fraction collector to the outlet of the column.

Step 2: Run an input signal, and after three periods or two hours, sample one

period length of the P solution.

Step 3: Switch to another input signal, and repeat steps 1-2.

203

Three repeated experiments were carried out in time, and the average was used for

data analysis.

3. Results

In the column experiments, the concentrations ofP and WIOff in the influent

solution were termed input P and input WIOff, respectively. Similarly, the

concentrations ofP and WIOff in the eflluent solution were termed output P and output

WIOff, respectively. The adsorption/desorption ofP at the gibbsite Iwater interface

when the input WIOff was at pH 4 was termed P adsorption/desorption process of

gibbsite-nitrate system at pH 4, and similar for other ions, minerals and input WIOff

conditions.

3.1 The dynamic changes ofH+ or Off concentrations in the influent solution.

Input W or input Off were constants since the pH of solution A and B were

equal.

3.2. The dynamic changes ofP and W or Off concentrations in the eflluent solution.

The output P and W or Off ofgibbsite, goethite, hematite systems at pH 4 and

pH 10 are shown in Figures 6.1 to 6.6. The output P and W or Off of kaolinite systems

at pH 4 and pH 10 are shown in Figures 6.7 to 6.8. In Figure 6.7, the output W of

kaolinite system at pH 4 showed obvious distortion.

3.3 The spectral component of output H+ or output Off.

204

A spectral analysis of the output H+ or output OKwas carried out using a FFT

transform. The relative amplitudes of subharmonics of output W or output OK are

shown in Figures 6.9 to 6.16. The maximum relative subharmonic amplitudes ofmineral

systems at both pH 4 and pH 10 are shown in Table 6.2. The systems were all

approximately linear.

3.4 The relationship ofamplitudes and phases of fundamental frequencies between output

W or OK and output P.

In order to eliminate the slight difference of pH of solutions A and B among

repeated experiments, the following method was used to adjust the amplitude of output

WorOK

and

A =A 0.1OHk OHk [OH-]

(2)

(3)

where AHk is the amplitude of the f(h fundamental frequency of output W at pH 4, AOHk is

the amplitude of the f(h fundamental frequency of output OK at pH 10, [W] and [OK]

are the measured concentrations ofW and OK of the solutions A and B designed as pH

4 and pH 10, respectively. The relationships between the adjusted amplitudes of output

W or OK and the amplitudes of phosphate are shown in Figures 6.17 to 6.24 for systems

at pH 4, and pH 10. The amplitudes of nitrate (Ap ) and adjusted amplitudes ofW or OK

(AH/OH ) were regressed with model

205

(4)

The regression coefficients are shown in Table 6.3.

The relationships of phases between output H+ or aIr and output P are shown in

Figures 6.25 to 6.32 for systems at pH 4 and pH 10. The phases of output P (({Jp) and

those of output W or aIr (((JHIOH) were regressed with model

The regression coefficients were shown in Table 6.4.

The output P (x) was predicted from:

x =Ap cos(OJt + (fJp) + Xo

where OJ is the frequency in radians min-I, and Xo are the average concentration ofP in

solutions A and B.

The output W or aIr (y) was predicted from:

y =AH10H cos(OJt + (fJH 10H) +Yo

=(ao + a]Ap)cos(OJt + Po + fl](fJp) + Yo

whereyo is the average concentrations ofH+ or OH- in solution A and B. Since the

values of ao, fli and flo are not significantly different from or very close to 0, 1,0 or -7t ,

(5)

(6)

(7)

ao is set as 0, PI as 1, and flo as 0 or -7t. Then, the relationships ofaqueous concentrations

deviated from their averages according to the following:

where

(8)

if flo =0

if flo =-1((9)

The ratios of output P and output W or OIr, deviated from the average status are shown

in Table 6.5.

206

4. Discussion

A possible interpretation ofthe signs of the ratios of concentration oflr or OIr

and P deviated from their averages in eftluent solutions in Table 6.5 is as follows.

For systems at pH 10, the net surface charge is negative, and sodium and P should

be adsorbed from aqueous solution into the diffuse layer and the J3-layer, and there should

be negative adsorption of nitrate from bulk solution into the diffuse layer and the J3-layer

should result in the decrease of OIr in aqueous phase to balance charge of the bulk

solution. On the contrary, desorption of sodium and P from the J3-layer to aqueous

solution should result in the increase of OIr in aqueous phase to balance charge of the

bulk solution.

For gibbsite and goethite systems at pH 4, the net surface charge should be

positive, and the adsorption of nitrate and negative adsorption of sodium should be from

the bulk solution into the diffuse layer and the J3-layer, and adsorption ofP from aqueous

solution to the J3-plane should result in the decrease ofIt to balance charge of the bulk

solution. On the contrary, the desorption of nitrate from the diffuse layer and the J3-layer

into aqueous solution, and desorption ofP from the J3-plane to the bulk solution should

result in the increase ofIt to balance charge of the bulk solution. Besides the

electrostatic attraction, there may be release of OIr due to the adsorption ofP (parfitt et

at, 1975) at the J3-plane

(10)

207

For hematite and kaolinite systems at pH 4, the reason why the signs are positive

is not clear.

5. Conclusion

The output W or OIr were sinusoidal functions of time. The spectral analysis of

output W or OIr showed that gibbsite, goethite, hematite, and kaolinite system at pH 4

and 10 were approximately linear. The relationships ofamplitudes and phase between

output W or OIr and output P were linear, and the relationships between concentrations

ofW or OIr and P in effluent were linear. These linear relationships indicated that the

system identification ofP only was valid.

References

Bowden, J.W., S. Nagarajah, N.J. Barrow, AM. Posner, and J.P. Quirk. 1980. Describing

the adsorption ofphosphate, citrate, and selenite on a variable charge surface.

Australian Journal of Soil Research 18:49-60.

Breeuwsma, A, and J. Lyklema. 1973. Physical and chemical adsorption of ions in the

electrical double layer on hematite (alpha-Fe203). Journal ofColloid Interface

Sciences 43:437-448.

Goldbrg, S., and G. Sposito. 1985. On the mechanism of specific phosphate adsorption

by hydroxylated mineral surfaces: a review. Communications in soil science and

plant analysis 16:801-821.

208

Hingston, F.J., A.M. Posner, and J.P. Quirk. 1972. Anion adsorption by goethite and

gibbsite. 1. The role of the proton in determinating adsorption envelopes. Journal

of soil Science 23: 177-192.

Parfitt, RL., RJ. Atkinson, and RS.C. Smart. 1975. The mechanism of phosphate

fixation by iron oxides. Soil Science Society of America Journal 39:837-841.

Rajan, S.S.S. 1975. Adsorption ofdivalent phosphate on hydrous aluminum oxide.

Nature 262:45-46.

Rajan, S.S.S. 1976. Changes in net surface charge of hydrous aluminum oxide. Nature

262:45-46.

Rajan, S.S.S., K.W. Perrott, and W.M.H. Saunders. 1974. Identification ofphosphate

reactive sites of hydrous alumina from proton consumption during phosphate

adsorption at constant pH values. Journal of soil Science 25:438-447.

White, RE. 1981. Retention and release ofphosphate by soil and soil constitutes. In.

P.B. Tinker (ed.) Soils and Agriculture. Halsted Press, New York.

209

Table 6.1. Sampling intervals of pH for different periods of signals

Period ofInput Signals of Sodium Nitrate Interval ofpH sampling

(min) (sec)

1.2 2 or 3

1.4 2 or 3

1.6 2or3

1.8 2or6

2 2or6

2.2 2or6

2.4 6

3.0 6

4.8 6or9

7.2 6 or 12

10 6orl2

12.8 6or12

20 6or12

25.6 6 or 12

40 12

60 12

210

Table 6.2. The maximum of relative subharmonic amplitudes ofmineral systems and pH

4 and 10.

Mineral Maximum of relative subharmonic amplitude (%)

pH4 pH 10

Gibbsite 20 9

Goethite 13 7

Hematite 31 16

Kaolinite 44 12

211

Table 6.3. The regression coefficients and their 95% confidence intervals oflinear

relationships between adjusted amplitudes ofW or OlI" (AH IOH ) and amplitudes ofP

(Ap ), AHloH =ao +a)Ap

Mineral pH <Xl <X.o

Estimate 95% Confident Interval Estimate 95% Confident Interval

Lower Upper Lower Upper


Gibbsite 41.3196 1.206 1.4333 0.0002 -0.0022 0.0026

Goethite 41.1429 1.1084 1.1773 -0.0006 -0.0012 0.0001

Hematite 40.6827 0.5725 0.7928 -0.0004 -0.0018 0.001

Kaolinite 40.4955 0.4384 0.5526 -0.0048 -0.016 0.0065

Gibbsite 100.2893 0.2519 0.3267 -0.0002 -0.0007 0.0004

Goethite 100.4223 0.4075 0.4372 -0.0003 -0.0007 0.0001

Hematite 100.4332 0.3983 0.4682 -0.0001 -0.0017 0.0014

Kaolinite 100.3848 0.3732 0.3965 -0.0406 -0.0477 -0.0336

212

Table 6.4. The regression coefficients and their 95% confidence intervals oflinear

relationships ofphases between W or OIr (f/JHIOH) and P (cpp), ((JHIOH =Po + p\((Jp

Mineral pH ~1 ~o

Estimate 95% Confident Interval 95% 95% Confident Interval

ConfidentLower Upper Lower Upper

IntervalBound Bound Bound Bound

Gibbsite 40.9846 0.9667 1.0025 -3.205 -3.3111 -3.099

Goethite 41.0026 0.9994 1.0057 -3.2893 -3.3149 -3.2636

Hematite 40.9787 0.9092 1.0482 -0.0197 -0.5918 0.5524

Kaolinite 41.0271 0.9419 1.1123 -0.4204 -1.0584 0.2175

Gibbsite 100.9832 0.9434 1.023 -3.0004 -3.2523 -2.7485

Goethite 101.014 0.9715 1.0565 -3.3179 -3.5692 -3.0667

Hematite 101.0144 1.0044 1.0243 -3.0839 -3.1631 -3.0047

Kaolinite 101.0717 1.0547 1.0887 -3.3761 -3.491 -3.2612

213

Table 6.5. The ratios ofconcentration ofP in effluent solution deviated from its average

(x-xo) and concentration ofW or OK in effluent solution deviated from its average (Y-Yo)

for mineral system at pH 4 and pH 10

y- Yo

Mineralx-xo

pH4 pH 10

Gibbsite -1.32 -0.29

Goethite -1.14 -0.42

Hematite 0.68 -0.43

Kaolinite 0.50 -0.38

214

0.04

0.03 T=60 min ~-t- P

~0.02

Sc 0.010

~c 0Q)(,,)c0o -0.01

-0.02

-0.030 5 10 15 20 25 30


X 10-3

8

6

4 T=10 min

~ 2Sc 00

~-2C

Q)(,,)c -400

-6

-8

-100 5

Figure 6.1. The concentrations ofIr and P deviated from their averages in the effluent ofthe gibbsite system at pH 4, T is the period ofP concentration in the influent solution

215

0.04

0.03 1-: Off I-+- PT=40 min

~0.02

Sc: 0.010

~1: 0CD(,)c:

8 -0.01

-0.02

-0.030 5 10 15 20 25

Sampling Sequence

x 10.3

4

3T=10 min

2

~S1c:0

~ 0-c:CD§ -1t>

-2

-3

-40 5 10 15 20 25

Sampling Sequence

Figure 6.2. The concentrations of Olf and P deviated from their averages in the effluentofthe gibbsite system at pH 10, T is the period ofP concentration in the influent solution

216

0.03

T=40 min-- H+

0.02 -+- H2P04

~ 0.01Sc:0:; 0!!!CQ)(,)c:8 ~.01

~.02

~.030 5 10 15 20 25

X 10-3

4

3 T=7.2 m

2

~S 1c:0:; 0l!!-c:Q)

g -1

8-2

-3

-40 2 4 6 8 10 12

Sampling Sequence14 16 18

Figure 6.3. The concentrations ofW and P deviated from their averages in the effluent ofthe goethite system at pH 4, T is the period ofP concentration in the influent solution

217

0.04

0.03 1=12.8 min 1-:- Off I-t- P

0.02~S 0.01c0

~ 0...1:

CDg ~.01

8~.02

~.03

~.040 5 10 15 20 25 30 35

1816146 8 10 12Sampling Sequence

X 10-3

8

6 1=3.6 min

4

~S2c0

~ 0-cCD6~2

04

-6

~0 2 4

Figure 6.4. The concentrations ofOIf and P deviated from their averages in the eflluentofthe goethite system at pH la, T is the period ofP concentration in the influent solution

218

0.04

0.03 T=60 min

~-+-p

:E'0.02

Sc 0.010

~1: 0Q)(Jc

8 -0.01

-0.02

-0.030 5 10 15 20 25 30

X 10.34,---------,-----o----r-----;-----,----------,------,------,

3

2

:E' 1S§ 0~1: -1Q)(Jc8 -2

-3

-4T=20 min


5-5'----------'-------'----------'----------'--------l

o

Figure 6.5. The concentrations oflr and P deviated from their averages in the eftluent ofthe hematite system at pH 4, T is the period ofP concentration in the influent solution

219

0.08

0.06 T=25.6 I~:- Off I--t- P

0.04

~S 0.02c:0

~ 0-c:G)

g -0.02

8-0.04

-0.06

-0.080 5 10 15 20 25 30 35

0.015.---------,--------,------,-------,-------,

0.01

~E O.OOS.....,c:oi 0""1:G)(,)

<3 -0.005

-0.01 T=2.2 min

25205-0.015 '-- ----'--- '-- ---'- -'----- ------.J

o

Figure 6.6. The concentrations ofOIf and P deviated from their averages in the eftluentof the hematite system at pH 10, T is the period ofP concentration in the influent solution

220

0.02,--------,-----,------,------,---------,------,

0.015 T=60 min

0.01

~S O.OOSc:o~ 01:Q)

g -0.005

8-0.01

-0.015

30252015105-0.02 L.__ L.__ L- -'-- --'--- ---L- ---'

o

X 10.46,-----,-----------,-----,--------,------,-----------,-----,

4

2

Ic: 0 T=12.8 mino~~ -2o50-4

-6

353025105~L.-----'---------'----'------'-------'------------'------'

o

Figure 6.7. The concentrations onr deviated from their averages in the eflluent of thekaolinite system at pH 4, T is the period ofP concentration in the influent solution

221

O.OS

0.04

0.03 T=25.6 min

:?e 0.02......,c: 0.010

~0c

Q)(,,)8 -0.01

-0.02

-0.03

-0.040 5 10 15 20 25 30 35

0.015 ,----,----,---,----,-------,---,-------.------,---,------,

0.01

:? T=2.0 min

S 0005c: .o~CQ) 0

~-O.OOS

2018168 10 12 14Sampling Sequence

642-0.01 L-==--_--L__L-_----'-_--"-__-'----_-'-_-----"-__...l..-_---'

o

Figure 6.8. The concentrations ofOIr deviated from their averages in the eflluent of thekaolinite system at pH 10, T is the period ofP concentration in the influent solution

222

0.110.10.090.080.05 0.06 0.07

Frequency f (min-1)

0.040.030.02oL----l_----l_------.L_------.L_--L----==::L===::±::c====:::L==~_~0.01

20

5

25r---.-----,-----.-------,----.-----,------,,-----,--~~:::;l

-.- 2f

--e-3f--- 4f-+- Sf--+- Sf

Figure 6.9. Spectral components of output If" of the gibbsite system at pH 4

223

70 i-----=---,------,---,---,-----,--,-------,---,----;:r====;l

-- 2f--e-3f--- 4f-+- Sf--+--- Sf

60

10

o~~~~~;c~~0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11

Frequency f (min-1)

Figure 6.10. Spectral components of output H+ of the gibbsite system at pH 10

224

14-.- 2f

-e-3f-- 4f

12 ---+- Sf-+- Sf

~ 10UI

~.~

III

£ 8::JII)

'0

~6~

!c(CD

j4CD

0::

2

0~~~~~S==~0.02 0.04 0.06 0.08 0.1 0.12 0.14

Frequency f (min-1)

Figure 6.11. Spectral components of output W of the goethite system at pH 4

225

8

--- 2f

7--e- 3f-41--+- Sf-+- 6f

~6

~l/I

.~~ 5III

:fi~

(/)

'04-8~

:t:=

13GI

~liIt: 2

0.30.250.15 0.2

Frequency f (min-1)

0.10'---------'--- '-- -1.- '-- ----l

0.05

Figure 6.12. Spectral components of output W of the goethite system at pH 10

226

0.050.0450.040.03 0.035

Frequency f (min-1)

0.0250.02

35---+-- 2f--B-- 3f

-- 4f30 -t- Sf

-+-- 6f

~25...~~IIIi5 20:::J

(J)

~

~!i! 15

tIII

~{j 10n:

5

Figure 6.13. Spectral components ofoutput W ofthe hematite system at pH 4

227

0.50.450.40.350.2 0.25 0.3

Frequency f (min-1)

O'--------'----"----'---~L.L-----'------'-------'---~'----_ ___'____'o 0.1 0.15

18 i ----,-------r--,-----,-----r--,-------,-----,r---;:r===:::;l

----- 2f--e- 3f-.<- 4f-t- Sf-li'- 6f

16

2

4

14

~til

~ 12.~

~ 10l/)

'0

~ 8""IGl 6

!

Figure 6.14. Spectral components of output W of the hematite system at pH 10

228

45,.-------,r------,------,----.-----,----.---------=----,

40

10

5

-.- 2f--e-3f--- 4f-+- Sf-+- Sf

oL~====:======:L======~=_-----l- _ _____L_~0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Frequency f (min-1)

Figure 6.15. Spectral components of output H+ of the kaolinite system at pH 4

229

14---e- 2f-e-3f----- 4f

12 -t- 51-+- Sf

~ 10III

l!.~

~ 8:J

lJ)

'0-8:J

6JGI>.,! 4

2

0.05 0.1 0.15 0.2 0.25 0.3

Frequency f (min-')

0.35 0.4 0.45 0.5

Figure 6.16. Spectral components ofoutput H+ of the kaolinite system at pH 10

230

0.04 ~-------.-----........----.----------,--------."

+

0.035 +

0.03~s:c 0.025'0(I)

~ 0.02'!.Ec( 0.015

+

++

y =1.3196 * x + 0.0002, R2 =0.9926

0.01 +

0.030.0250.010.005 '- ---1- --'- -'---- --'- --'

0.005

Figure 6.17. The linear relationship between adjusted amplitude ofW and amplitude ofP ofgibbsite system at pH 4.

x 10.39,---------,-------,---------,--------,--------.-------,

+8

+7

~e 6...,

as'0~4:t::

~3c(

2+

++

+

+

1 +'-I-

+ Y= 0.2893 * x - 0.0002, R2 = 0.9634

O'-----'--------l------I-------'-------L--------'o 0.005 0.01 0.015 0.02 0.025 0.03

Amplitude of P (mM)

Figure 6.18. The linear relationship between adjusted amplitude ofOlf and amplitude ofP ofgibbsite system at pH 10. The circle points were not included for regression.

231

0.035 r----~---____r_---___._---___r_---,____--___,

0.03 +

~ 0.025

S:c 0.02'0

CD-g 0.015'KE<C 0.01 y =1.1429 * x - 0.0006, ~ =0.9975

o.oos

0'---------'------'-----"----...1..-----'-------1o 0.005 0.01 0.015 0.02 0.025 0.03

Amplitude of P (mM)

Figure 6.19. The linear relationship between adjusted amplitude ofIr and amplitude ofP ofgoethite system at pH 4.

+ ++

+

+

y =0.4223 * x - 0.0003, R2 =0.9953

0.02 ,----,-----,-------.------,----,----,-------,,--------,

0.018

0.016

i' 0.014Si: 0.012o'0 0.01

CD

'B... 0.008~

~ 0.006

0.004

0.002

0'--------'--_----1..-__--'-__-"-__...1..--__l.--_-----''--_---I

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045Amplitude of P (mM)

Figure 6.20. The linear relationship between adjusted amplitude of OIr and amplitude ofP ofgoethite system at pH 10.

232

0.025 r-----,.----.-----,---------,-------,------,

+0.02

~e 0.015.....-:c'0 0.01CD-g-~E 0.005c(

+

+

y = 0.6827 * x - 0.0004, R2 = 0.9502

-O.OOS '-------"------'------'------'-------'---------'o 0.005 0.01 0.015 0.02 0.025 0.03

Amplitude of P (mM)

Figure 6.21. The linear relationship between adjusted amplitude ofH+ and amplitude ofP of hematite system at pH 4.

+

+ ++ + +

+ ++

+ +or+

+ ++

y =0.4332 * x - 0.0001, R2 =0.9552

+

++

+

0.03

0.005

~ 0.025

S5 0.02

'0CD-g 0.015

:t:::Q.

~ 0.01

0.035 r-------,------,---.----,-----r----r---,--------,-----,

OL------'-------'-------'--------'---------'---------'----------'--------'o 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Amplitude of P (mM)

Figure 6.22. The linear relationship between adjusted amplitude ofOIr and amplitude ofP of hematite system at pH 10.

233

0.2

0.18

0.16

:E' 0.14.§.j: 0.12"-0 0.1Q)

-g! 0.08E« 0.06

0.04

0.02 +

+

y =0.4955 * x - 0.0048, R2 =0.9961oC--------'-----'-------'--------'--------'---------'---------'-------'o 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Amplitude of P (mM)

Figure 6.23. The linear relationship between adjusted amplitude ofW and amplitude ofP of kaolinite system at pH 4.

0.3,----,-------r---,------,------r--..,---,------,

0.25

:E'.§.j: 0.2

'0Q)

-g! 0.15E«

0.1y = 0.3848 * x - 0.0406, R2 = 0.9991

0.05 C---__---'-----__-----'-- -'-----__---'----__-----'- ...l--__--'

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Amplitude of P (mM)

Figure 6.24. The linear relationship between adjusted amplitude ofOIf and amplitude ofP of kaolinite system at pH 10.

234

-4

-5

-6

- -7~IV

i -8,:;..

i:'0 -9(I)

!e.r:. -10a..

-11

-12

y =0.9846 * x - 3.2050, R2 =0.9997

-13 l--_-----'-__--'--_--L__--'-__"'---_--'-__--'--__l--_----'

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1Phase of P (radians)

Figure 6.25. The linear relationship of phases between H+ and P ofgibbsite system atpH4.

-2

-3

-4-IIICIV

-5II -60-0(I) -7IIIIV.r:.a..

-8

-9+

++

y =1.0133 * x + 3.0182, R2 =0.9963

-10 L-__....L-__------L L--__---'----__-----'- -"-----__-----'

-12 -11 -10 -9 -8 -7 -6 -5Phase of P (radians)

Figure 6.26. The linear relationship of phases between OIr and P ofgibbsite system atpH 10

235

-4,-----------,----------r----------,

-6

-8

i~ -10-:c'0 -12Q)

lQoJ:£:L. -14

-16

y =1.0026 * x - 3.2893, R2 =1.0000

-18 '-- --'- ---1- -----'

-15 -10 -5 0Phase of P (radians)

Figure 6.27. The linear relationship of phases between H+ and P ofgoethite system atpH4.

+

++

y = 1.0140 * x - 3.3179, R2 = 0.9933

-14 '---_-'--_--'--__'---_---'--_---'-__-'--_-'-_---'-__--L-_-----'

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1Phase of P (radians)

Figure 6.28. The linear relationship ofphases between OIr and P ofgoethite system atpH 10.

236

-2,-----,---,----,--------r------,---,----,------,

-4

-6

i ~I"J: -10'0G)

~ -12~

ll.-14

-16

++ +

+

y =0.9787 * x - 0.0197, R2 =0.9912

-18 L-_--'-__-----'-__-----'-__-----'-__---'--__---'--__---'--__--'

-18 -16 -14 -12 -10 -8 -6 -4 -2Phase of P (radians)

Figure 6.29. The linear relationship of phases between H+ and P of hematite system atpH4.

-4

-6

~-IIIC.~ -10

~"J: -12'0G)~ -14~

ll.-16

-18

+

+y = 1.0144 * x - 3.0839, R2 = 0.9993

-20 "--------'-------'--------'--------'------'-------'-------'------'-16 -14 -12 -10 -8 -6 -4 -2 0

Phase of P (radians)

Figure 6.30. The linear relationship of phases between OIr and P ofhematite system atpH 10.

237

-3r------.-----,-------,------,----r------r-----,-----r----,

-4

-5

~ -6

i -7

"'J: -8'0Q) -9lQit -10 y =1.0271 * x - 0.4204, R2 =0.9980

-11

-12

-13 l--_-----'--__---'-----_------'__--'---__-'----_--l.__-'-__-'--_----.J

-12 -11 -10 -9 -8 -7 -6 -5 -4 -3Phase of P (radians)

Figure 6.31. The linear relationship of phases between H+ and P of kaolinite system atpH4.

-4,-----r-------,-------,-------,-------r----,--------,

-6

~ -8

~......"'J: -10't5Q)

~ -12a..

y =1.0717 * x - 3.3761, R2 =0.9997

-14

-16 '----'__------' --l. ---L------'----------'---------'

-12 -10 -8 -6 -4 -2 0Phase of P (radians)

Figure 6.32. The linear relationship of phases between OIr and P of kaolinite system atpH 10.

238

Chapter 7

The Properties of Minerals

Four minerals, gibbsite, goethite, hematite, and kaolinite were used in the column

experiments in which acetone, nitrate and phosphate were tracers to determine the

processes of convection-dispersion through porous media, nitrate adsorption/dsorption,

phosphate adsorption/desorption at the mineral/water interfaces. The properties of

minerals including X-ray diffraction analysis (XRD), surface area, and phosphorus

adsorption isotherms were reported.

7.1 X-ray diffraction (XRD) analyses

The patterns of XRD of minerals are shown in Figures 7.1-7.4, and the

corresponding peaks are shown in Tables 7.1-7.4. The results showed that the goethite,

kaolinite, and hematite contained quartz.

7.2 Surface areas

The surface areas of minerals were determined by single point ofBET adsorption

ofN2. The surface areas ofgibbsite, goethite, hematite, and kaolinite were 12.02, 9.04,

20.53, 16.91 m2got, at PlPo=0.3106, 0.3093, 0.3052, and 0.3014, respectively.

7.3 Acetone adsorption isotherms

239

Four 0.2 g minerals were added together with 20 ml 0, 0.625 1.250 1.875 ml L-1

acetone in water solutions in centrifuge tubes. The suspensions were gently shaken for

18 hours, and then centrifuged at 3000 rpm for 10 minutes. The acetone concentrations

in the supernatant were analyzed by UV method at 264 nm. The results shown in Table

7.5 indicated that the adsorption of acetone on minerals was negligible.

7.4 Phosphorus adsorption isotherms

7.4.1 Phosphorus adsorption isotherms when minerals were previously adjusted to pH 4.0

by nitric acid.

Five 0.2 g minerals were added together with 40 ml 0.1 M nitric acid in centrifuge

tubes. The suspensions were gently shaken for 12 hours, filtered with 0.2 Ilm nylon

membrane, and the solids were washed back into the tubes by 0.1 M NaOH solution and

completed to 40 ml. These suspensions were gently shaken for 12 hours, filtered through

0.21lm nylon membrane, and the solids were washed back into the tubes by 2.5 ml L-1

acetone in water solution and completed to 40 ml. The suspensions were gently shaken

for 12 hours, filtered through 0.2 Ilm nylon membrane, and the solids were washed back

into the tubes by pH 4.0 RN03 and completed to 40 ml. These suspensions were gently

shaken for 12 hours, and then checked pH. If the pH was less than 3.95 or greater than

4.05, we resumed washing the minerals with 40 ml pH 4.0 RN03solution, otherwise,

filtered through 0.21lm nylon membrane. The solid were transferred to tubes using 30.0

ml pH 4.0 RN03, and then added 0.0, 0.4, 0.8, 1.6, 2.4 ml102.5 mg L-1 P stock solution

to the tubes and completed to weight 40.2 g of the mineral and aqueous solution. Gently

240

shook for 12 hours, and centrifuged at 2000 rpm for 10 minutes. 1 m1 supernatant was

sampled for P analysis by Murphy-Riley method (Murphy and Riley, 1962). The pH

values of the supernatant were also measured.

The relationships between the initial P concentration and that after 12-hour

adsorption are shown in Figure 7.5. The nonlinear plateau-linear model

y =al min(x, xo) +ao was used to describe the relationship between initial aqueous P

concentration (x) and measured aqueous P concentration after 12- hour adsorption (y).

The coefficient estimates are shown in Table 7.6. The relationship between 0.0, al and Xo

are

2ao =-0.7974xo - 0.0274, R =0.9979

and

2al =-O.0670xo +0.9935, R =0.9791.

The sequence ofP adsorption capacity was gibbsite>goethite>kaolinite>hematite.

The pH values after 12-hour P adsorption, shown in Figure 7.6, were greater than

the original pH values due to the release of hydroxide in P adsorption reaction. The

sequence of pH changes was gibbsite>goethite >kaolinite>hematite. The sequence of pH

change was the same as that ofP adsorption capacity, and this result indicated that the

mechanism of releasing hydroxide when P was adsorbed to the mineral surfaces (parfitt

et aI., 1975; Rajan, 1975, 1976).

7.4.2 Phosphorus adsorption isotherms when minerals were previously adjusted to pH 9.6

by sodium hydroxide.

241

The procedure ofP sorption isotherm was similar to P adsorption isotherm

experiments at pH 4.0. Added pH 10.0 NaOH solution together with 0.2,0.4,0.8, 1.2,

1.6 mI KH2P04 solution with concentration 102 mg P Lol to 40 mI. The pH values of the

P solution were not constant due to phosphate hydrolysis. The relationships between

initial aqueous P concentration (x) and measure P concentration after 12 hours (y) are

shown in Figure 7.7, and linear modelsy =f31xwere fitted. The estimated coefficients

are shown in Table 7.6. The sequence ofP adsorption capacity was

gibbsite>goethite>kaolinite>hematite.

The pH values after 12-hour P adsorption are shown in Figure 7.8, and they were

greater than the original pH values due to the release of hydroxide in P adsorption

reaction (Parfitt et aI., 1975; Rajan, 1975, 1976). The sequence ofpH changes was

gibbsite>goethite>hematite>kaolinite. The sequence of pH change was close to that ofP

adsorption capacity, and this result also indicated that the mechanism ofreleasing

hydroxide when P was adsorbed to the mineral surfaces.

References

Murphy, J., and H.P. Riley. 1962. A modified single solution method for the

determination of phosphate in natural waters. Anal. Chim. Acta 27:31-36.

Parfitt, RL., RJ. Atkinson, and RS.C. Smart. 1975. The mechanism ofphosphate

fixation by iron oxides. Soil Science Society of America Journal 39:837-841.

Rajan, S.S.S. 1975. Adsorption ofdivalent phosphate on hydrous aluminum oxide.

Nature 262:45-46.

242

Rajan, S.S.S. 1976. Changes in net surface charge ofhydrous aluminum oxide. Nature

262:45-46.

243

Table 7.1. Peak search report for XRD analysis ofgibbsite-----

SCAN: 2.0/59.99/0.03I2r/m), Cu, l(max)=3524, 11126/0212:44

PEAK: 9-pts/Parabolic Filter, Threshold=3.0, Cutoff=0.1 %, BG=3/1.0, Peak-Top=Summit

NOTE: Intensity = Counts, 2T(0)=0.0(O), Wavelength to Compute d-Spacing = 1.54056A (Cu/K-alpha1)

# 2-Theta d(A) BG Height 1% Area 1% FWHM

1 12.412 7.1251 105 52 1.5 686 3.3 0.336

2 12.669 6.9816 102 79 2.3 812 3.9 0.262

3 14.486 6.1096 98 65 1.9 502 2.4 0.197

4 18.326 4.8371 129 3395 100.0 20637 100.0 0.155

5 20.332 4.3642 158 1398 41.2 9183 44.5 0.168

6 20.569 4.3145 16C 707 20.8 7169 34.7 0.259

7 24.756 3.5934 115 66 1.9 954 4.6 0.369

8 24.950 3.5659 115 45 1.3 969 4.7 0.549

9 25.167 3.5356 115 64 1.9 1385 6.7 0.552---

10 25.403 3.5034 115 99 2.9 1385 6.7 0.357

11 26.575 3.3514 119 206 6.1 1394 6.8 0.173--_.... _....__..

12 26.959 3.3046 117 359 10.6 2636 12.8 0.187

13 28.045 3.1790 122 238 7.0 1781 8.6 0.191

14 28.738 3.1039 114 121 3.6 582 2.8 0.123

15 32.069 2.7886 78 394 11.6 2348 11.4 0.152

16 33.136 2.7013 77 59 1.7 558 2.7 0.241

17 36.472 2.4615 117 309 9.1 2942 14.3 0.243

18 36.678 2.4481 122 537 15.8 4734 22.9 0.225

19 37.109 2.4207 109 157 4.6 1731 8.4 0.281

20 37.704 2.3839 135 669 19.7 4504 21.8 0.172

21 38.363 2.3444 110 102 3.0 525 2.5 0.131

22 39.382 2.2861 90 142 4.2 831 4.0 0.149

23 40.189 2.2420 85 219 6.5 1842 8.9 0.214

24 41.719 2.1632 77 283 8.3 3176 15.4 0.286

25 44.246 2.0454 96 411 12.1 3522 17.1 0.219

26 45.502 1.9918 93 336 9.9 2726 13.2 0.207

27 46.251 1.9613 87 59 1.7 201 1.0 0.087

28 47.421 1.9156 74 233 6.9 2269 11.0 0.248--

29 50.634 1.8013 68 308 9.1 3193 15.5 0.264

30 52.248 1.7494 76 275 8.1 3183 15.4 0.295

31 52.880 1.7300 82 60 1.8 399 1.9 0.170

32 53.999 1.6967 81 53 1.6 743 3.6 0.357

33 54.388 1.6855 83 235 6.9 2906 14.1 0.315

34 55.468 1.6552 86 55 1.6 643 3.1 0.298

35 55.965 1.6417 74 36 1.1 976 4.7 0.691

36 57.941 1.5903 75 72 2.1 668 3.2 0.237

37 58.141 1.5853 78 44 1.3 676 3.3 0.392

38 58.661 1.5725 82 53 1.6 213 1.0 0.102

244

Table 7.2. Peak search report for XRD analysis ofgoethite--

SCAN: 2.0/59.99/0.0312(0/m), Cu, l(max)=862. 11/25/02 14:55

PEAK: 35-pts/Parabolic Filter, Threshold=3.0, Cutoff=1.1%, 8G=3/1.0, Peak-Top=Summit

NOTE: Intensity = Counts, 2T(0)=0.0(0), Wavelength to Compute d-Spacing = 1.54056A (Cu/K-alpha1)

# 2-Theta d(A) 8G Height 1% Area 1% FWHMi---

1 5.567 15.8628 52 161 58.5 4071 100.0 0.645

2 8.182 10.7972 49 100 36.4 1957 48.1 0.499

3 12.800 6.9101 45 51 18.5 2120 52.1 1.060

4 17.747 4.9936 40 60 21.8 1179 29.0 0.501

5 19.905 4.4568 38 42 15.3 604 14.8 0.367

6 21.202 4.1870 37 275 100.0 3936 96.7 0.365

7 22.264 3.9896 36 44 16.0 354 8.7 0.205

8 24.132 3.6849 34 27 9.8 188 4.6 0.178-

9 26.602 3.3481 32 123 44.7 1170 28.7 0.243

10 28.606 3.1179 30 157 57.1 1624 39.9 0.264

11 33.227 2.6941 25 163 59.3 2175 53.4 0.340

12 34.726 2.5812 24 63 22.9 824 20.2 0.334

13 36.050 2.4894 23 63 22.9 1863 45.8 0.754

14 36.705 2.4464 22 189 68.7 2775 68.2 0.374

15 37.485 2.3973 21 56 20.4 1139 28.0 0.519

16 40.100 2.2467 19 44 16.0 622 15.3 0.360

17 41.150 2.1918 18 88 32.0 1676 41.2 0.486

18 42.804 2.1109 16 20 7.3 285 7.0 0.363

19 47.253 1.9220 12 22 8.0 383 9.4 0.444

20 50.748 1.7975 9 37 13.5 597 14.7 0.411

21 53.324 1.7166 6 88 32.0 1762 43.3 0.511

22 54.169 1.6918 5 56 20.4 1323 32.5 0.602

23 56.623 1.6241 3 40 14.5 1167 28.7 0.744

24 57.706 1.5962 2 44 16.0 1013 24.9 0.587

25 59.236 1.5586 0 81 29.5 2170 53.3 0.683

245

Table 7.3. Peak search report for XRD analysis of hematite

~---_._-

-----~---~._--_.- ----_._~._---~--

SCAN: 2. 0/59.99/0. 03/2("/m), Cu, l(max)=2780, 11125/02 16:29

PEAK: 25-pts/Parabolic Filter, Threshold=3.0, Cutoff=1.1 %, BG=3/1.0, Peak-Top=Summit

NOTE: Intensity = Counts, 2T(0)=0.0("), Wavelength to Compute d-Spacing = 1.54056A (Cu/K-alpha1)


1 20.734 4.2804 59 426 15.6 2170 16.4 0.130

2 21.163 4.1947 52 118 4.3 1667 12.6 0.360

3 24.024 3.7012 49 69 2.5 941 7.1 0.348

4 26.540 3.3557 43 2737 100.0 13251 100.0 0.123

5 33.081 2.7057 51 169 6.2 3508 26.5 0.529._.

6 35.543 2.5236 56 320 11.7 2721 20.5 0.217

7 36.443 2.4634 29 214 7.8 2131 16.1 0.254

8 39.361 2.2872 31 161 5.9 778 5.9 0.123

9 40.192 2.2418 39 63 2.3 704 5.3 0.285. --._ ..__.-- __ ··_·_~· _______ ·___ ._"n___·___····

10 40.790 2.2104 30 65 2.4 1492 11.3 0.585

11 42.352 2.1323 26 85 3.1 419 3.2 0.126- ----- _.-_._--..... "_.- -------~-_._-_._------------------ ._-

12 44.310 2.0425 22 137 5.0 689 5.2 0.128

13 45.694 1.9839 21 123 4.5 858 6.5 0.178

14 49.395 1.8435 34 69 2.5 1180 8.9 0.436

15 50.056 1.8207 23 456 16.7 2769 20.9 0.155

16 53.184 1.7208 27 51 1.9 614 4.6 0.307

17 54.013 1.6963 49 61 2.2 1400 10.6 0.585

18 54.774 1.6745 27 145 5.3 1282 9.7 0.225

19 58.944 1.5656 28 24 0.9 229 1.7 0.243

246

Table 7.4. Peak search report for XRD analysis of kaolinite-~-~---------_.__.--. __ . __._-~._-- -- ---~----,-- --------

SCAN: 2.0/52.67/0.0312(o/m), Cu, l(max)=4440, 11125/02 15:28

PEAK: 35-pts/Parabolic Filter, Threshold=3.0, Cutoff=1.1%, BG=3/1.0, Peak-Top=Summit

NOTE: Intensity = Counts, 2T(0)=0.0("), Wavelength to Compute d-Spacing = 1.54056A (Cu/K-alpha1)


1 12.296 7.1926 117 687 16.0 6721 33.1 0.249

2 20.251 4.3815 140 292 6.8 6464 31.8 0.564

3 20.854 4.2561 315 937 21.9 5129 25.2 0.140

4 21.337 4.1607 287 112 2.6 1416 7.0 0.322

5 23.093 3.8483 208 127 3.0 971 4.8 0.195

6 24.863 3.5781 180 686 16.0 7123 35.1 0.265

7 26.630 3.3446 159 4281 100.0 20321 100.0 0.121

8 35.000 2.5616 89 223 5.2 4238 20.9 0.485

9 35.959 2.4954 163 201 4.7 1476 7.3 0.187

10 36.553 2.4562 126 147 3.4 916 4.5 0.159

11 38.419 2.3411 163 322 7.5 3865 19.0 0.306

12 39.445 2.2826 149 297 6.9 2394 11.8 0206

13 40.281 2.2371 105 311 7.3 1514 7.5 0.124

14 42.446 2.1278 89 309 7.2 1316 6.5 0.109

15 45.792 1.9798 90 137 3.2 1698 8.4 0.316

16 46.851 1.9376 92 41 1.0 524 2.6 0.326

17 50.130 1.8182 86 400 9.3 2025 10.0 0.129

247

Table 7.5. The acetone concentration in the supernatant after I8-hour adsorption.

Control Gibbsite Goethite Hematite Kaolinite

(ml r 1) (ml L-1) (ml L-1) (ml L-1) (ml L-1)

0.000 0.000 0.000 0.152 0.000

0.630 0.630 0.608 0.805 0.625

1.256 1.247 1.239 1.410 1.256

1.869 1.865 1.869 2.036 1.874

248

Table 7.6. The coefficients of y =ao +aI max(x,xo)for P sorption isotherm when

minerals were previously adjusted to pH 4.0 and y =PIx for P sorption isotherm when

minerals were previously adjusted to pH 9.6.

Mineral pH al au Xo PI

Gibbsite 4 0.8114 -2.2288 2.7800

Goethite 4 0.9145 -0.8318 0.9307

Hematite 4 0.9838 -0.2070 0.2648

Kaolinite 4 0.9680 -0.3711 0.4502

Gibbsite 10 0.763

Goethite 10 0.941

Hematite 10 0.997

Kaolinite 10 0.952

249

Figure 7.1. X-ray diffraction pattern ofgibbsite 250

d=4.8371

3500

3000

2500

Cil§2000o~>

:1::fI)CQ)

1: d=4.3642

1500

1000d=4l3145

d=2.3839

40

d=2.448

d=2.7888

302-Theta(O)

d=3.3046

2010

d=2.0454

d=1.9918

l..' d=1.80••. g~17....494. . '.'1·:ll'9158-lJ~··'~WM'_. : i ! 1\ •. .

.. ... • -1"'" .. • ,,; ~ . • ,'- , .,~" .,

o I I I I i: liti I!: I j I / , i ]: : ! I! :.] !, ,j i ,1 I I ;! I50

500

Figure 7 .~. X-ray diffraction pattern of goethite 251

350

d=4.1870

300

250

VI-c:5 200

~>:t:II)c:CD-c:

150

d=11.8628

10.7972

d=3.1179

d=3.3481

d=2.4464

d=2.6941

oI! (' lit :1;. I: j: I ' i I l i ,: I I I

d=1.

d=1.7166

50

,WlJI' 'I!~~ riM l'

d:2.1918

40

=2~.,

302-Theta(O)

2010

50

100

Figure 7.3. X-ray diffraction pattern of hematite 252

d=3.3557

2500

2000

,......Vl"E::Ja(,) 1500->-:1::Vlc:Q)-c:

1000

d=1.B207

302-Thpta(O)

20

d=4.2804

10

~,\

~~~~

Figure 7.4. X-ray diffraction pattern of kaolinite 253

4500 d=3_3446

4000

3500-

3000

§'c:g 2500

8>:t:C/lc:<I)

E 2000~

50

I

d=1.818J

d=1.9798

J.'=1.9376 J..\~·'t..~ ...../

40

'~j d.41' d=3.5781d=7.1926

1000

j~j~

30

~~o.,...;.;,;

I I

2-Thetan20

r ro~ I r

10

61r=======~--'---'-------'-------'---"'-:;:~-1o Gibbsitex Goethite+ Hematite... Kaolinite

5

~3c.2

I~2Q.

o 2 3 4

Initial P Concentration (mg L-1)

5 6 7

Figure 7.5. The relationship between the initial aqueous P concentration and measuredaqueous P concentration after 12-hour adsorption onto the mineral surfaces. Theminerals were previously adjusted to pH 4.0 by nitric acid.

254

5.8

5.6 --- Gibbsite-e- Goethite-I- Hematite~ Kaolinite

5.4

5.2

5

:I:Do

4.8

4.6

6.5

4.2~==========t=:-==============t====~4L--_----L__--L-_----.l__--'-__-'-----_--l__---l.-__-'-----_--l.__---'1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Initial Concentration of P (mg kg-1)

Figure 7.6. The relationship between the initial P concentration and the pH values ofaqueous solution after 36-hour P adsorption onto the minerals. The minerals werepreviously adjusted to pH 4.0 by nitric acid.

255

4.5 r.=:=:=r:::=:::;-.-----.-----.---,-----,---,---r--io Gibbsitex Goethite

4 + Hematite... Kaolinite

3.5

0.5

0.5 1.5 2 2.5 3

Initial P Concentration (mg L- l )

3.5 4 4.5

Figure 7.7. The relationship between the initial aqueous P concentration and themeasured aqueous P concentration after 12-hour adsorption onto the minerals. Theminerals were previously adjusted to pH 9.6 by sodium hydroxide.

256

10,--------.------,-----,------,-----,-----71

-- Gibbsite-e-- Goethite-+- Hematite

9.5 --4- Kaolinite

9

8

7.5

7""-- :-'-::- ----'- -"----- -'- L- -'

7 7.5 8 8.5 9 9.5 10pH of Control

Figure 7.8. The relationship ofbetween pH values of control and P adsorption onto theminerals after 12 hours. The minerals were previously adjusted to pH 9.6.

257

UNIVERSITY OF HAWAI'llIBRARY...outputW orOlI'and output P 205 Discussion 207 Conclusion 208...

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