UNIVERSITY OF HAWAI'llIBRARY...outputW orOlI'and output P 205 Discussion 207 Conclusion 208...
Transcript of UNIVERSITY OF HAWAI'llIBRARY...outputW orOlI'and output P 205 Discussion 207 Conclusion 208...
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
Materials and Methods
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
Interface by System Identification Approach
Abstract 157
Introduction 158
Materials and Methods
x
Chemicals 160
Minerals " .160
Experimental setup 161
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
Materials and Methods
Xl
Experimental setup 202
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
3.3. Estimated dispersion coefficients (D) and ratio of sorption and
desorption rates for goethite system at pH 10 82
3.4. Estimated dispersion coefficients (D) and ratio of sorption and
desorption rates for hematite system at pH 10 83
3.5. Estimated dispersion coefficients (D) and ratio of sorption and
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
3.8. Estimated dispersion coefficients and rates of adsorption and desorption
for goethite system at pH 4 87
3.9. Estimated dispersion coefficients and rates of adsorption and desorption
for hematite system at pH 4 88
3.10. Estimated dispersion coefficients and rates of adsorption and desorption
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
1.11 Change of relative amplitude of subharmonics of output signals
with fundamental frequency for goethite system '" 30
1.12 Change of relative amplitude of subharmonics of output signals
with fundamental frequency for hematite system. .. .. . . .. . . . . . . . . . . . . . . . . . . . . ... . ..... 31
1.13 Change of relative amplitude of subharmonics of output signals
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
for bauxite system at pH 4 133
4.3. Typical dynamic concentrations ofH+ and N03• in effluent solution
for goethite system at pH 4 134
4.4. Typical dynamic concentrations ofW and N03- in effluent solution
for hematite system at pH 4 135
4.5. Typical dynamic concentrations ofW and N03- in effluent solution
for kaolinite system at pH 4 136
4.6. Typical dynamic concentrations ofH+ and N03• in effluent solution
for bauxite system at pH 10 137
4.7. Typical dynamic concentrations ofH+ and N03- in effluent solution
for goethite system at pH 10 138
4.8. Typical dynamic concentrations ofW and N03- in effluent solution
for hematite system at pH 10 139
xx
Figure Page
4.9. Typical dynamic concentrations ofIr and N03- in eflluent solution
for kaolinite system at pH 10 140
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
4.20. Linear relationship between amplitudes of output N03- and
output It of hematite system at pH 4 150
4.21. Linear relationship between amplitudes of output N03- and
output H+ of kaolinite system at pH 4 150
4.22. Linear relationship between amplitudes of output N03- and
output H+ ofbauxite system at pH 10 '" 151
4.23. Linear relationship between amplitudes of output N03- and
output It of goethite system at pH 10 151
XXi
Figure Page
4.24. Linear relationship between amplitudes of output N03- and
output W of hematite system at pH 10 152
4.25. Linear relationship between amplitudes of output N03- and
output H+ of kaolinite system at pH 10 152
4.26. Linear relationship between phases of output N03- and
output W ofbauxite system at pH 4 153
4.27. Linear relationship between phases of output N03- and
output W ofgoethite system at pH 4 153
4.28. Linear relationship between phases of output N03- and
output W of hematite system at pH 4 154
4.29. Linear relationship between phases of output N03- and
output H+ of kaolinite system at pH 4 154
4.30. Linear relationship between phases of output N03- and
output W ofbauxite system at pH 10 155
4.31. Linear relationship between phases of output N03- and
output H+ ofgoethite system at pH 10 155
4.32. Linear relationship between phases of output N03- and
output W of hematite system at pH 10 156
4.33. Linear relationship between phases of output N03- and
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
of transfer function derived from Triple Layer ModeL 195
5.16. Frequency response of hematite system at pH 4 and curve fitting
of transfer function derived from Triple Layer ModeL 196
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
6.19. Linear relationship between adjusted amplitude ofW and amplitude
ofP ofgoethite system at pH 4 232
6.20. Linear relationship between adjusted amplitude of OK and amplitude
ofP ofgoethite system at pH 10 '" '" 232
6.21. Linear relationship between adjusted amplitude ofW and amplitude
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
6.23. Linear relationship between adjusted amplitude ofW and amplitude
ofP of kaolinite system at pH 4 234
6.24. Linear relationship between adjusted amplitude of OK and amplitude
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)
Figure 1.11. The change of relative amplitude of subharmonics of output signals with
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
Figure 1.12. The change of relative amplitude of subharmonics of output signals with
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
Figure 1.13. The change of relative amplitude of subharmonics of output signals with
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
Interface by System Identification Approach
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.
2. Materials and Methods
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%
Repeated Parameter Estimate Confidence Interval
Experiment Lower Upper
#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
Table 3.4. Estimated dispersion coefficients (D) and the ratio of rates of sorption and desorption
for hematite system at pH 10.
Asymptotic 95%
Repeated Parameter Estimate Confidence Interval
Experiment Lower Upper
#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
Table 3.5. Estimated dispersion coefficients (D) and the ratio of rates of sorption and desorption
for kaolinite system at pH 10.
Asymptotic 95%
Repeated Parameter Estimate Confidence Interval
Experiment Lower Upper
#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.
Asymptotic 95%Confidence Interval
RepeatedExperiments Parameter Estimate Lower Bound Upper Bound
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.
Asymptotic 95%Confidence Interval
RepeatedExperiments Parameter Estimate Lower Bound Upper Bound
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)
Figure 3.7. Normalized concentration of nitrate, deviated from the averages, varying with time
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)
Figure 3.9. Normalized concentration of nitrate, deviated from the averages, varying with time
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)
Figure 3.10. Normalized concentration ofnitrate, deviated from the averages, varying with time
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)
Figure 3.11. Normalized concentration of nitrate, deviated from the averages, varying with time
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)
Figure 3.12. Normalized concentration ofnitrate, deviated from the averages, varying with time
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)
Figure 3.13. Normalized concentration of nitrate, deviated from the averages, varying with time
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
fitted using model (18) and (20) derived from the Two-Plane Model.
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
fitted using model (18) and (20) derived from the Two-Plane Model.
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
using model (18) and (22) derived from the Three-Plane Model.
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
using model (18) and (22) derived from the Three-Plane Model.
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.
2. Materials and Methods
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
Bound Bound Bound Bound
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
252010 15Sampling sequence
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
2018168 10 12 14Sampling sequence
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
"
8 10 12 14Sampling sequence
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 "
252010 15Sampling sequence
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
Amplitude of NO; (mM)
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
Amplitude of NO; (mM)
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
Amplitude of NO; (mM)
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
Amplitude of NO; (mM)
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
Amplitude of NO; (mM)
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
Amplitude of NO; (mM)
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
Phase of NO; (radians)
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
Phase of NO; (radians)
-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
Phase of NO; (radians)
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
Phase of NO; (radians)
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
Phase of NO; (radians)
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
Phase of NO; (radians)
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
Phase of NO; (radians)
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
Interface by System Identification Approach
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.
2. Materials and Methods
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
Repeated Parameter Estimate 95% Confidence Interval
Experiment Lower Bound Upper Bound
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
Repeated Parameter Estimate 95% Confidence Interval
Experiment Lower Bound Upper Bound
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
Repeated Parameter Estimate 95% Confidence Interval
Experiment Lower Bound Upper Bound
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
Repeated Parameter Estimate 95% Confidence Interval
Experiment Lower Bound Upper Bound
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
Repeated Parameter Estimate 95% Confidence Interval
Experiment Lower Bound Upper Bound
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
Repeated Parameter Estimate 95% Confidence Interval
Experiment Lower Bound Upper Bound
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
Repeated Parameter Estimate 95% Confidence Interval
Experiment Lower Bound Upper Bound
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
35302515 20Sampling Sequence
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.
2. Materials and Methods
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
Bound Bound Bound Bound
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
252010 15Sampling Sequence
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
252010 15Sampling Sequence
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)
# 2-Theta d(A) BG Height 1% Area 1% FWHM
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)
# 2-Theta d(A) BG Height 1% Area 1% FWHM
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