develop mathematical models able to simulate solute transport through ... SIL model by assuming that air displaced the solution in a piston-type ...... efficiency was considerably higher with intermittent than with continuous ...... dSIdt =A exp(- BS) ..... than with manual methods, because of the greater standardisation of the.
UNIVERSITY OF NEWCASTLE UPON TYNE Department of Civil Engineering
Movement of Solutesin StructuredSoils During IntermittentLeaching: A TheoreticalandLaboratory Study.
NEWCASTLE
UNIVERSITY ---------------------------096 50019 5
LIBRARY
7 09
by Mahmoud Al-Sibal
B. Sc.Civil Engineering(Al-BaathUniversity,Syria) M. Sc.IrrigationEngineering(Universityof NewcastleuponTyne,UK)
A ThesisSubmittedfortheDegreeof DoctorofPhilosophy
September1996
Abstract
Soil salinity is one of the major problemsin and and semi-aridzones,affecting up to 50% of arable land in Syria. Salt-affectedsoils are usually desalinizedby leaching the excesssalts out of the soil profile. Somestudieshave shown that applying the leachingwater intermittently insteadof continuouslymay result in more efficient leaching. This thesis aims to investigate, theoretically and experimentally, the benefits and limitations of intermittent leaching and to develop mathematical models able to simulate solute transport through structuredsoils under such conditions.
Laboratory leaching experiments were conducted on bi-continuum media, as an analogueof structuredsoils, createdby packing porous aggregates (ceramic spheres or soil aggregatesof uni- or multi- diameters) in glass columns. The columns were either leachedcontinuously or intermittently and with different pore-water velocities. Intermittent leaching was undertaken either under saturatedor drained conditions. Under "saturated conditions" the column remained saturatedthroughout the experiment, while under "drained conditions" the column was allowed to drain at the beginning of each rest period and remainedlike this until being saturatedagain for the next leaching period. The solute concentrationin the leachatewas monitored continuously (either using a flow-through conductivity cell, or by using ion-selective electrodesfor Ký and Br' ) to producebreakthroughcurves.Thesecurveswere used to investigate solute transport through such media and validate the developedmodels.
R
The experimentsshowedthat water savingsof up to 22% underintermittent leachingfrom a soil aggregatecolumn were possibleunder saturatedconditions. Such saving increasedwith aggregatesize, flow velocity and duration of rest period. Under drained conditions, for ceramic spheres,12% more solute was leachedwith the sameamountof water under intermittent leaching.
Two models were developed,the SIL (SaturatedIntermittent Leaching) and the DIL (Drained Intermittent Leaching) models, for saturatedand drained in The SIL transport conditions respectively. structured model simulatedsolute soils under intermittent leaching.The governing equationsduring displacement During the the period were mobile-immobile convection-dispersionequations. is by diff-usion flow transfers the the solute only rest period stopped, and betweenimmobile and mobile water regions. The DIL model simulated solute transport when the soil drained. Here, during the displacementperiod, the mobile water was drained.The model simulatedthis using the equationsof the SIL model by assuming that air displaced the solution in a piston-type displacement.During the rest periods the solute difluses within the aggregates establishing a more uniform concentrationin the immobile water across the aggregate.
The modelscanbe usedwith a wide rangeof columnconditionsandfor both sorbed and non-sorbedsolutes. Both models were verified against experimental results.
Acknowledgments
I would like to express my gratitude toward Dr. M.A. Acley and Dr. D.A. Rose for the invaluable guidance and supervision provided during this study, moreover for their readiness to listen and to advise on all matters.
Many thanks are due to the staff and colleagues of salinity research group for the valuable and lengthy discussion provided during the group meetings.
Thanks are due to Al-Baath University,Syria, for sponsoring my research study.
Thanksalso to the staff of the Soil Physics& Soil Chemistry Laboratory, for their help and assistance.
I wish to thank my parents, brother and sisters for the inspiration, continuous concern and love they always gave.
Finally, I wish to thank my wife for her constant encouragement, support and her patience in enduring many inconveniences while I embarked on thisstudy.
Table of contents
Abstract
i .....................................................................................................................
Acknowledgement
....................................................................................................
iii
iv Table of contents ...................................................................................................... x List of symbols .......................................................................................................... Chapter One: Introduction ChaRter Two: Literature review 3 2.1 Miscibledisplacement...................................................................................................... 3 Introduction 2.1.1 .................................................................................................................... 5 2.1.2 Breakthrough curves...................................................................................................... 6 In 2.2 Solutetransport soils ................................................................................................... 6 2.2.1 Convection equation.................................................................................. -Dispersion 7 diffusion hy 2.2.1.1 Solute transport .................................................................................................. 8 by 2.2.2.2 Solutetransport convection.............................................................................................. 9 The 2.2.2.3 combinedsolutetransport equation............................................................................... 2.2.2 Modelling approaches for solute transport in soils ...................................................... 11 12 2.2.2.1 Mechanisticmodels ............................................................................................................. 13 2.2.2.2 Functional models ...............................................................................................................
14 in Solute transport 2.3 structured soils .............................................................................. 14 Soil 2.3.1 structure................................................................................................................ 15 2.3.LI Characterisation ofsoil structure........................................................................................ 16 2.3.1.2 Pore sizesandfunctions ...................................................................................................... 16 2.3.1.3 Structureand hydaulic conductivity ..................................................................................
16 2.3.2 Bi-continuumconcept .................................................................................................. 20 2.3.3 Salt leachingin structuredsoils ..................................................................................
21 2.3.3.1Modelling approach ............................................................................................................. 24 2.4 Salt leaching In practice .................................................................................................
25 2.4.1 Intermittentand continuousleaching ........................................................................... 28 2.4.2 Factorsaffectingintermittentleaching .........................................................................
29 2.4.2.1.Effect ofevaporation rate .................................................................................................... :. 29 2.4.2.2 Effect ofplot size . ............................................................................................................... 30 2.4.2.3 Effect ofsoil hydraulic conductivity . ................................................................................... 30 2.4.2.4 Effect ofcrop . ...................................................................................................................... 31 2.4.2.5 Effect ofsodicity . ................................................................................................................. 32 2.4.3 Management advantage of intermittent leaching ........................................................ . 32 2.5 Proposed programme of the study .............................................................................. .
2.5.1 Leachingunder saturatedconditions 33 ............................................................................ 2.5.2 Leachingunder drainedconditions 34 ..............................................................................
PART
ONE
Solute transport through columns of ceramic spheres under saturated condition
Chapter Three: Experimental work 35 3.1 Accessory experiments ..................................................................................................
3.1.1 Aim of the experiments 35 ................................................................................................ 3.1.2 Particledensity 35 ............................................................................................................. 3.1.3 Porosity the 36 of
ceramic porous spheres .......................................................................
3.2 Diffusion from porous ceramic spheres experiment 36 ..................................................
3.2.1 Materialand method 36 .................................................................................................... 38 3.2.2 Experimentalresults .................................................................................................... 3.3 Continuous leaching experiments 39 ................................................................................ 3.3.1 Aim of the experiments 39 ................................................................................................ 3.3.2 Materialsand methods 39 ................................................................................................. 3.3.3 Experimentalresults 43 .................................................................................................... 3.4 Intermittent leaching experiments 43 ................................................................................ 3.4.1 Aim of the experiments 43 ................................................................................................ 3.4.2 Materials and methods 43 ................................................................................................. 3.4.3 Experimental results and discussion 45 ............................................................................
Chapter Four: Modelling work 4.1 Diffusionfrom porousspheres 49 ..................................................................................... 4.1.1Introduction 49 ................................................................................................................. 4.1.2Aimofthemodel 51 .......................................................................................................... 4.1.3Governing 51 equations .......... . ........................................................................................ 4.1.4Testingthediffusion 53 model.......................................................................................... 4.2 Solutetransportthrougha columnof saturatedporousspheres 53 ............................. 4.2.1Aimof themodel 53 .......................................................................................................... 4.2.2Continuous leaching 55 ..................................................................................................... 4.ZZI Governing 55 equations .................................................................................................... 422.2 Initialandboundary 56 conditions ....................................................................................... 4.2.3Intermittent leaching 58 ..................................................................................................... 4.2.3.1 Governing 58 equations ............................ .............................................................................. ... 4.3 Numericalsolution 59 ................ .......................................................................................... 4.3.1 Finite-difference 59 methods ............................................................................................ 4.3.1.1Crank-Nicolson 62 method ................................................................................................ 4.4 Modelstabilityandconvergence 65 ..................................................................................
Y-i 68 4.5 Model validation ..............................................................................................................
Chapter Five: Simulation work 5.1 Estimating of effective diffusion coefficient De .....................................
70
71 5.2 SIL model simulation ...................................................................................................... 71 5.2.1 Continuous leaching ..................................................................................................... 71 5.2.1.1 Estimatingthe SIL modelParameters .................................................................................. 77 5.2.2 Intermittent leaching ..................................................................................................... 79 5.2.2.1 Running the SIL model with different onlofftimes ............................................................... 83 5.2.Z2 Running the SIL model with different inflow concentrations ...............................................
85 5.3 Effect of OnlOff time on Intermittent salt leaching .................................................... 85 5.3.1 Using the same sphere diameter with different interstitial velocities: ............................
86 5.3.1.1 SIL model predictionsfor IL ................................................................................................ 89 5.3.2 Using different sphere diameters with the same interstitial velocity .............................. 91 5.3.2.2 SIL model predictionsfor IL ................................................................................................
PART TWO Solutetransportthrough columnsof soil aggregatesunder saturatedcondition.
Chapter Six: Transport of sorbing solutes through soils 6.1 Definitions
.....................................................................................................................
103
104 6.2 Equilibrium adsorption Isotherms .............................................................................. 105 6.2.1 Langrnuir equilibrium isotherm ................................................................................... 106 6.2.2 Freundlich equilibrium isotherm ................................................................................. 107 6.2.3 First-order kinetic isotherm ........................................................................................
108 6.3 Modelling transport of sorbing solutes through soils .............................................. 6.3.1 Equilibriummodel 109 ....................................................................................................... 6.3.2 Non-equilibriummodels 110 ............................................................................................. 6.3.2.1 Chemicalnon-equilibriummodels 110 ..................................................................................... Hf 6 3.2.2 Physical non-equilibriummodels .....................................................................................
Chapter Seven: Ion-selective electrodes 7.1 Introduction
119 ...................................................................................................................
7.2 Essential components 119 .................................................................................................. 7.2.1 Ion-selective electrode types 121 ..................................................................................... 7.2.1.1 Glassmembraneelectrodes 121 ............................................................................................... 7 2.1.2 Solid-statemembraneelectrodes 121 ....................................................................................... 7.2.1.3 Plastic membraneelectrodes 122 ............................................................................................. 7.2.1.4 Gassensingelectrodes 122 ...................................................................................................... 7.2.2 Reference electrodes 122 .................................................................................................
Yii 7.3 Operation principles 123 ..................................................................................................... 7.4 Calibration
126 .....................................................................................................................
7.5 Selectivity and Interference 126 ......................................................................................... 7.6 Using ISE In continuous flow measurements 127 ............................................................ 7.6.1 Continuous flow measurements of soil leachate 128 ........................................................ 7.7 Conclusion
....................................................................................................................
128
Chapter Eight: Experimental work 8.1 Soil analysis 130 .................................................................................................................. 8.1.1 Soil texture 130 ................................................................................................................. 8.1.2 Soil pH 131 ........................................................................................................................ 8.1.3 Aggregate stability 131 ...................................................................................................... 8.1.4 Cation exchange capacity CEC 132 ................................................................................. 8.1.5 Linear shrinkage of the soil 132 ........................................................................................
8.2 Selected tracers
133 ............................................................................................................
8.2.1 Potassium in soils 133 ...................................................................................................... 8.2.1.1 Potassiumfixation 134 ............................................................................................................. 8.2.1.2 Releaseoftotassium 134 ..........................................................................................................
8.2.2 Potassiumadsorption-desorption isotherms 134 .............................................................. 8.2.3 Bromideadsorption-desorptionisotherms 138 ................................................................. 8.3 Electrode characteristics 138 ............................................................................................. 8.3.1 Potassium ISE 140 ............................................................................................................ 8.3.2 Bromide ISE 141 ............................................................................................................... 8.3.3 Reference electrodes 141 ................................................................................................. 8.3.4 Calibration of ISEs 141 ..................................................................................................... 8.3.5 Flow-through cells 143 ...................................................................................................... 8.3.6 Response time 143 ........................................................................................................... 8.4 Using ISEs with soil columns 146 ......................................................................................
8.4.1 8.4.2 8.4.3 8.4.4 8.4.5
Preparingsoil columns 148 ............................................................................................... The ISEs' connections 149 ............................................................................................... Leachingexperiment 149 .................................................................................................. CalibratingISEs with soil solution 152 .............................................................................. Experimentalresultsand discussion 155 ..........................................................................
Chapter Nine: Modelling work 9.1 Modelgoverningequations 164 ......................................................................................... 9.2 Model parameter estimation 165 ........................................................................................ 9.2.1 Mobileand immobilewater content 165 ............................................................................ 9.2.2 Fractionof adsorptionsites ( f) 166 ................................................................................ 9.2.3 Usingthe best-fitoptimisationprogram 168 ......................................................................
Mid 169 9.3 Model stability and convergence ................................................................................
Chapter Ten: Testing the model 171 10.1 Estimatingmobile and Immobilewatercontent...................................................... 173 10.2 Validating the model ................................................................................................... 174 10.2.1 With experimental results of bromide leaching ........................................................ 174 10.2.1.1 For continuousleaching .................................................................................................. 175 10.2.1.2 For intermittentleaching ................................................................................................. 180 10.2.2 With experimental results of potassium leaching ..................................................... 180 10.2.Zl For continuousleaching .................................................................................................. 185 10.2-2.2 For intermittentleaching ................................................................................................. 187 10.2.2.3 Fitting theparametersusingthe SIL model, .................................................................... 192 10.3 Conclusions ................................................................................................................
PART THREE Solutetransportthrough columnsof ceramicspheresunderdrainedcondition
Chapter Eleven: Intermittent leaching under drained conditions 11.1 Introduction
........................................................
.......................
195
196 11.2 Experimental set-up ....................................................................................................
196 11.2.1 Aim of the experiment .............................................................................................. 196 11.2.2 Methodand materials ............................................................................................... 200 11.2.3 Resultsand discussion ............................................................................................ 205 11.3 Model modifications and simulations ....................................................................... 206 11.3.1 EstimatingDIL model parameters ............................................................................
Chapter Twelve: Reversibility of intermittent displacement 12.1 Experimental method
214 ...................................................................................................
12.2 Results and discussion 215 .............................................................................................. 12.3 Conclusion
........................................................................
217 ................
Chapter Thirteen: Conclusions and recommendations 13.1 Conclusions 218 ................................................................................................................ 13.2 Recommendations
220 and further research ...................................................................
ix References
.....................................................................................................................
223
Appendices Abbreviations Appendix A
used in the models .................................................................................
........................................................................................................................
Appendix B Appendix C Appendix D Appendix E Appendix F
....................................................................................................................... ....................................................................................................................... ....................................................................................................................... ........................................................................................................................ ........................................................................................................................
240 242 250 258 263 271 278
List of main symbols
Symbol a
Definition
Units
sphereradius
[L]
sphereor aggregatediameter
[L]
the fraction of adsorptionsitesthat equilibratewith the mobile water phase
kf
kb
the forward andbackwardratesof reactions
[1-11
q
water flux
[L3 T-1]
r2
coefficient of determination(linear regresion)
t
time
IT]
V
the averagepore-watervelocity
[L Tl
,
q A
Vd
Darcyvelocity (vd
VM
averagepore-watervelocity in the mobile water region
X
distance
z
spaceco-ordinate(positive downwards)
T-11 [L T-11
A
cross-sectionalareaof the column
2] [L
A
activity of the measuredions
[M L-3]
C
soluteconcentration
[M L-3]
concentrationat equilibrium
[M L-3]
Ce
Cm(t)measured concentration the of externalsolution (for diffusion model) Cim(r, 0
[M L-3]
concentrationof solutein the solution within the ceramicspheres(for diffusion model)
[M L-3]
xi
CM
averagesoluteconcentrationin the mobile water region
[M L-3]
Cim
soluteconcentrationin the immobile water region
[M L-3]
Cinp
concentrationof solutein the addedwater
[M L-3]
CO
initial concentrationof solutein both the mobile and immobile solution
[M L-3]
Do
ionicdiffusioncoefficientin freewater
[L2 -rl]
D,
effectivediffusioncoefficient
[L2 1-1]
D,,
dispersion mechanical coefficient
[L2 T-1]
hydrodynamic dispersion coefficient
[L2 -r I
D
s
F
massfractionof type I "equilibrium"sites
K
distributioncoefficient
Lr
lengthof theporouscolumn
P
dimensionless variable(Table6.2)
R
factor totalretardation
Rm
retardationfactorsin themobilewaterregions
Rim
retardationfactorin theimmobilewaterregion
R2
(non-linearregresion) coefficientof determination
S
of adsorbate, expressed asmassadsorbate concentration perunit massof dry soil
[mM-1]
S2
adsorptionon type 2 "kinetic" sites
[mM-11
Sa
initial amountof potassiumin the column including the adsorbedone
IM]
so
initial saltmassin thecolumn
[M]
SS (t)
saltmassin thecolumnat timet
[M]
V
outflowvolume
[0]
VO
total volume of mobile and immobile water in the column
[0]
Va
volume of addedwater neededto remove90% of the initial salt load of the column
[L]
[L31
2di Ve
volume of externalsolution (for diffusion model)
VWS
volumeof solutioninsidetheceramicspheres
(X
masstransferrate coefficient betweenthe mobile and immobile
101 3]
[L
water regions
[1-11
CCk
first-order kinetic rate coefficient
IT" I
P
dimensionlessvariable(Table 6.2)
0
sphereporosity
[L3 L-3]
0
volumetric water contentof the column
[L3 L-3j
OM
volume of mobile water as a proportionof total column volume
[L3 L-3]
0
volume of immobile water as a proportion of total column volume [L3 L-3]
SP
im
dispersivity
[L]
solutesourceor sink
[M L-3 T-1]
P
soil bulk density
[M L-3]
PP
averageparticle densityof the spheres
[M 1.,-3]
(D
fraction of soil water that is mobile (= 00
CO
dimensionlessmasstransfercoefficient.
(z, 0
The abovesymbols representthe main variables and parameters,otherswill be defined through the text as necessary.
Chapter One
6rmwwfýý
The provision of adequate drainage and the accompanying problem of accumulation of salt in soil has plagued irrigated agriculture for centuries. Different factors can causesuch salt accumulation.Saline irrigation water, low soil permeability, inadequate drainage conditions, low rainfall, and poor irrigation managementýall contribute to the tendency of salt to accumulatein (Yaron, 1981). Excess salts in the soil solution influence the growth of soils by plants osmotic effects and toxicity of specific ions, and by changing the physical properties of the soils. Over time, salts may concentrateto such an extent that they hinder germination, seedling,vegetativegrowth, and the yield and quality of crops (Tanji, 1990),andultimately renderland sterile.
Historical records for the past 6000 years reveal that numerous based irrigated One failed due to societies of on agriculture salinity problems. This the mosthighly publicisedis that of ancientMesopotamia, Iraq. once now productiveland appearsto have sufferedprogressivesalt damagefrom about 2400 BC to 1700 BC which contributedto decline of this civilisation (Gelbured,1985;Tanji, 1990).Nowadays,salt-affectedsoils are to be found on all continents,covering about 10% of
land dry the total surface of
(SzaboIcs,1980),andaboutone-thirdof all irrigatedland(Yarot; 1981).
2 The only way known to effectively remove excesssalts from soil is by leaching (Shalhevet,1973).Pondingwater on the soil surfaceis the traditional leaching During but large such method, consumes quantitiesof valuable water. the water does not uniformly flow through the soil, but preferentially through inter-aggregate be (Tanji, 1991). These 1990; Jury, to tend pores macropores in flow a reduced results within a structured soil and such preferential leaching More leaching from uniform effectiveness of within aggregates. is but if in is leached this the slower and occurs soil unsaturatedcondition requires greater managementcontrol. In practice this requires a sprinkler irrigation system,which is unusualin and and semi-arid environments. Alternatively, if pondedleachingwere intermittent, with a "rest period" during which the profile drained, solutes could diffuse to the exterior of the flow during the though was not macropore aggregates rest period even occurring. During the subsequentponded-irrigation phasesuch solutes would be leached. Such diffuse to the a strategy macroporeregion and readily quickly should result in more efficient use of water for leachingpurposes.
This study aims to optimiseleachingusing such intermittentponded leachingof the soil profile.
ChapterTwo
YteYatuQ
øview
This chapter consists of four main parts. The first will explain the idea of illustrate displacement different the shapesof the resulting miscible and will breakthroughcurves.The secondwill considerthe equation of solute transport through the soil and a general overview of the modelling approachof solute transport. The third part will introduce the idea of soil structure and the bifourth for leaching from The continuum concept modelling salt structuredsoils. part will
summarise the comparative studies between continuous and
intermittent leaching and will identify the most important factors affecting intermittent leaching.
2.1 Miscible displacement 2.1.1 Introduction Miscible displacement(MD) is the processthat occurswhen one fluid mixes displaces with and anotherfluid (Kirkham & Powers, 1972).Leachingsalts from a soil is an exampleof MD, becauseaddedwater mixes with, and displaces,the solutionin the soil. Anotherexampleof MD is the movementof dissolved fertiliser or herbicides into and throughthe soil. water containing One of the best known introductionsof MD techniquesto the field of soil by sciencewasmade Nielsen& Biggar in a seriesof papersin 1961 1962and
4
1963.Sincethen MD techniqueshave gainedwidespreadacceptanceanduse in the field of soil science . Solutesin the displacing fluid are transferredthrough the soil by mass transport of the moving fluid and by diffusion (Bear, 1972; Kirkham & Poivers, 1972; Rose, 1973). In a soil column this flow can be expectedto be rather "erratic". It varies in magnitudeand direction from point to point due to the complex pore geometry,and this erratic flow causesthe solute to disperse between the displacing and the displaced fluid. The term "mechanical dispersion" is used to differentiate this spreadingmechanismfrom that due to diffusion. This distinction is made becausediffusion is causedby the random thermal motion of solute molecules, whereas dispersion is due to the erratic flow of the fluids through complex pore systems.Mangold & Tsang (1991) defined three processesoccurring in the pore channels causing mechanical dispersion: 1) mixing within individual pore channelsdue to differencesin velocity of the molecules between those in the middle of the channel and those subject to draggingforces along the pore walls, 2) mixing causedby differencesin the sizes of the pore channelsand hence velocities along the flow paths,and 3) differences causedby the branching of flow channel paths in the soil matrix.
The separationbetweenmechanicalanddiffusiondispersionis, actually,rather artificial as they are inseparable.However,moleculardiffusion dependson concentration differences (Bear, 1972), whereas mechanical dispersion dependsonly on velocity (Wagenet,1983).
The coefficientof hydrodynamicdispersion(sometimescalledeffective diffusion coefficient (Jury et al., 1981), or apparentdiffusion coefficient
(Biggar & Nielsen, 1980; Wagenet,1983),or simply the dispersioncoefficient (Nielsen et al., 1986)) is a term which is used to describe the spreading resulting from both mechanical dispersion and molecular diffusion (Rose & Passioura, 1971; Passioura, 1971; Bear, 1972;Freeze& Cherry, 1979).
2.1.2 Breakthrough curves Becauseit is difficult to study the shapeof the boundarybetweendifferent fluids asthey emergefrom a pipe or soil column,it is commonto monitor the concentrationchangeof the displacingsolutein the effluent.The mannerin which the concentrationchangescan give someinformationaboutthe porous mediumandthe physicalbehaviourof the fluid movement.Dataarepresented in a standardform calleda breakthrough curve(BTC). The BTC is a graphof concentrationin the effluent relativeto some standardconcentration(usuallythe concentrationof influent ), plotted against theratio of thevolumeof the collectedeffluentrelativeto thetotal porevolume in the column(Krupp& Erick, 1968). In the absenceof dispersion(i.e., immiscibledisplacement), the BTC form is line. This in take the the there would of vertical no model, which dispersion,is called the piston-flow model of solute transport(Fig. 2.1), so namedbecausethe solute is displacedthrough the soil like a piston. Such displacementis seldomif ever encounteredin practice(Hillel, 1980).What happens boundary (or the front betweenthe two solutions)is a the normally at gradual mixing resulting from the hydrodynamicdispersion so that the boundary becomesincreasingly diff-use about the mean position of the advancingfront and the BTC will take a form of S-shapedcurve (Bear & Bachmat, 1991; Rose, 1977). Shifting of the curve to the left indicates exclusionor "bypass"from a significantportion of the soil solution, while shifting to the right indicatesadsorptionor solute retention by soil. The symmetricalbreakthroughcurve which passesthrough the point of relative
6
in 2.1) line Fig. dashed by (showed 0.5 the concentration= at pore volume =1
discussions the ideal Comprehensive dispersion (or MD). of typical an called Rose (1962) & Biggar by Nielsen have been BTCs and shapesof presented (1977).
1.00 PlIelon IFow
0.75 Exclusion 0.50
0.25
C
00
0.5
LO PORE VOWME
1.5
2.0
ZI: SoluteBTC illustrating severalpatternsof soluteclution (after ffagenct, 1983)
2.2 Solute transport in soils
2.2.1 Convection -Dispersion equation Accordingto thepreviousmiscibledisplacement theory,thetotalflux of solute is the resultof the combinedeffectsof diffusion andconvection.Thatis: J
Sý -j D+JC
(2.1)
in is J the transported through where massof solute area a a cross-sectional unit time, andthe subscriptss,D, andc repiesenttotal solute,solutetransported by diffusion andsolutetransportedby convection,respectively. ,
7
ZZLI
Solute transport by diffusion
Ficles first equationstatesthat for one-dimensionaldiffusion in free solution: dC JD= -D " dr
(2.2)
where C=
soluteconcentration,
x=
distanceand,
Do
= the ionic diffusioncoefficientin freewater.(Valuesof Do for major
ions can be obtainedfrom Robinson& Stokes,1970.)
In a soil: Becauseof the tortuous flow path in the soil matrix, Eq. 2.2 becomes;
JD=-D.
dC 0 dx
(2.3)
is diffusion De the where effective coefficient, which takes into account the tortuosity of the soil matrix.
Estimationof D, (0 ) hasbeenthe subjectof a nwnberof studies(Nye, 1979).Bresler(1973)relatedD, (0 ) for anygivenion to Do by
D. (0) = Do0
(71, )
Cy
(2.4)
is (I )2 0 lie is a tortuosityfactorreflectingthe where volumetricwatercontent, tortuosity of pores within the soil matrix (I is the averagestraightpath of diffusion,le is the actualtortuouspathalongwhich diffusingmoleculesor ions move) and ý and y representthe effectsof anionexclusionandthe charged soil matrix on waterviscosityrespectively. Jury (1991) basedhis equation on the Millington & Quirk (1961) tortuositymodelto estimateD, i. e., ,
-a
DJO )= Do 0
10/3
/c2
(2.5)
is where c soil porosity. Other equationsare empirical. For exampleit has been found that, in a clay-water system,D, can be representedas (Kemper& van Schaik, 1966): D, (0) = D, aeb0
(2.6)
where a and b are empirical constants.Olsen& Kemper (1968) found a good fit for soils of texture ranging from sandy loam to clay, with b=10, and 0.001 )
(rr*jn)
EýJ.
3cac3l
100.8-
0504-
0.2-
0.04
C)
20
40
50 O; mp(oýqbýt
E30
100 tiýqv
120
140
ISO
C-)
Fig. 3.6: Relativeeffluent concentrationagainstdisplacementtime for intermittent (--o-) and leachingexperimentsIC&I, 2C&I, and 3C&I. continuous(-o-)
Chapter Four
kýalo4eA QAý(,
The aim of the modelling work was to write a computer code able to simulate the solute transportin a bi-continuum.systemduring intermittent leaching.This chapter consists of four parts. The first, dealing with diffusion out of inert spheres, and the resulting spread-sheetprogramme, will be useful later to estimate the effective diffusion coefficient De - In the second part the SIL (SaturatedIntermittent Leaching) model is constructed;this model is testedfor continuous leaching against an analytical solution in the third part and the validation of the model is discussedin the fourth part.
4.1 Diffusion from porous spheres 4.1.1 Introduction Diffusion is the processby which matter is transportedfrom one part of a systemto another down a concentrationgradient as a result of random molecularmotion. The transfer of heat by conductionis also due to random molecular This was motion, and thereis an obviousanalogybetweenthe two processes. recognisedby Fick (1855),who first put diffusion on a quantitativebasisby adoptingthe mathematicaltheory derivedsomeyears earlier by Fourier.The theoryof diffusionin an isotropicsubstance is thereforebasedon mathematical
5Q the hypothesisthat the rate of transfer of diffusible substancethrough unit area of cross-sectionis proportional to the concentrationgradient measurednormal to the area,i. e. for one-dimensionaltransfer: dC F= -Do dx
(4.1)
C is F the the transfer where cross-section, area of of unit rate per concentrationof diff-usingsubstance,x the spaceco-ordinatemeasurednormal to the cross-sectionalarea, and Do is the molecular diffusion coefficient. Eq. 4.1 is often called Fick's first equationof diffusion. The secondFick!s equationfor one-dimensionaldiffusion (i. e., if there is a gradient of concentrationonly along the x-axis) can be derived by combining i. 4.1 the equation with equationof continuity, e., ac
I- D,, -x Tx ax-
OF=_ =-
at5
a
aC
If Do is constant,i. e. not a function of concentration,then ac
Doalc 5XT
at=
(4.2)
Other forms of this equation follow by transformation of co-ordinates,or by considering elementsof volume of different shape Crank (1975) gave, in .
his comprehensivestudy on the mathematicsof difftision, the following equationfor diffusionin a sphere 0 C(r, t)
at-,
where radial co-ordinate.
fO'C
D 'Jar'
2aCl
+r
f ar
(4.3)
51
4.1.2 Alm of the model (e. KCI) The modelaimsto simulatethe diffusion of a non-adsorbed g. solute dilute inert into of out of a continuouslystirred volume porous spheres solution.From this, the difftision coefficientof the solutewithin the spheres maybe determined. 4.1.3 Governing equations A mechanistic model was developed based on Ficles second equation of diffusion (Eq. 4.2 and4.3) wherethe rate of solutetransfer is describedby: ,
fa Ici. D, +20
a C,. (r, t)
Ci. I
(4.4)
5rF r ar
cl t[
where time Ci (r, t) .. D,,
in = concentration of solute the solution within the sphere diffusion in coefficient = effective porous spheres.
D, replaces the molecular diffusion coefficient, Do, in order to account for the tortuous path followed by ions within the complex matrix in the porous spheres.
The total massof solute (MT) in the system (in the saturatedspheresplus the external solution, Fig. 4.1) is given by : M
T
=v
ws
c im
(t)
+ ve
CM(t)=vc ws
0
(4.5)
where V', ' V"
= volumeof solutioninsidethe spheres = volumeof externalsolution
C"'(t) = concentrationof solutein the externalsolution CO
initial concentrationof solutewithin the spheres. =
The averageconcentration in the sphere, C,. (t), is calculated as (Rao et aL, 1980a):
52 it
11
-) 2r
(4.6)
(r, t) dr
-f where a=
sphere radius.
Porous sphere
HIS c im
External
c
solution
Fig. 4.1 : Simple diagram showing diffusion model parameters
The initial and boundary conditions are: (',,,,(a, t) = C,,(t);
t ý! 0
(4.7a)
Ci,,(r, t) = CO
0:!ý r:!ý a, t= 0
(4.7b)
Qt)
t= 0
(4.7c)
=0
(t) can be calculated from the solution of Eqs. 4.4 to 4.6 under conditions (4.7a,b,c), given by Crank (1975):
ml M.
Co - c"ll CO
-
67 (7 + 1)exp(-
C,
n-19+
q-1 -'Iti-)
a2 97 + qt,y -2
(4.8)
where M, M, are the amountof solute in the sphereat times t and oo respectively, V
to and
53
3q,, tanq, = 3+,vq.'
are the non-zeroroots of
A table of q,,values can be found in Crank (1975 ). be (Q the ), will At equilibrium (i. e., as t--->oo the solute concentration
i. inside the andoutside poroussphere, e. same C. C. (00)= Ci.(Go) =
(at t-->oo)
andcanbe calculatedas C.
mr = V. V. +
(4.9)
The external solute concentration,C. Q), can be calculatedby
(Co ji. -(t» C. (t) = v. 4.1.4 Testing the diffusion model 4.8,4.9 Eqs. (t) C, C. An "Excel" spreadsheet to and using wasused calculate (tyC, C. For different the was 4.10 time, t, time steps. ratio each and at Fig. 4.2 time. showssuchresultscomparedwith calculatedandplottedagainst into (for diffusion CaC12 the dataof Rao et al. (1980a) out of ceramicspheres a continuouslystirredvolumeof water). Themodelsuccessfullysimulatedthemeasuredresults 4.2
Solute transport
through
0.985).
a column of saturated
l3orous
spheres 4.2.1 Alm of the model The SIL (SaturatedIntermittentLeaching)model was aimedto simulatethe solutetransport,under steadyflow conditions,througha saturatedcolumnof immobile inert porousspherescontainingboth mobile (inter-spheres) water and for (intra-spheres) and continuousor intermittentwaterapplication.
54
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
0 0
50
100
150
200
250
300
400
350
Time (min)
Ve De ý 0.012 CM2 h-1,C#= 0.01 N CaC1, cm, ,a=0.75
V= 100 CM3, =
WS
14.83CM3
Cm(t) = concentrationof solutein the external solution Ce
= concentrationof soluteat equilibrium
0
Fig. 4.2: A plot of C (tyC,.(t) againstdiffusion time (measureddata from Rao et al., 1980a). ..
55-
4.2.2 Continuous leaching In this casethe water is continuouslypondedon the surfaceof the porous medium.The leachinghere is similar to the miscible displacementprocess (Section2.1). 4.ZZI
Governing equations
Recalling Eqs. 2.17 and 2.19 for solute transportthrough bi-continuum.media, and assuming,from now on, that the solute concentrationin the water within the spheres(immobile water) is constantacrossthe spherediameter and equal to the averageconcentration:
0
aCmtZ2Z
D, =O.
-
Vd
a cim
a c. -0im
and ac 0 ý ýim=a (c, im at »
(4.12)
where z= C.
vertical distance,positive downwards = concentrationof solute in the mobile solution (betweenspheres)
Ci.. = concentrationin of solute in the immobile solution (within spheres) D,
hydrodynamic dispersion = coefficient for mobile solution
Vd
Darcy = velocity
0M= 0
volume of mobile solution as a proportion of the total column volume
immobile = volume of solution as a proportion of im
the total. column
volume
cc = masstransferratecoefficientfrom immobileto mobilesolution. Rao et aL (1980a) showed experimentally that the mass transfer rate coefficient, a, is not constant but changeswith time. They introduced the
56
for independently be transfer averagedmass coefficientwhich can calculated porous spheresfrom the equation: (0.1)"(DO,. ), Mý CC = ýF I-B,
or 2(1+
cc= (Dql
)(DOiln) OJBI (Ra T2 (I-B I
0.0001< T: 5 0.1
T
0.1
(4.13a)
(4.13b)
where
om+oim (167) B, = 0.144721n ý(Dqj2) is in depending for a constant q, = on (D, q, values various (D values are given Rao et aL (1980), a= sphere radius, and the dimensionless time, T, is given by
DAt
(4.14)
a2
is
is displacement For time e period over which a a calculated.
experimentit is consideredto be the mean column residencetime (i.e. At = Vd /Ont ) which hasbeenshownto give the bestresult(Raoet aL, 1980b). 4.ZZ2 Initial and boundary conditions
The initial conditionfor the soluteconcentrationof the mobile and immobile solutions is given by:
57
ci. (Z,t) = c, (Z,t) = Co »
(t = 0)
(4.15)
where CO = the initial soluteconcentrationof both mobile and immobile solutions. For the upper boundarycondition (z = 0), three possibilities are available (Javandelet aL, 1984): 1) Dirichlet boundary condition: which prescribes concentration along the boundary, 2) Neumann boundary condition: which prescribes the normal gradient of concentrationover the boundary,and 3) Cauchy boundary condition or prescribedflux boundary condition: which prescribesconcentrationand its gradient. Van Genuchten& Parker (1984) showedthat a prescribedflux boundary condition is a better representationof the physical reality than the others, since it gives a better prediction of the massbalanceby allowing for dispersionin the column at z=0. With this boundarycondition the massflux of the solute at the upper boundary at any time is equivalentto the total flux of that solute carried out by dispersionand convection(Javandelet aL, 1984),i. e.
at z=0, tý-.O Cinp Cm Vd Vd -0A -z
ac
m
az where Ci,, = is the concentration in the of solute added water. r For a finite column of length L, frequently used lower boundary ,a condition is
ac az Brenner (1962) gave the analytical solution for CDE under these two conditions. However, accommodatingthe presenceof a boundary layer at z=L, leads to a discontinuousconcentrationdistribution acrossthe lower boundary,
58
is in interior that thus the to to the transition zone and a gradient concentration it is Therefore, be Genuchten, 1984). & (Parker not constrainedto zero van often more convenient to solve the CDE equation for an "effectively" semiinfinite column rather than for a finite column (Selim, 1992). For
a* column
assumedto be semi-infinite, the lower boundarycondition is written as at z= oo,t>0 cl (00,1)= cm(00,t)= (", . Even though this condition does not exactly represent the physical reality (since the column is finite), it can be used if the length of the column is extended as shown in Fig. 4.3. Parker (1984) stated that " so long as back 4.17 is Eq. boundary is (which the the case), mixing at normally exit negligible for a semi-infinite case can be used with impunity and the resulting solution applied to the finite region 0 :!ý z !ý Lr as well". Accordingly, this boundary condition was used.
Fig. 4.3: flýpothetical expansion of the column.
4.2.3 Intermittent leaching 4.2.3.1 Governing equations 1) During the displacement period, "On Time"
J;o
The governing equationsare sameas in the continuousleaching,i. e., Eqs. 4.11 & 4.12. 11) During the rest period "Off Time" The governing equations can be obtained from Eqs. 4.11 and 4.12, by assuming that there is no mass flow, and that solute is only transferred by diflusion (i. e., vd= D, =0). Thus, from Eq. 4.11:
om
aC.aci. =Ojm atat
(4.18)
where a
0
im
ci. (C.- Ci.) at =a
In calculating ccduring the "on" time, the At in Eq. 4.14 is the sameas for the continuouscase;however, for the "off' time it is equal to the period of , the "off' Phase,sincethis is the real time of the difftision .
The two main numericalmethodsare the finite-differencemethod and the finite-elementmethod.Although thesemethodshave their own techniques, & (Bear they arebasicallyvery similar,especiallyin the one-dimensional case Verruyt, 1987).Both use a spatial discretizationand in one dimensionthis discretizationis the samein the two methods.Becausethe finite differenceis somewhatsimpler to derive, it is usedin this study.
4.3.1 Finite-difference methods In mostnumericalmethodsof solvingdifferentialequations(suchasEqs.4.11 & 4.12), the first step is to replacethe latter by algebraicfinite-difference equations.Theseare relationshipsbetweenvalues of the dependentvariable (here,concentration values)at neighbouringpointsin zt space.The numerical
60
solution of the seriesof simultaneousequationsthus obtained gives the values ( dependent discrete points grid of variables at a predeterminednumber of points ) through the domain investigated There are many possible ways of writing Eqs. 4.11 & 4.12 in a finitedifference form, according to the choice of the approximation of the terms. Usually the time and spacedomain is divided into a grid, where At represents the time increment and Az the spaceincrement(Fig. 4.4). Time and spacecan be expressedas multiples of At and Az I j= 0,1,2 n1 ........ A-7 Ii=0,1,2 1. z=i. M ...... So a z,t function ftz, t) is written as F(i. Az j. At , approximate solution of f(z, t) at the discrete point t=j.
At
is 'F the where jjF , i. Az J. At ) of the defined
domain.
Fig. 4.4: A grid of the numericalsolution with initial and boundarvconditions.
61
The differential equation is replacedby a finite-differenceequation The in dependent the terms the the grid points. written of variableat valuesof solutionof the differenceequation,or setof differenceequations,is carriedout by S, differential Denoting the the the numerically. equations exactsolutionof exactsolutionof the differenceequationsby D, and the numericalsolutionof IS-DI differenceequationsby N theterm is calledthe truncationerror and ID-NI the numerical,or the round-offerror The conditionfor convergence IS-D1 domain. in The 0 the of the solutionis that solution -4 everywhere IDNJ -> 0 conditionfor stability is that everywherein the solutiondomain (Bear, 1972). For example,if thepartialdifferentialequationwasof the style: 02C
19C
Ot
x
x
then the generalfinite-difference analogueof this equationwould be: J+l
J+l
iC-'IC=fl At
C-2 +,
J+lc+j+lc X2
'C C j-, +(l fl) j+1 -2' iC+i-, AX2 _
where00.95 in all predictionsand experimentaldatathroughoutthe simulation(R D, is interesting It the the to that, same also cases). notice as model used optimisedvaluesof Fig. 5.2a,c;, it inheritedthe similar BTC shapesshowing data breakthrough longer the tailing than alsoslightly earlier experimental and (Fig. 5.3a,c).
Table5.2: Parametervaluesfor SIL model. Spherediameter
om
olm
(MM)
Exp. IC
13
Exp. 2C Exp. 3C
13+3 1 16
0.486
0.230
0.320 0.310 11 1 1 1 1 0.404 0.268
Vd
D, *
(mm/min)
(mm2/min)
2.31
0.0575
0.96
& 0.0575
D, (=2/min)
2.824 2.641
0.0528
5.47
0.0422
28.
* values are obtained from Table (5.1) for relevant diameter # values are optimised using CXTFITI program I values are optimised using SIL model.
5.2.2 Intermiftent leaching The intermittent leaching (IL) experimentswere performedunder similar leaching & 3.3) (CL) (Tables 3.2 to the with continuous conditions experiments the inclusion of several interruptions to the leaching flux. Therefore the same determined for the CL experiment (Table the parameterspreviously values of 5.2) were used.The main advantageof the CL experiment is the estimation of D,
be estimatedindependently.The SIL model will which cannot otherwise ,
but different (Table these the time with parameters with values of run onloff
5.3). Fig. 5.4 showsthe resultsas plots betweeneffluent concentrationand time. Experimental data points follow the same general pattern as for the leaching experiments (Fig. 5.3) with a decrease in effluent continuous Superimposed time. a certain after concentration on this pattern are successive
78
0ý1
--L
111
ON
3-006 ,
OFF
004
0.02
SiL Exp
0.0-+ 50
0
100
150 Time
200
2ýO
300
350
(a)
(min)
]Exp-
&
f3771"I
0.08-
20
20
21
3-MWL
min
ON
rnirt
OF'F
0.04-
0.02
1
SIL Exp
0-0
0
50
1ý0
150 Time
2L
2L
300
3ý0
(b)
(min)
rxp.
31
01
d=
30 30
6
772.rn
min min
ON OFF
0ý04
0.02
0
SIL Exp
ao0
50
100
150 Time
200 (min)
250
300
350
Fig. 5A Plot of effluent concentration vs. time for SIL model and experimental results for Exps. 11,21,and 31
Z2 Table5.3: Parametervaluesfor SIL model. Spherediameters
om
olm
13
Exp. 21
0.486
13+3
'
(MM2/Min)
(mm/min)
(MM)
Exp. 11
Vd
0.300
0.230 0.316
6
0.398
0.271
(MM2/Min)
2.150
0.0575
2.82
0.862
0.0575 &
2.64
1
Exp. 31
D, 4
De *
5.240
1
0.0528
0.0422
28.21
values are obtained from Table (5.1) for relevant diameter values are taken from Table (5.2) of continuous leaching simulations.
increases in short concentrationcorrespondingto measurementsmade after the end of each off period. The amplitudes of these peaks progressively decrease with successiveoffperiods with the most rapid such decreasein Exp. 31and the least rapid in Exp. 21.Thesereflect the temporary increasein concentrationof the mobile water during the off time due to continuing difftision out of the spheres. The model simulatedthe IL experimentaldata (Section3.4) without any
optimisationand excellentagreementbetweenthe model and experimental 2>0.95). for The small differences results all the threecaseswas obtained(R betweenthemodelpredictionsandexperimentaldatacouldbe becauseof some differences between the experimentalconditionsof CL andIL (Table5.3 small 5.2).Themodelwasableto simulatethe increasein effluentconcentrationat eachoff time using Eq. 4.18 for uni-sphereand mixed-spherecolumns.
Using the parametersof Exp. 11the modelwas run with two different conditions:
1)with differentonloff times 2) with differentinflow concentrations.
5.ZZI
Running the SIL model with different onloff times
The model was run for different combinationsof onlofftimes (5115,15/25, and 25/15 min). The results of the model were plotted as CICO (relative VIVO (pore (Fig. 5.5), where volumes) vs. concentration)
80
13 mm
1.0
15 min OFF
0.8
25 min OFF 15 min OFF
0.6
us 0.4
0.2
0.0 34 V/Vo
Fig. 5-5:Plotof CICOvs. VIVOfor differentonloff timesandunderExp. 11conditions.
81
C= the concentration in of solute the effluent CO= theinitial concentration in immobile both the of solute mobileand solution V =the oufflowvolume VO= thetotalvolumeof mobileandimmobilewaterin the column(i.e., 1 porevolume). With this dimensionlessunit on x-axis, the off times will appear as an instantaneousincreasein concentrationbecauseno flow occurred. Every such jump will indicate the end and the beginning of a leaching cycle. Fig. 5.5 in CO initially that to the the shows solute concentrationof equal effluent was all experiments (CICO=1). The difference in effluent solute concentration betweenCL and IL will appearfirst with the experimenthaving the shorteston time (appears in red colour in Fig. 5.5). As the on time increased such difference betweenIL and CL decreased.After leaching most of the solute in the mobile solution (i. e., after about one pore volume has passed)the BTC of CL showeda long tailing (towards the zero value of CICO)as the solute in the immobile solution continued to slowly diffuse out. By contrast in IL, as most in immobile the the solute of solution diffused out during the off times, shorter tailing was observed for all the cases, with a consequent greater effluent beginning the concentrationat and a shorter tailing. More discussionabout the effect of on and offtimes will comein Section5.3. However, this is not the whole story since the previous results showed only the effluent concentrationand gave no information about how the solute concentration in the immobile solution was changing. Fig. 5.6 is a plot of simulated solute concentrations of the mobile, C. (z,t), and immobile, C,, (z,1)31 depth for both (with CL IL 5115 min onloff cycles) at two vs. solutions and (0.5 VIVO and 1.8). The concentration of both mobile and immobile values of increased depth, with solutions with the concentration of the immobile solution
82 "'orr, cent-ction.
0.0
O.D2
0.04
(ki KCI)
O.DB
0.06
o4-
50
(5/15
ntermittent l)
ý2)
--
E
-ir, )
-100
CL
V 0 150
(0)
200
Concentration (M KCI)
0.0
0.005
0.01
0.015
0.02
0.025
0.03
04-
50
Intermittent --R
E E
--
2)
100 0-
150
(b) 200
M C. (Z' (2) C,.(z,
Fig. 5.6: Concentration profile of solutein mobile(blue)and immobile(red)waterat VIVOequalto (a) 0.5 (6 IL cycles)and (b)1.8(22 ELcycles)respectivelyfor intennittentand continuousleaching. Notedifferentscalesfor thex-axis.
83
increasingmore rapidly. The solute concentrationof immobile solution was lower always under IL than under CL. As the leaching progressedthe IL than concentrations of the immobilesolutiondecreased rapidly under more underCL. At VIVO= 1.8 (after 22 IL cycles)both soluteconcentrationin the mobileandimmobilesolutionswasfive to six timeslower underIL thanunder CL. The off times also help in keepingthe differencebetween Qz, I) and C,.(z,1) smallunderIL astherewasmoretime availableto achieveequilibrium betweenmobileandimmobilewater.
S.ZZ2 Running the SIL model with dyf'erent inflow concentrations IL remainsmore efficient than CL evenwhen water of lower quality is used for IL. Fig. 5.7 showsa plot of Ss(tySovs. VIVOwhere: So = the initial salt massin the column, equal to: So
Co Vo =
Ss (t) = the salt massin the column at time t, which is calculatedby: Ss(t) = So+ 0 where QLt)
(C,., - C. (L,, t)) A vd dt p
is the effluent concentration.
The plot is for two different application methods. One is continuous leachingwith distilled water, the other two are IL with 5115min onloff cycles but different solute concentrationsof the leaching water. The three applications were otherwiseunder the sameconditions. The figure showsthat there was less in the column, all the time, under CL with distilled water than remaining salt IL under with water containing 1g solute A. However, leaching of about 80% (Ss(tySo solute of = 0.2, requiring VIVO;4-.1.2) can be achievedwith the same by leaching of quantity water continuously with distilled water or intermittently A (point 1g B in the graph). with water containing solute
B
.............................................. d= 13 -7nm
0.6--
0.4-
.............. ..............
............ ................Continuouir Intermittent
..........
6
Oil
15 mill
Off
ý..... Inflow concentration
0.2-------......
0.01 .............. 0.0
0.5
A.
............ 1.0
1.5 V/Vo
............... ------------------------
....
0.013 M KCI 0 g/1) 0.007 M KCI (0.5 g/1) 0M KCI
........................ 2.0
2.5
3.0
Fig. 5.7: Plot betweenthe relative amountof solute remaining in the column vs. the number of pore volumesfor leaching continuouslywith distilled water and intermittently with low quality water (see text for point A and B).
K Whenthe columnwas intermittentlyleachedwith watercontaining0.5 g solute /1, IL was more effective in leaching than CL after about 0.8 pore (Ss(tySo 0.1 leached 90% volumes, and remainedso until about = of solute and VIVO;zý1.8, point A in the graph). The salt load in the column remained constantunder IL as solutekept enteringwith the inflow water. Leaching salt-affectedsoil with low quality water will be of particular importance in and and semi-arid areas where water of good quality is frequentlyunavailable.
5.3.1 Using the same sphere diameter with different interstitial
velocities: Five experimentswere conductedto continuously leach columns containing 2.4 mm diameter spheres (Table 5.4), each experiment with different interstitial velocities. The outflow rate was controlled by using a peristaltic pump connectedto the column outlet (Fig. 3.3). The experimental conditions in Table 5.4. are shown Table SA Continuousleachingexperimentconditions. Erp.
d
VM
1 2 3 4 5
(MM) 2.4 ei ei ei ei
(mm/min) 1.89 5.56 22.78 48.11 101.00
om 0.498 0.484 0.486 0.514 0.496
olm
Ds *
tolt,
0.226 0.232 0.231 0.190 0.227
(MM2/min) 1.61 7.70 90.30 256.70 294.50
(Lr=250nun) 73.98 25.15 6.13 2.90 1.38
Vm is mobile water velocity (V. = Vd / 0. ) * The valuesare optimisedusing CXTFITI program.
These experimentswere aimed both to test the SIL model at different determine hydrodynamic dispersioncoefficients (D,, ) for to the and velocities these experimentalconditions (in the same manner as in Section 5.2.1). The be later D, values will used with intermittent leaching simulations. estimated
86
Fig. 5.8 showsthe results of SIL, CXTFITI, and experimentaldata as effluent concentrationagainsttime for Exps. 1,2,3,4,and 5. The experimental datapoints follow the samegeneralpattern for all the experimentswith a rapid decreasein effluent concentration after a certain time depending, for these experiments,on the interstitial velocities. As the interstitial velocity increases, the less concentratedwater reachesthe column end sooner and the effluent concentrationdecreasesearlier. The experiment with the highest interstitial longer in its BTCs, becausethe solute of the mobile tailing velocity showeda water leachedout much faster than the solute of the immobile water, which then continuedto slowly diffuse out causingsuch tailing. The SIL model was in to the the simulate all able experimentsvery well using parameters Table 5.4. However, when the velocity becameextremely high ( i. e., Exp. 5), some behaviour occurredin the numerical results (Fig. 5.8e). oscillatory This oscillation (usually in the form of an overshoot) is typical of many higher-order difference equationsdesignedto eliminate numerical dispersion (such as the Crank-Nicolson method). The oscillations slowly die away with
time (Peaceman1977; Smith, 1985) and were found to occur when the disPlacement " residence" time (t. =L) becamevery small and closeto the V.
diffusion (t, time characteristic
a) as shown in Table 5.4. However, by 15D,
decreasingthe time stepsthis problem disappeared(Fig. 5.8f
5.3.1.1 SIL modelpredictionsfor IL The SIL model was run with each experimentalset of parameters(Table 5.4) but with different onloff time combinations. Twelve different intermittent leaching cycles were used to explore the effect of different on and off times and pore-watervelocities on the water requiredto leach 90% of the initial salt
87 Fxp
.:
01 La
0 1-1 0, -16
SiL CXTFtT Ep
0.0
0
300
250
200
50
350
400 (0)
(min)
Tima
1
2
ExpW'-
cot
CXTFIT
1
Exp 0 C)
ýo 0
10
20
30
5,0
40 Time
6,0
7,0
90
100
(b)
(min)
Exp.
3
ol o os
ii -------
SIL CXTFIT
I
0.0 10
20
is Time
25
30
(min)
Fig. SA A plots of effluent concentrationsagainst time for experiments 1,2,3,4 and 5. Note (e) and (f) corresponedto the same experimental data set but having time steps of I min (SEL,e) or 1s (SELI, f) in the numerical integrationsin the simulations. Note also the different x-axis scales.
88 Exp
4
t_).
J
"M -
SIL CXTFIT1 Exp 0.0-0
5
10
is
20 Timýa
30
25
35
40 (d)
(rnin)
Exp.
1-) 1
5
2-9 cýF,
SIL CXTFIT Exp
24
IS Tirne
10 (min)
12
14
16
1
18
Exp.
5
0.
li
0.
SIL 1 CXTFIT Exp 10
0 Time
(min)
12
14
is
iß
(f)
1
89
load of the column. The results are presentedin Table 5.5 and in Fig. 5.9 / V,, V,, time, the respectively,where offtime, and where xyz axesare on V,, = the volume of addedwater neededto remove 90% of the initial
load salt of the column. TableS.S: Valuesof V-IV- at different velocitiesfor sr)heresof diametercl-'-2.4mm. VM (mnVmin) On Time (min) Off Time (min) 1 5 10
22.78
48.71 11
1.047 1.030 1.028 1.028
51 1.162 1.151 1.149 1.149
I
10 1.206 1.205 1.205 1.205
11 0.995 0.985 0.983 0.983
5 1.070 1.064 1.063 1.063
10
I
S.S6
11
1.087 1 0.924 1.084 0.921 1.083 0.920 1.083 1 0.920
5ý 0.945 0.942 0.941 0.941
1.89
10
10
11
51
0.951 0.950 0.950 0.950
0.909
P.915 1 0.917 0.914 0.917 0.914 0.917 0.914 0.917
0.907 0.907
It is apparentfrom Fig. 5.9 and Table 5.5 that the effect of off time was is because in IV, This (2% V,, the sphereswere change at most). small very five in first diffuse 80% the to the minutes of out solute was able of small and the rest period (Fig. 5.1). The effect of on time was more distinct especially at high velocities. At
15 I 48.11 than time time rather mm/minand off =1 min, using min on v,,= low in leaching However, 15% to the at amount of water. min can save up benefit both time the of and were small overall effects and on off velocities,
intermittentleachingdecreasedbecausethere was ample time avail-ablefor diffusion during the displacementperiod. This is consistentwith the field in (1989) found decrease Verma & Gupta who only a marginal observationof intennittent low for their with application salinity over continuous soil
hydraulicconductivityclayeysoil. 5.3.2 Using different sphere diameters with the same interstitial velocity Three continuous leaching experiments were performed (Table 5.6) with
interstitial but different the velocity same with almost spherediameters.A
90
d=
2.4
m7n
rn irn "m in ..
V. IV,, = is number of pore volumes neededto remove 90% of the initial salt load of the column.
Fig. 5.9: Relationshipbetween"a IV, and on and off times for Exps. I to 4.
21
mixture of 13 and 2.4 mm. diameter sphereswas used in Exp. 7 with mass proportionsof 0.56 and 0.44 g g" respectively. TableSA Continuousleachinp,exr)crimcntconditions.
Exp.
VM
6 7 8
om
(MM)
(mm/min)
13 13+2.49 2
4.18 3.83 3.94
Ds
0im
(MM2/Min)
0.425 0.337 0.485
0.259 0.298 0.232
3.2 * 3.7 3.7 *
i ne vaiuesare optimiseausing uxuýn-i program the massproportionsof spherediameters13 &3 were 0.56 & 0.44 g g7lrespectively.
5.3.Z2 SIL modelpredictionsfor
IL
The SIL model was then run with theseprevious conditions (Table 5.6) and for different onloff times. The results are shown in Table 5.7 and presentedas a three-dimensionalgraph in Fig. 5.10. It is clear that mixing sphereswith large and small diameters increasedthe required amount of leaching water (i. e., increasedthe values of V,,/ V,,) significantly and becamemuch closer to the casewhen only large sphereswere used. Table S.7. Valuesof V, / V,, for different spherediametersat v- =-,3.89 mm/min.
13
d (mm) On Time (min) Off Time (min) 5 10 20 40
11
13+2.4
51 10 115 120
948 1 1.07 . 31 1.03 . 923 996 . . 918 969 . .
1.17
1.23
1.11
1.18
1.07 1.04
1.13 1.09
1.28
932 . 1.28 1 926 . 1.18 922 . 1.13 1 919 .
2
15 110-1 15 120 1.02 994 . 978 . 967 .
1.08 1.05
1.12
1.15 1.12
1.09 1.03 - E.O6 1.09 1.01 1.041
11 57
10 115 20
912 . 912 . 912 912 .
976 . 976 . 976 . 976 .
923 . 923 . 923 . 923 .
927 . 927 . 927 . 927 .
928 . 928 . 928 . 928 .
The effectof changingon loff timeson a columnof small spheres(d=2 mm) was small as the velocity is low which allowed sufficient time for difftision. With large spheres,the effect is clearer. The effect of On time was found to be much greaterthan off time. For largest time the =5 min, off spheres(13 mm) would require 26% less leaching
92
N
3.98 = -v.
/Tnin rnrn.
d= 13 in! -n 3 2
M777.
-ll
VaIVO = is number of pore volumes neededto remove 90% of the initial salt load of the column.
Fig. 5.10: RelationshipbetweenUaIV, and on and off times for Exps. 6 to 8.
014 %LV.
had been if been 20 if had I time the than the time min on used water min on in less leaching 19% the caseof a mixture of spheres. to water used,andup The effect of Off time was very small for the smallest spheres(almost for IV,, in V,, time the sphere varied values when off no changewas noticed diameter=2 mm). This effect was clearer for large spheres,though it was still (largest diameter, "best" For the the time. than case effect of on smaller longeston time) 13% of the leachingwater could be savedif an offtime of 40 instead had been 5 of min. used min Fig. S.10 also can be transformedto show the effect of onloff times on the total leachingtime neededto remove the 90% of the initial salt load of the Comparing demonstrates (TT). 5.11 Table 5.8 Fig. these results. and column Fig. 5.11 with Fig. 5.10 shows that saving water is always accompaniedby longer leachingtimes. Table5.8: Valuesof total leachingtime (min) for different spherediametersat v. =-!3.89 mm./min.
d (mm On Time (min) Off Time (min) 5 10 20 40
13+2.4
13 5
10 15 120 51
215 311 504 882
177 1 167 225 198 324 266 405 526
162 186 238 344
2
10 11
205 1 164 1151 302 212 184 311 495 251 880 509 385
51 146 170 221 323
10 115 20
187 1 140 280 187 467 281 840 468
125 1 117 156 141 219 ! 88 344. . _ 281
Tables5.7 & 5.8 show that for the largestspheresat 20/5 min onloff time combination,thenumberof porevolumesrequiredto leach90% of solute for leaching 162 With 5/40 1.28, time taken the was min. a and onlofftime was
leaching 0.918, the time was the and of pore volumes was combination number 882min. This meansthat,to save28% of the leachingwater,the total duration Using increased 5.4 leaching times. the sameargumentwith the sameonloff of it be for that, the mixture of spheres,increasing time combinations can shown the total leachingtime six-fold saved20% of water. For the smallestspheres,
94
-
L,-=
3.9B mm/min
11 (13 92
4)
yron.
771.7n
TT = is the total time neededto remove 90% of the initial salt load of the column.
Fig. 5.11: Relationshipbetweentime to remove 90% of the initial salt load and on and off times, Exps. 6-8.
OR 3LV-
increasing the leaching time by 718% saved only 1.7% of water. It can be concludedthat using IL is much more worthwhile if the proportion of large in aggregates the soil is high. However, the time taken by IL is still very large is and the main disadvantageof this method.
9 Constructing a dimensionless graph:
The off time is the period when diffusive flow within the spheresallows a movement of solute towards their surfaces. The total amount of such movementwill dependon D, and a. The characteristicdifftision time, ti, of a sphere is given (Passioura, 1971)by: a2 15D, If the diffusion of solutewithin the sphereis assumedto be well approximated by a first-order process,then the characteristicdiffusion time is the inverse of the first-order mass-transfer rate coefficient (Passioura, 1971; Hayot & ( Of() ) ft ime Lafolie, 1993). Therefore, using the ratio as a dimensionlessunit tj insteadof off time might be more appropriatebecausethe effects of D, and a diffusion resultswill be minimised. on The mobile phase"residence" time is the averagetime required for the
displacingfluid to reachthe columnend (assumingpiston flow in the mobile phase)and is calculatedby: Lr V,
It alsomightbe moreappropriateto usethe ratio instead of on time. unit
( Ontime
asa dimensionless
96
To calculate the characteristic diffusion time ( t,
a) the sphere 15D,
is be known. For radius, a, should a mixture of spheres,a taken as the mean radius of mixture.
Estimatinga meanradiusfor a mixture ofsphere diameters: Five different estimatesof meanradii havebeenusedin the literature to model the diffusion out of mixed sizesof sphericalaggregates:
1)MSR= the meansquareradius for the mixture (Passioura, 1971) j(j: SR M =,
a,
2p
where ai = the radius of spherei Pi = the mass proportion of spheres of radius ai
2) MVR= meanspherevolume radius (Han et al., 1985) 3p )1(Za, ) MVR=(I: a, P,
3) MER= mean exchange area radius (Hayot & Lafolie, 1993) (Z Si. Pj) MER =3/
Si is the surfaceto volmneratio for spheresS =3 /a. , 4) WAR=weighted-average radius(Raoet al., 1982) WAR
P,-Tý
5) VWR=volume-weightedaverageradius (Rao et al., 1982) VWR=Za,. P, .
97
To ascertainwhich meanradius is the best to averagea mixture of spherediameters,a FORTRAN-77 computer code depending on Eqs. 4.8 & 4.10, for diffusion of solute out of inert spheres into a fixed volume of solution, was developed(Appendix Q. The equationswere:
C=. 0CO
D, q,,t 6y(, v+1)exp(---j-) -
C.
1-E 1 K--
9+9y+q
2
y2
',
and v
(C.*2 V.
is V,,,, thetotal volumeof waterinsidethe spheres. where Thecodewasusedin two differentways: A)- The code was run with the real medium which consistedof a mixture of sphere diameters.At each time step, Eqs. 4.8, and 4.10 will be
solvedfor eachdiameter. B)- The codewasrun assuminga "simple:e' mediumconsistingonly of onemeanradius.The modelwasrun with eachof the five definitionsof mean radii above. The values V,,, Co and V, were kept the samein each case.CaseA is , , considered asthereferenceto comparethe utility of the resultsof caseB. Six testsweredonewith thesetwo cases: Test 1 (Tl): Assumingtwo spheresof diameter13 mm.,and 177 spheresof diameter2.4 mra (giving a massproportionfor largespheresP,=0.727,andfor P2=0.273). smallspheres
98
Test 2 (T2): Assuming two spheresof diameter 13 mm., and 370 spheresof diameter2.4 mm (giving a massproportion for large spheresPI=0.56, and for small spheresP2=0.44).
Test 3 (T3): Assuming 12 sphereswith diametersnormally distributed around a deviation 5 a=0.05. value of standard meanvalue of mm, and a
Test 4 (T4): Assuming 12 sphereswith diametersnormally distributed around a meanvalue of 5 mm, and a value of a=0.1
Test 5 (T5): Assuming 12 sphereswith diameters log-normally distributed 5 of arounda meanvalue mm, and a value of a=1.
Test 6 (T6): Assuming 12 sphereswith diameters log-normally distributed arounda meanvalue of 5 mm.,and a value of cr= 1.5.
For eachtest, the codewas run as A and B aboveand valuesof the B determination between A the resultof case andcase aregiven. coefficientof The resultsare shownnumericallyin Table 5.9 and as an examplethe results in Fig. first time test are shownas a plot of externalconcentrationagainst of 5.12.
It is clear from the valuesof coefficient of determination(Table 5.9) that,for the caseof diffusionfrom spheres,the meansquareradius(Passioura, 1971) MSR,wasthe bestto averagethe mixture of radii for tests1,2,3and4. , However,for the last two tests (5&6) the volume-weightedaverageradius, VWR,wasslightly betterto averagethe mixture. The mixture in Exp. 7 was similar to that of test T2, thus the best is i. MSR. e., averageradius
99
CN
N-1 c 0
0.727
41 0
=0.273
c 0
Reference MSR MVR MER WAR VWR
x Ld
0
50
100
200 Time(min)
150
250
300
350
seetext fbr expidnation
di = the diameter of sphere i Pi = the massproportion of spheresof diameter di
Fig. 5.12: A plot betweenthe external solution concentrationand diffusion time.
100
Table 5.9: Valuesof different aveme radii and their Rý. I
I
Meanradius (MM)
MSR*
I MVR
MER*
I WAR*
I V"*
5.83
6.74
2.82
2.04
5.25
TI
R2
0.958
0.881
0.536
0.326
0.952
5.14
6.68
2.08
1.63
4.29
T2
Meanradius (MM) -R2
0.352
0.219
0.048
0.810
T3
Meanradius (mm)
6.03
6.22
5.70
5.54
5.95
R2
0.998
0.994
0.993
0.983
0.998
7.37
7.47
6.98
6.59
7.31
0.999
0.998
0.991
0.962
0.999
29.93
32.60
26.19
24.02
28.88
R2
0.991
0.957
0.970
0.922
0.993
Meanradius (mm)
25.98
28.21
21.95
19.95
24.98
R2
0.990
0.892
I ---- -0.991
Meanradius (mm) R2
T4
Meanradius (MM) T5
T6
-
0.811
I
0.962
0.952
I
* seetext for explanation.
MSR
'P, a,
j(670.56
+122 0.44) = 4.93 mm.
The meanradiusvaluecalculatedabovewas then usedto calculatethe diffusion for diameters time, the tj characteristic of mixture sphere value of , Exp. 7. However,using a meanradius to predict the BTCs from miscible by displacement instead distribution, the experiments, of real size was shown Hayot & Lafolie (1993) to give a very different result. They statedthat an averageradiuscannotbe found suchthat this "homogeneoue'mediumbehaves similarly to the porousmediumpresentingan aggregatesize distribution,and that the possibility of fmding an averageradius dependsnot only on the distribution but also on the velocity. Therefore,the the of size characteristics previouslycalculatedmeanradiuswasusedonly to calculatethe valueof h-
101
The SIL model was run again with the same conditions of Exps. 6,7, and 8, but for new onloff times as shown in Table 5.10. The new onloff times dimensionless have the the that same were chosenso all experimentswould in 5.13 Fig. The the where the x and y values on results are shown xy axes. (2nttime) Offtime become axes and respectively.
TableS.10: Valuesof (ontimelt-) and (off timelt. ) for Fijz. 5.13.
2
13+2.4
d(mm)
13
On Time 5 (min) On Timelto 0.083 Off Time 14 (min) Off Timelti 0.262
10
20
5
10
20
5
10
20
0.167 38
0.334 76
114
0.077 8
0.153 23
0.306 46
69
0.078 1
0.157 2
0, 315 . 3
0.712
1.425
2.137
0.261
0.749
1.499
2.248
0.792
1.584
2.376
Representingthe results on this dimensionlessgraph did not eliminate the differencesbetweenthe results of the experimentsas was expected.It can be noted that the number of pore volumes hardly changedfor small diameter large it for in the the and mixture samemanner spheres,while changedalmost
is than This because time the rather used graph residence spheres. mainly "characteristic displacemenf' times similar to the characteristic other using diffusion time where the influence of hydrodynamic dispersion and the lateral diffusion from the spheresare taken into account. Such a characteristictime is in is because leached during time the t a very amount of solute out of reach form, especially with the existenceof off times and complicatedmathematical different spheresizes,and will include the integral of the analytical solution of Eq. 2.17 with time.
102
V-= 3,98 m.m/'min
(13 &- 2.4) rim
,-2, '3'
Va0IV
= is number of pore volumes neededto remove 90% of the initial salt load of the Column.
Fig. 5.13: Relationshipbetween
gýlime Va IVOfor Exps. 6 to 8. and , 1
PART TWO
Solute transport through columns of soil aggregates under saturatedcondition
ChapterSix
d4w, 74 SOW4
This chapter introduces the theory of solute adsorption/desorption from soil introduced Equilibrium kinetic and the surfaces. adsorption models are and different principles of modelling transportof sorbedsolute through the soils are explored.
6.1 Definitions Adsorptionis the net accumulation between interface the a solid of matterat phaseand an aqueoussolution phase(Sposito,1989). It differs from include because it development does the precipitation of a threenot dimensional in sucha twomolecularstructure.The materialaccumulating dimensional interface is The the at an moleculararrangement adsorbate. solid is the adsorbent. A moleculeor anion in the surfaceon whichit accumulates ( be is Sposito, that termed soil solution canpotentially adsorbed anadsorplive 1989). The processesof adsorptioninvolve changesin the compositionof the bulk phasefrom which the amountadsorbedcanbe described.It is necessary to distinguishbetweenchangesof bulk phasecompositionbroughtby absorption, when a given chemicalsubstanceis partitionedbetweentwo bulk phases,and
IDA by adsorbtion at an interface (Burchill et d, 1981). Desorption refers to the reverseof the processof adsorption. A generalterm sorption (or sometimesretention) is used when it is not desired,or is experimentallyimpossible,to distinguish betweenadsorption and absorption.
The partitioning of solutesbetweenliquid and solid phasesin a porous medium determined by laboratory as experiments is commonly expressed in twoordinate graphical form where the mass adsorbedper unit mass of dry solids "adsorbent" (S) is plotted against the concentrationof solute in solution (C). This graphical relation of S versus C and the equivalent mathematical description are known as isotherms. This tenn. derives from the fact that (Freeze temperature adsorptionexperimentsare normally conductedat constant et aL, 1979). Solute sorption-desorptionprocessesin soils have been quantified by
different lines. One representsequilibrium two scientists several along reactions and the second representskinetic or time-dependenttypes of reactions.A comprehensive surveyof sorptionrelationshipsfor reactivesolutes in soil hasbeenundertakenby Travis& Etnier (1981).Table 6.1 showssome kinetic-type isotherms. selectedequilibriumand Equilibriumisothermsarethosefor which solutereactionis assumedto be fast or instantaneous.The Langmuir and Freundlich isothenns are perhaps
the most commonly used equilibrium isotherms(Rubin & Mercer, 1981). Kinetic isothermsrepresentslow reactionsin which the amount of solute is function of contact time. Most common is the first-order kinetic sorption (Selim, 1992). reaction
105 Table 6.1: Selectedequilibrium and kinetic-type models(from Selim, 1992). Model Equilibriumtype Linear
Formulation' S-K. C
Freundlich(nonlinear)
S=K. C'
Langmuir
S-W CS,.,.,,/(l +WQ
Langmuir with sigmoidicity
S=W CS..,, /(l +WC+0, /C)
Kinetic type First order n-th order Irreversible (sink/source) Langmuir kinetic Elovich Power Mass transfer
dSIdt - ki(Olp)C - k, S dSIdt = kf(Olp)C4 - kbS dSlOt = k, (elp)(C - C, ) dSlat - kj(E)1p)C(S,,,.,,- S) - kbS dSIdt =A exp(- BS) W(E)lp)C-SdSIdt -, e(elp) (C -C asIdt =
6A, B, b, C*, Cp, le, Kd, kb, kt, k,, n, m, S,,,,., w, and a are adjustable model parameters.
6.2.1 Langmuir equilibrium Isotherm Langmuir (1918) describedthe relationshipbetweenS and C as: K, C
1+K2
C
(6.1)
whereK, andK2are constants, S= C=
dry is the massof adsorbate adsorbent,and per massof is the equilibrium concentrationof solute in the solution after
has adsorption occurred. This isothermwas developedto describethe adsorptionof gasesby solids finite assummingthat the surfaceof a solid possesses a numberof adsorption it it is If the sites. gasmoleculestrikesan unoccupiedsite, adsorbed,otherwise is reflectedback.The derivationof this equationcanbe foundin mostPhysical (e. Adamson, 1976;Rubin& Mercer, 1981). textbooks chemistry g., This isothennhasbeenusedextensivelyin the literature(seeTravis & Etnier 1981) e.g., for sorptionof Pb, Cd, Zn andP amongotherelements. .
iM 6.2.2 Freundlich equilibrium isotherm Freundlich (1926) suggestedthe following empirical equation for describing the sorption of ions or moleculesfrom a liquid onto a solid surface:
(6.2)
S=KC' where K= is the distribution coefficient, found by Baes & Sharp (1983) to be a
from very variable and unpredictableparameter,which may range 3 I dm g-1to many ordersof magnitudegreaterdependingon solute and soil characteristics,including pH. b= is a constant,typically having a value of boo),when the rate of sorption approaches zero,the aboveequationyields kf
s=0
p
kb
C=KC
which resultsin a linear equationsimilar to that for linear isotherms(Eq. 6.3) in which equilibrium conditions were assumed. Eq. 6.4 has been used frequently to describe the sorption kinetics of several different organic chemicals, some heavy metals and more frequently the sorption kinetics of P in soils (Travis & Etnier, 1981).
Adsorption reactions are important processesgoverning the fate of dissolved solutes.Models of solute transport must therefore incorporate a mathematical description of the chemical processesof adsorption as well as the -physical describes dispersion The that processesof convection and classical equation . one-dimensionalsolute transport by saturatedflow (with no solute source or sink) is ( Eq. 2.14): a Ic
ac
ac pas-z'D -V +0 at at aZ2 az S
where V=
is the averageporewatervelocity
C=
is soluteconcentration
Ds z=
= is thehydrodynamicdispersioncoefficient the spaceco-ordinate(positivedownwards).
(6.5)
109 The transportof reactive solutethrough the soil is dependenton the rate of adsorption-desorptionbetween the soil solution and the solid phase. In a general sense,this reaction can be either a kinetic one, in which the relative it is in in time, amount of solute soil solution and soil matrix changingwith or can be an equilibrium situation in which the above relationship is attained rapidly andthereafterremainsconstant(Travis & Etnier, 1981). The modelling of adsorptionwithin the transport models therefore takes one of two directions depending on the acceptanceor not of the local equilibrium assumption "LEA", defined by Yalocchi (1985) as: "If the microscopic processesare "fast enough" with respect to the bulk fluid flow rate, then reversiblesorption reaction can be assumedto be in the stateof local in is In chemical equilibrium7'. other words, an equilibrium situation one which the rate of adsorptionbetweenthe soil solution and solid phaseis much faster than the rate of changein concentrationof solute in the soil solution becauseof any other cause(Travis& Etnier, 1981).
6.3.1 Equilibrium model If the LEA is accepted,the adsorption reaction is considered to be instantaneous, and may be describedby one of the equilibrium adsorption isotherms. The most common approachfor modelling the adsorption term
at
(in
Eq. 6.4) has been to assume instantaneousadsorption and a simple linear
between (Parker S C relation and et al., 1984;Nielsenet al., 1986).i.e.; S=KC
sothat
as= as ac=K ac Tt ac at at Eq. becomes: 6.4 and
110
a 2c
ac
R
DT-j--v ":
atz
ac
az
(6.6)
by; is factor, R the where given retardation pK 0
The validity of LEA is found to depend upon a complex interplay between macroscopic transport properties (flow velocity, hydrodynamic dispersion, time variation of mass input) and microscopic properties (e.g. effective diffusion coefficient, aggregate size, distribution coefficient) (Valocchi, 1985).Nielsen et al. (1986) found that the equilibrium model did not it is likely For that chemical these aggregated soils perform well with soils. transport is not at equilibrium and the equilibrium model fails. Various kinetic diffusion-limited and
have laws (i. models) rate e., non-equilibrium
consequentlybeenproposedto describethis non-equilibrium transport.
6.3.2 Non-equilibrium models As reviewed by van Genuchten& Cleary (1979), most models of nonbeen have flowing through equilibriumadsorptionof solutes soils andaquifers is rate baseduponthe assumptionthat only oneof the microscopicmechanisms limiting. Thesemodelsareusuallygroupedin two classes: I- chemicalnon-equilibriummodels,or 2- physical non-equilibrium models.
6.3.ZI Chemical non-equilibrium models
A chemical non-equilibrium model that did lead to improved transport description is the two- site model (Selim et aL, 1976). The model assumesthat adsorption sites can be divided into two fractions; adsorption on one fraction (type I site) is assumedto be instantaneous(linear Freundlich isotherm), while
ill
adsorptionon the other fraction (type 2 site) is assumedto be time-dependent. This leads to the following equation (Nkedi-Kizzý et aL, 1984; Parker et aL, 1984);
a C+
1+
P OS2
cl t0
19 t
"C-v
'C
OZ2
az
=D,
(6.7)
and, assuming first-order kinetic reaction in site 2 similarly to Eq. 6.3, one
obtains: as
ýL = cc [(I F)KC *
-S2]
(6.8)
t where CC k=
first-order kinetic rate coefficient
F=
is the massfraction of type I "equilibrium" sites
S2
is = the adsorptionon type 2 "kinetic" sites.
functions in be The parametersF and CC found to of most studies were k
from independently be derived pore-watervelocity, and generallycould not batch equilibrium studies.They usually neededto be adjustedfor different experimentscarried out on the samesoil column (Nielsen et al., 1986).
6.3.2.2 Physical non-equilibrium models
Theseare sometimesalso called two-region models. In such models fluid insidethe porousaggregateis assumedto be stagnant,andthusthe total liquid phase is partitioned into mobile (inter-aggregate)and immobile (intraaggregate)water regions. The soil also is divided into two regions (van Genuchten & Wierenga,1976):
112 located the water soil region, sufficiently close to the a) mobile mobile water phase for equilibrium (assumed)between solute in the mobile liquid and that adsorbedby this part of the soil mass,and b) the immobile water soil region, located mainly around the immobile water inside the aggregates.Adsorption occurs here only after the chemical has diffused through the liquid barrier of the immobile liquid phase (diffusion controlled). Transport models are based on first-order exchangeof solute between the mobile and immobile water regions. Van Genuchten& Wierenga (1976) extendedthis concept of mobile-immobile water to include Freundlich-type equilibrium adsorptionprocesses.Their equationsare of the form; a2C.
cc
2-C'" 'D '9 " 0. R. D, O. Ri. ' =O. -v. -0j. at az, az at C` 'a (C. Ci. ) 0,. Ri. =a at
(6.9)
(6.10)
where a=
masstransferratecoefficientbetweenthe mobileandimmobilewater regions
C',
in the mobilewater(within the inter-aggregate = soluteconcentration region)
Ci,,, = soluteconcentration in the immobilewater(within the intra-aggregate region)
0m= volumeof mobilewaterasa proportionof total columnvolume immobilewaterasa proportionof total columnvolume. volume of = jn
113 . R,, and
Ri.. = retardation factors, which account for equilibrium-type
adsorption processes in the mobile and immobile water regions, respectively. For a Freundlich adsorption isotherm, they are given by: fpK bC, b-1
R =l+
9.
f where
0.
l+
(6.11)
(1-f)pKbCjmb-'
01.
representsthe fraction of adsorption sites that equilibrate with the
mobile liquid phase. As f
increases,more adsorption occurs in the mobile
water region and relatively less inside the aggregates,the total adsorption remaining the same, and the chemical will appear later in the effluent (Fig. 6.2). Whenf =I,
adsorptiontakes place only in the mobile water soil region.
The influence of all the model parameterson the shapeof the breakthrough curves (BTCs) was comprehensivelystudied by van Genuchlen & Wierenga (1976). 1.0 INFLUENCE
f q- 10 cm/day 0-0.40
.8
P. 1.30 g/cms D-30 CMZ/day
r55//,
#. 0.65 6-0.15 T'. 3
0.25 /
0.40 0.70 0.70
K, d,.
I/day 0.50
0.55 0.40
,x
0.25 .2
0 0234? POREVOLUME-T
Fig. 6.2: Influence off valueson the BTCs (from van Genuchten& Mierenga, 1976).
114 Comparisonof the two types of non-equilibrium model (Section 6.3.2.1 & 6.3.2.2) showsthat they havethe samemathematicalstructureand can be put into the same dimensionlessform by means of model-specific dimensionless parameters(Nkedi-Kizza et aL, 1984; Parker et aL, 1984). Thus Eqs. 6.7 and 6.9 may both be expressedas
Rý-C+(IaTaT
c'I )o P
a2c
I
paZ2
ac
1
aZ
where c' (Cl (I-P)RO =co -C2) aT
(6.13)
The relations betweentheseparametersand those of the previous two models are given in Table 6.2. Table 6.2 : The relations betweenthe dimensionlessparametersand the physical and chemical non equilibrium modelsDarameters
Parameter
Nonequilibrium chemical model
z
ZIL
ZIL
V.t L v,"L D, O+FpK O+pK
T p
(0
ak(I - P)RL vM
C,
C- CO CMp CO -
C2
Nonequilibrium model physical
v,"t(D L L v. D, 01"+fp K 0+ pK L (x V"IOM
C,-CO
Cinp
CO
S2 -(I - F)K CO
Ci.
(Cjp F)K CO -
Cl"p CO
(I
in theinputwater. Cj,, is thesoluteconcentration P 0. (D= 0. +O/M
CO
m
The retardationfactor R describesthe effects of adsorptionduring transport through the soils (R < 1.0
ion means exclusion or negative
adsorption).The parameterP is directly related to the value of f
F, Or and
reflects the fraction of adsorptionoccurring in the mobile liquid phaseor in site I for physical and chemical non-equilibrium models respectively. Finally, the mass-transfercoefficient co describesthe rate at which equilibrium is attained from an initial non-equilibrium situation; the larger is co the sooner is , equilibrium obtained (Nkedi-Kizzaet aL, 1984). Because both models can be described in exactly the same dimensionlesstransportequation(Eq. 6.12 & 6.13), and the BTC curve in both models could be describedby the samedependentvariable (CI), it follows that the two models are equivalent with respect to their BTCs. Therefore both models were equally successfulin describing measuredBTCs (Nkedi-Kizza et 1984). dependent define However, the two aL, conceptually'different variables in quantities the two models. For example, C2 in the physical non-equilibrium describes the average solution concentration of the immobile water model region, whereasC2in the chemicalnon-equilibrium model definesthe adsorbed concentrationassociatedwith type 2 (kinetic) non-equilibrium sites. Anamosa et aL (1990) modelled displacement of 3H20 from an
undisturbed soil column of a structured soil consisting of a gravelley Oxisol (with aggregatediameters 1 as was expected.With negative charge on the Nkedi-Kizza be the surfaceof soil particles, cationswill adsorbedand retarded. ions during for (1982) found factor R=3.03 calcium et aL a retardation However, displacement Oxisol. through such values miscible an aggregated 0 (Eq. 10.2). depend (i. interactions), K and p, will on e. soil particle-ion For the remaining parameters,since there is retardation, P will not only depend on the mobile water fraction (as for bromide) but also on the distribution coefficient KL (Table 6.2). The value of cocould be different from
that of bromidedueto the smallerdiffusionrate. Using the experimentaldataof continuousleaching (Figs. 8.16a & 8.17a), the fitting program CXTFIT I was usedto optimise the values of P, co,and Ds . The fitted parametersco and P were then used to calculate the parameters in Eq. SIL 9.2. The the and ccof model using optimisation results are shown
Fig. 10.5& Table 10.4. Table 10.4 : Optimisedparametersfor potassium(with estimatedR) using the optimisation program of Parker & van Genuchten(1984) (CXTFIT 1).
I Optimised
Parameters
2 Ds(mm/min)
R*
I
ß
# CL
f#
R'
(Ilmin)
Exp. la
.
3.43
1 3.28 Exp. 2a
1 1
67.55
0.044
129.63
0.032
0.243 0.00039 0.102 1 0.994 , . ý , 1 1 1 0.251
0.00058 0.091 0.9971
R7 coefficient ot determination * values are calculated using Eq. (10.2) # values are calculated from optimised parameters using Eq. (9.2).
The lower values of the dimensionless parameter co for potassium comparedwith bromide (Table 10.2)were expectedbecausethesevalues are
182
Exp,
0.8
ry-TFITI.
Rest fit
1.3
frr
SILY: f=1.0 0.6
SILZ f "Caivuiatcl!
"
0 0.4
SL 1 SIL2 CATFIT1 Exp
0.2
0.0-t 0
2 (a)
Pore voýjmes
Exp. Za CXWn: STI.1ý
Dcýt fýt for (D ft, w) y 10 0.0,91
0.4-
SL 2 CXTFIT1 Exp
o.o4 0
234 Pore volumec.
(b)
Fig. 10.5 : Relative potassiumconcentrationagainst number of pore volumes for: (a) Exp. la and (b) Exp. 2a.
lu
(0)
directly related to the values of the mass transfer coefficient, a
ý!
L)
Vd d
which describesthe diffusion of the ions through the aggregate. Bromide ions will be excluded from the surfacesof soil particles (which were previously found to have a net negative charge, Section 8.1.2) enhancingtheir mobility and resulting in a greatervalue of masstransfer coefficient. The values of a for both Ký and Br' ions increasedas the pore-watervelocity increased(i. e., from Exp. Ia,b to Exp. 2a,b). Rao et aL (1980 a,b) showed,both theoretically and experimentally, that cc (and so (o ) is not constant but a function of residencetime (in addition to effective difftision coefficient, aggregate.size and mobile water content).The residencetime (Eq. 4.14) is a measureof the length of time available for solutes to diffiase into or out of immobile water regions during miscible displacement.They found that cc increasedas the residence time decreased.The residencetime is inversely related to pore-water velocity, thus a should increasewith an increasein pore-watervelocity (Nkedi-Kizza et aL, 1983).
ThePecletnumberfor moleculardiffusionis givenby vd D, where
the mobilewatervelocity diameter the meanaggregate D,,
2/inin diffusion 0.118 for Ký' and =molecular coefficient( and 0.125 min
Br' respectively (Shacketrord& Daniel, 1991) ). Using this equation Peclet numbers are approximately 527 and 1110 for Exps. I a,b and 2a,b respectively.Thesevalues are high enoughQ, >> 20) to ensurethat mechanical dispersion dominates the hydrodynamic dispersion coefficient D, for both experiments (Perkins & Johnson, 1963; Kutilek & Nielsen, 1994). Mechanical dispersion depends only on pore-water velocity
J-U
(Section 2.1.1) which is the same for both ions. This means that similar values of Ds are expected for both bromide and potassium leaching. Tables 10.2 & 10.4 show that the hydrodynamic dispersion coefficients, Ds, used to describe potassium displacement (Fig. 10.5) are higher tfian those for bromide displacement through the same columns at the same velocities. Theoretically, similar values for Ds would be expected. Similar results were found based by Ds Genuchten (1977). They that the values obtained van et aL on 2,4,5-T displacement through Glendale clay loam were smaller than those determined from tritium data (no retardation) under similar conditions. They suggested that one reason was that Ds was influenced by the fact that a linearized adsorption relation was used (here by CXTFITI program), instead of the non-linear one observed. When thef and (x values were calculated from the optimised parameters
(Eq. for both 9.2) in SIL the the and (o and used model predicted, model, lower experiments, effluent potassium concentration than actually observed (Fig. 10.5).This is due to the fact that the SIL model usesthe actual desorption (Eq. instead by linearized in 8.2) CXTFITI the equation of equation used which theseparameterswere optimised. To improve the fit, the value off
was altered using the SIL model ( the
effect of varying (x was marginal). The best optimised value was found to bef high implies This that most of the desorption occurred from the value =1. mobile water soil regions with very little desorption from immobile water soil The desorbed amount of regions. solute, S, depends on the equilibrium concentration,C, of the surrounding solution ( S=K Cb). The desorption curve for this soil (Fig. 8.5) showed a very small desorption at high concentrations increase in desorption with a sharp at very low concentrations ( L, and the model, in effect, describesa semi-
infinite column).
206
At t=O
