Hindered settling and consolidation of mud - analytical results

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~~(

,~l*~,.-TU

Delft

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Delft Universityof Technology

Department of Civil Engineering

Hydraulic and GeotechnicalEngineeringDivision HydromechanicsSection

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Hindered settling and consolidation of mud - analytical results

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C. Kranenburg Report no. 11-92

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Hydromechanies Section

Department of Civil Engineering Delft University of Technology Delft, The Netherlands

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1992

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_Abstract

The theory of hindered settling from a suspension developed by Kynch (1952) is known to satisfactorily describe the settling and primary consolidation of mud. This theory, in which effective stresses are disregarded, is reviewed for this purpose, and some analytical results for interface formation and consolidation are obtained. In particular it is shown, for an empirical expression relating settling velocityw to concentration by volume c given by w/wp

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= (1 - c)fJ, that for sufficiently large times %a> - 1 := (1 -1I(3)({3w;/0a>t1/CfJ-1). Here wp is the settling velocity of a single partiele or floc, {3an empirical coefficient, 0 (oa» the (final)

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layer thickness, and t time. This result is in accord with observations described in the literature (Krone, 1962). An experiment to further test the applicability of Kynch' theory to the consolidation of mud is proposed.

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CONTENTS

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Abstract

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1. Introduetion

2. Review of Kynch' theory 2.1. Equations

2.2. Continuous solutions 2.3. Interfaces

2.4. A boundary-value problem: consolidation on a rigid bottom

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3. Position of the upper interface 3.1 Primary consolidation

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3.2. Application to cohesive sediment 4. Intermediate interface 5. Discussion

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Acknowledgement References Notation Figures

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1. Introduction

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The settling velocity of cohesive sediment (mud) particles or mud flocs dispersed in quiescent water is known to depend on the sediment concentration. At low concentrations it increases with concentration as a result of flocculation. At sufficiently high concentrations (e.g., larger than 2 gil, Ross & Mehta 1989), however, particles interfere so that the settling velocity decreases with increasing concentration, and even becomes equal to zero at a certain (maximum) concentration. Introducing an empirica1relationship between settling velocity and local sediment concentration, Kynch (1952) developed a theory of this process of hindered settling.

When the mud partic1es are deposited on a rigid bottom, a high-density layer builds up in which pore water is expelled very slowly through the influence of gravity. During this consolidation process the density and strength of the deposited layer increase, and effective stresses, however small, develop gradually. Pane &Schiffman (1985), Toorman (1992) and Toorman & Berlamont (1992) showed that, although Kynch' theory assumes zero effective stress, it can be applied not only to the settling stage but also, to a certain extent, to the consolidation process. The applicability to consolidation may be partly due to the empirica! input to the theory. Nevertheless, this approach is appealing because of its simplicity and the difficulties encountered when modelling the effective stress. For reviews of these more advanced theories see Alexis et al. (1992) and Kuijper (1992).

All theories referred to consider physical aspects of consolidation only (primary consolidation). In addition physico-chernical and organic processes may affect the strength of the deposited layer in later stages of the consolidation process.

In this note Kynch' theory is reviewed and some analytical solutions are presented that deal with the primary consolidation process and the formation of interfaces in the settling sediment. These solutions do not add to the theory of settling and consolidation as such, but may be useful when interpreting experimental and numerical results.

This work is part of a research project on the dynamics of fluid mud, which is carried out in the framework of the Netherlands Centre for Coastal Research and the EC MAST-2

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programme.

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2. Review of Kynch' theory

2.1. Equations

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The one-dimensional volume balance for a settling suspension can be written as

oe

+

as

=

0

at az

(2.1)

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where e is the sediment concentration divided by its value in the fully consolidated state

(e=1),S the vertical transport of sediment, t time and z the vertical coordinate (positive in

the downward direction). Assuming a monodispersive sediment, which is a reasonable

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approximation for cohesive sediment in a flocculated state (Migniot, 1968), settling is

characterized by a single settling velocity w. The transport S then becomes

S = wc (2.2)

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The assumption of a single class of settling veloeities could be relaxed (e.g., Kranenburg & Geldof, 1974). The effect of hindered settling is introduced by assuming that the settling

velocity is a decreasing function of the local concentration,

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w

=

wpf(e) (2.3)

wher wp is the settling velocity of a single partiele (that is, at zero concentration), and f(e)

anempirical function satisfyingj(O)

=

1,./(1)

=

0 and djlde :5 0 (the equal sign holding for C =

1). Substituting from (2

.

2) and (2.3), equation 2.1 becomes

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oe

oe

- +w F(e)-

=

0

at

p

az

(2.4)

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where

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₃

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F(e) =~[cj{e)]

de (2.5)

Figuré 2.1 schematically shows functionsJte) andF(e) that have been found to apply to the hindered settling of various sediments including mud (Richardson & Zaki, 1954; Toorman,

1992). The function

f

is convex to the horizontal axis, the function F has a point of

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inflexion, and its derivative at e = 1 vanishes. It should be noted that Kynch (1952)

considered this behaviour to be anomalous (hls case b), and did not analyze it in further detail. However, this is the very case that represents the sustained consolidation of mud observed in experiments.

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2.2 Continuous solutions

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kinematic wavesEquation 2.4 is a first-order quasi-linear partial differential equation describing. Itcanbe solved by integrating along characteristic lines in the (z.z) -plane (Whitham, 1974, Chapter 2). The characteristic lines are given by

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de

o

(2.6)

which implies thateis constant along characteristic lines given bydzldt = wpF(e). Because

F(e) is also constant along these lines, the characteristics are straight lines which for an initial-value problem are given by

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z

= zore)

+

wpF(e) t (2.7)

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where the function Zo(e) represents the concentration distribution at t = O.Itis the inverse of the function, e

=

Co =ZO·I(Z), which gives the initial concentration as a function of z. It is assumed that

ac/az ~

0 at t = O. As an example, flgure 2.2a shows two characteristic lines. Note that characteristic lines do not coincide with partiele paths.

As noted by Kynch, concentration gradients continue to increase when the characteristic lines converge. At a certain instant a discontinuity, or interface, then emerges. The condition for convergence is that along a characteristic the derivative

o

z

l az

o decreases

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as time elapses, see figure 2.2b. Equation 2.7 gives

az =

1 + w dF dc t

azo

P dc dz,

(2.8)

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If the concentration increases with depth (dc/dz,

>

0), an interface will therefore develop if

dF <0

dc

(2.9)

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Figure 2.1 shows that this condition is satisfied for concentrations up to the concentration cm

where Fis minimal (and the function cj(c) shows an inflexion point). The instant t = tj at

which the interface comes into existence is given by

az/azo

= O. Equation 2.8 then gives

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(2.10)

Equation 2.7 gives the associated value of z (= zJ as

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(2.11)

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2.3. Interfaces

A condition goveming the vertical movement of an interface can be obtained by

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integrating (2.1) over a small interval containing the interface. This gives, see figure 2.3a,

(2.12)

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where (ZII

zJ

is the interval mentioned. The first term in this expression is given by

z,

!

f

cdz

= -

U (c2 - Cl)

t,

(2.13)

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where Uis the velocity of the interface andC1•2

=

C(ZI.2,t). Substituting from (2.2), (2.3) and

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₅ (2.13), equation 2.12 gives

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(2.14)

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The velocity U corresponds with the slope of the chord connecting the points 1 and 2 on a diagram of the transport S = wpcj{c), see figure 2.3b. Characteristic veloeities dz/dr = wp F(c) are given by the slopes of the tangents to the curve.

A condition for an interface to endure is that infinitesimal disturbances travel towards it from both sides, or are stationary with respect to it. Since these disturbances travel at the characteristic velocities, the stability condition for the existence of an interface is

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(2.15)

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Referring to figure 2.3b this means that, for the interface to be stabiebe below the curve representing the transport. For a certain concentration, the whole chord mustCl

<

Cmthe chord

that touches the part of the curve that is convex to the horizontal axis gives the maximum

concentration c

2 (=ct)

below the interface at which it is still stabie.

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2.4. A boundary-value problem: consolidation on a rigid bottom

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nontrivial problems the initial concentrations are less than one for zThe boundary condition at a rigid bottom (at z

=

h) is w

<

=

h, an interface starting

°

or C

=

1. As for at z =h, t =

°

may form in certain cases. IfCl

=

c(h-,0)

<

Cm, an interface will arise, the

concentration immediately below the interface being Ct (cm

<

Ct

<

1).However, if Cl ~ Cm

an interface will not form because (2.15) is not satisfied, and the concentration distribution will remain continuous. These two modes of deposition are shown in figure 2.4 for an initially uniform concentration distribution (c = co),

In both cases there is a domain in the (z,t)-plane where the solution no longer depends on the initial condition. In this domain the characteristic lines form a fan starting at z = h, t =0. Since length and time scales are absent here, the solution of (2.4) now is self-similar. Dimensional analysis shows that it can be written as

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6 c

=

c(y) where (2.16) h=z

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Y

=

-wpt

On substitution from (2.16), equation 2.4 gives as a solution in this domain (note that

integration of (2.4) is not needed here)

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y

+

F(c) = 0 (2.17)

Equation 2.17 is in agreement with the solution according to the method of characteristics:

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equation 2.7 becomes

z

=h

+

w

pF(c)t, which result is the same as that given by (2.16) and

(2.17).

This solution, which describes the primary consolidation of deposited sediment, is

discussed further in sections 3.1 and 3.2.

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3

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Position of the uwer interface

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Consider a suspension the concentration of which increases from zero at the upper

side to positive values at greater depths. In this case dF/dc

<

0 and dc/dz,

>

0 locally so

that, according to (2.10), an interface will develop sooner or later. Once an interface has

formed, the concentration Cl above it vanishes. Equation 2.14 then gives for the velocity U,

of the upper interface the familiar result

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(3.1)

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Solving for the concentration c2 immediately below the interface, equation 3.1 gives an

expression of the form

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₇

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1 1

dz"

C =Î (--) 2

w

dt p (3.2)

EliminatingC2between (3.2) and (2.7), in which c

=

c2, gives an equation that can bewritten

as

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z,

= w;G(p)

+

hH(P) (3.3)

where

G(p) = F[fl (P)] , _P=--1 dl"

wp dt (3.4)

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Equation 3.3 is d'Alembert's differential equation, the solution of which is known. Upon integration this equation gives the levelz" of the upper interface as a function of time.

For a uniform initial concentration distribution (c = co) integration of (3.3) in the domain in the (z.n-plane where the solution depends on the initial condition simply gives z

= Zo

+

wp t./{co), because in this domain the concentration equals co.

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3.1. Primary consolidation

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Equation 3.3 reduces to a sirnpler one in the domain where the boundary condition at the bottom determines the solution, because in this case H(P) = 1. Introducing new variables according to

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"1 = z" - h ("1

<

0)

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T = wpt (p

>

0)

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the equation for this domain becomes

"1 = TG(P) (3.5)

8 Integrating (3.5) gives

T

=

C

exprf

dG/dp dp] P - G(p)

(3.6)

where C is an integration constant, which depends on the amount of sediment that is consolidating. Equations 3.5 and 3.6 give

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and Tas functions of p, which variabie now plays the part of a parameter.

As shown in section 2.4, this solution holds good for concentrations cm < c

<

1 so that dF/dc ~ 0, where the equal sign applies to the concentration Cm • At the interface the

concentration, and hence the function F = G (c

>

cJ, increase with time. The parameter p decreases with increasing time (because p = dfl/dt). As a consequence dG/dp ~ O.

Defining p =Pm =j{cm), it follows that dG(pJ/dp

=

0 and from (3.6) and (3.5) that dT ~ 0 ,

dp

dfl < 0

dp and 0

<

p

s

e:

(3.7) These results are illustrated in figure 3.1. The concentration immediately below the interface is given by (3.1),

(3.8) The concentration c2 increases with time.

3.2. Application to cohesive sediment

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A functionj{c) that fairly describes the settling and primary consolidation of mud is

(Ross & Mehta, 1989; Toorman, 1992)

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f(c) = (l-cl (3.9)

where {j is an empirical constant equal to 2 to 5 depending on the mud (and possibly the concentration range) considered. Equation 2.5 gives the function F as

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₉

F(c)

=

(l-c)P-I [l-(P +l)c] (3.10)

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F is equal to zero for c = 11({3

+

1) and c = 1. The concentration cm becomes

(3.11)

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_{Solving (3.9) for c gives c = 1 -flIP. The function G(p) is obtained by substituting}

c = 1 -P liP in (3.10). This gives

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G(p)

=

({3+l)p -BpIrP-I (3.12)

The maximum value Pm of P follows from dG/dp

=

0, or

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(3.13)

The solution of (3.6) is

c

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T = ---={3p

-Y(r

-p

if

(3.14)

Equation 3.5 then gives

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The constant C follows from the value of 11for T - 00 (p - 0). Introducing the thickness 5 of the consolidating layer (5 = -11), the above equation becomes

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5

=

5Q> I

P

+ 1 l 1 -

P

P

(r -

p

ir

(3.15)

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where 500 is the value of 5 for T - 00. The constant Cis equal to 500,

These solutions are shown in figure 3.2 (thickness of consolidating layer) and figure

3.3 (concentration immediately below the interface, C2 = 1 -pIlfl). The results indicate that

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consolidation is faster for smaller (3, which was to be expected because according to (3.9) settling veloeities are less reduced for smaller {3.

The solution obtained can be simplified for large times by approximating for small

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p. This gives T = and Eliminating p gives

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[ ] 1 5 _ 1 {3 - 1 500 T=-I 000 - + {3

{3wl

(3.16)

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This solution, which is also shown in figure 3.2 (straight lines), is of the same form as the

empirical expression due to Krone (1962).

Eliminating p between (3.15) and the equation for c2 (c2 = 1 - pllP), an explicit

relationship between layer thickness and concentration at the interface is obtained,

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(3.17)

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Plotting experimental results as shownin figure 3.2, empirical values of{3 and wpCan be easily determined. Equivalent floc sizes can be calculated from wp and measured bed densities using Stokes' formula.

This procedure was applied to the consolidation experiments on Dutch cohesive

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sediments made by Winterwerp et al. (1992). The duration of these experiments varied from

50 h to 190 h and the values of 000 from 27 mm to 64 mmo Some of the samples contained

considerable amounts of fine sand, the settling velocity of which should be substantially

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larger than that of the mud fraction. To restriet the analysis to this fraction as much as possible, it was carried out for relatively large consolidation times only (equation 3.16 indicates that t - l/wp for a certain ratio 0/000), The results thus obtained are listedin table

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3.1. The accuracies of the {3 values and settling veloeities wp are

±

5% and

±

20%,

respectively. The values of {3 obtained are somewhat lower than those reported in the literature (Ross & Mehta, 1989; Toorman, 1992). The measured excess density of the mud near the bottom was selected to calculate equivalent diameters d, because at the bottom most water not captured in flocs has been expelled so that the bed density becomes approximately equal to the floc density. The equivalent diameters calculated are small and indicate that either little flocculation occurred in these experiments, or effective stresses built up in the consolidating layer.

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Table 3.1

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Site

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{3

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wp

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d

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(J.l.m/s) (J.l.m)

Hollandsch Diep (Moerdijkbrug) 1.9 2.9 5.0

Western Scheldt (Breskens) 1.9 1.0 2.6

Lake Ketel 2.0 9.9 7.7

River Meuse (Belfeld) 2.3 4.3 5.1

Loswal Noord (North Sea) 2.5 1.0 2.4

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4. Intermediate interface

Asindicatedin section 2.4, the presence of the bottom will procuce a rising interface if near the bottom the concentration is less than Cm' In the case of a nonuniform initial

concentration distribution an intermediate interface may arise between this interface and the upper interface and the interface caused by the bottom have arrived. The conditions for an intermediate interface to be formed therefore are

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where U, is the velocity of the bottom geverated interface (U,

<

0), and tiand z,are given by (2.10) and (2.11).

Sufficient conditions for the development of an intermediate interface can be derived

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as follows. Denoting the minima! concentration in the suspension as Ca, condition 4.1a is

certainly satisfied if, see (3.1),

(4. la)

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Since -U,

<

-wpF(c,J, see figure 2.3b, condition 4.1b is certainly satisfied if

(4.1b)

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Substituting from (2.9) and (2.10) this condition becomes

-dF/dc h - zo(c)

>

1

F(c) - F(cm) dzo/dc

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which expression can also be written as

d F(c) - F(cm)

<

0

dc h - zo(c) (4.3)

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The numerator in (4.3) represents the influence of hindered settling, and the denominator that of the initia! condition.

If near the bottom c

>

cminitially, no interface will be generated by the presence of

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the bottom, but nevertheless the "intermediate" interface may come into existence.

As an example consider an initia! concentration distribution given by, see figure 4.2,

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(t = 0) (4.4)

where Cb' a and k are positive constants; c, is the concentration at the bottom, ex a parameter

(the initia! concentration distribution shows a point of inflexion if ex

>

1), and k a length

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determining the level at which the intermediate interface develops. It is assumed that

h ~ k so that (4.2a) is satisfied.

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₁₃

Adopting (3.10) for the function F(c), the quantities in (4.2b) are given by

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_ F(c,,)

=

[{3 - 1] fJ- 1 {3+ 1

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_k

=

«Be

_[

1

a-I C - C ... -In a (l-c)fJ-2[2-({3+1)c]

c

b - Ca (4.5)

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h - Zi k [ C - c

1 ~

wt.

=

-In a - (1 - c )fJ-1 [ 1- ({3+ 1 )C]---L.!

~-~

k

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According to (2.9) the last two equations will give, for a certain concentration c, values of

tiand z, provided c

<

Cm = 2/({3

+

1).However, onIy the resuit with the minimal value of

ti is physically realistic, since, before intersecting, all characteristic lines related to larger tj

values cross the path of the interface that has developed in the meantime. As a consequence, condition 4.2b has to be tested for this minimal ti and the value of

z,

pertaining to it.

Figure 4.3 shows some results for the location in the (z.rj-plane where the intermediate interface starts to develop. The values of {3and

ca

were kept constant, whereas

c,

and Cl! were varied. This figure shows that at small values of

c,

an intermediate interface

will develop only ifCl!is large, that is, if a sharp transition in concentration is already present

initially. All curves for constant

c,

converge at z, = h - k, ti = 0 for Cl! - 00.

It is beyond the scope of this report to calculate the subsequent movement of the intermediate interface.

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5. Discussion

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The comparison of Kynch' theory with the consolidation experiments of Winterwerp et al. (1992) is incomplete in that the influence of the amount of consolidating mud was not

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examined systematically. The effect of effective stresses, which is ignored in the theory, should increase with the amount of mud. Therefore, the applicability of the theory may decrease with increasing amount of mud.

The predicted effect of the amount of mud that is consolidating, is as follows. Equations 3.5 and 3.6 give (for a certain value of the parameter p) values of 7J/rand r/ooo,

or, equivalently, of 0/000 and r/ooo• This means that according to Kynch' theory the

consolidation time r increases linearly with 000 for a fixed value of 0/000, This analytica1

result can be used to further test, in consolidation experiments, the applicability of Kynch' theory to the consolidation of mud.

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Acknowledgement

This work was funded in part by the Commission of the European Communities, Directorate General for Science, Research and Development under MAST 2 Contract no. CT 92-0027 (G8 Morphodynamics research programme).

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₁₅ References

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Alexis, A., P. Ie Hir & C. Teisson (1992) Study of consolidation of soft marine solids:

unifying theories and numerical modelling. In: MAST - G6M, Coastal

Morphodynamics, Final Workshop (abstracts - in - depth).

Kranenburg, C. & H.I. Geldof (1974) Concentration effects on settling-tube analysis. I. Hydraul. Res. 12,3, 337-355.

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Krone, R.B. (1962) Flume studies of the transport of sediment in estuarial shoaling processes. Hydraulic Engineering Laboratory and Sanitary Engineering Laboratory , University of California, Berkeley.

Kuijper, C. (1992) Sedimentation and consolidation of mud, a literature review. Cohesive Sediments Report 41, Delft Hydraulics and Rijkswaterstaat.

Kynch, G.J. (1952) A theory of sedimentation. Trans. Faraday Soc. !a,166-176.

Migniot, C. (1968) Étude des propriétés physiques de différents sédiments très fins et de leur comportement sous des actions hydrodynamiques.

Pane, V. & R.L. Schiffman (1985) A note on sedimentation and consolidation. Geotechnique,

.3..5.,

1, 69-72.

Richardson, J.F. & W.N. Zaki (1954) The sedimentation of a suspension of uniform spheres under conditions of viscous flow. Chem. Eng. Sci. .3" 65-72.

Ross, M.A. & A.J. Mehta (1989) On the mechanics of lutoclines and fluid mud. J. Coast. Res., special issue, 5, 51-61.

Toorman, E.A. (1992) Modelling of fluid mud flow and consolidation. Ph. D. Thesis, University of Leuven.

Toorman, E.A. & J.E. Berlamont (1992) Mathematical Modeling of cohesive sediment consolidation. Coastal and Estuarine Study Series 42, American Geophysical Union,

167-184.

Witham, G.B. (1974) Linear and nonlinear waves. Wiley, New York.

Winterwerp, J.C., J.M. Cornelisse & C. Kuijper (1992) Erosion of natura! sediments from the Netherlands. Cohesive Sediments Report 38, Delft Hydraulics and Rijkswaterstaat.

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Notation

c ratio of concentration and concentration in fully consolidated state

C1,C2 concentrations just above and below an interface

ca

constant background concentration C constant of integration

d diameter

j{c) function representing hindered-settling effect on settling velocity

F(c) function defined by (2.5) G, H functions defined by (3.4)

h initial height of suspension

k length scale of initial concentration distribution in (4.4)

p =

w/

dzldt

S vertical transport of sediment

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w

time

vertical velocity of interface

settling velocity of particles in suspension settling velocity of single partiele

= (h - z)/(w/)

vertical coordinate (positive downwards) parameter in (4.4)

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t U y

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z

{3 parameter in (3.9)

(J,o", thickness of consolidating layer, and value for t - 00

t7

=z-

h

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Subscripts

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b bottom

i

intersection of characteristics

m

minimum of function F(c) (= inflexion point of function cj{c»

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u

upper interface

o initial value

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1

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\

\

\

t,

F

_\

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_i

\

\

\

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\

_\

\F

\

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0

0

Cm

/

₁

~c

\

_I

P

\

/

/'

<,

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-.L- _.../

Fig. 2.1. _{Typical functions./{c) and F(c) for mud.}

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0 -7 t _{0 ~t} t·I Z = Zo (C1)

F>O

C = C1 C ~ C2 Z =Zo(C2)

zt

_-

a

Z

t

b

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Fig. 2.2. _{Characteristic lines in the (z,t)-plane.}

_a

_{influence of functionF, 12 intersection}

of characteristic lines.

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) C C1 C2

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0

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21

_--

_{--_} 22 ---

-2 ~ U

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o

1

I

--)~ C

Fig. 2.3. Moving interface. ~ definition sketch,

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velocity of interface (slope of chord) and stability boundary (clash-dotline).

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z~

I ...

"'

a

,-1

t2

t--J:- ___

---1-,

t1 \

L

' I ~

h

0

t1 t2 _t3 tI.

0

Co Ct ₁

₀

_{Ct 1} ~ t ~C _~C

-)

-

-

-

-

-

- - -

-o

o

t1

tI.

-.!f. __

t3

h

o

t1 t2 _t3 tI. 1

o

Co ~C 1

o

Co ~C ~ t

Fig. 2.4. _{Uniform initia! concentration distribution. Characteristic lines (marked with}

arrows), interfaces (solid lines in (z.rj-diagrams), and concentration distributions at four time levels. i!Co

< cm

,

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Co

>

Cm.

Q

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P-4Q

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Fig. 3.1. Diagram showing values of parameter p.

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- - - -

10

0

J~~~~~

10-

1

o

-1

0

00

t

10-

2 ~ ~

10-

3 Fig. 3.2. / / a

P=

P

ml / / I

"

~ ~

~2

10

1

10

2

10

3

10

4 Wp t

>

---::;I"

5

00

Layer thickness versus time during consolidation. The dash-dot lines represent the asymptotes given by (3.16).

-C2

t

'1.0

0.5

o

\ \ C2

=

Cm \ \ -,

"

-,

-10°

101

102

103

wpt -7

-000

Fig. 3.3. _{Concentration at interface versus time during consolidation.}

-

-

-

-

-10

4

-0

I

~

z

I

h

0

Ca Cb

I

~CO

o

z

intermediate interface

z

·

I

h

o

t·I

Fig. 4.1. _{Initia! nonuniform concentration distribution and diagram of resulting} intermediate interface in the case where c;

<

cm.

I

5

I

h-z

-

_k

3

I

i

2

I

1

I

I

Fig. 4.2. _{Initia! concentration distribution given by (4.4) for a = 0.5, 1, 1.5,2 and 3.}

h-z

j k

i

-1

2

1.5

-

-o

-

_j

-0.4

-o

1

2

~

Fig. 4.3. Location of the point in the (z.zj-diagram where the intermediate interface comes into existence. Results for {3

=

2.5, ca

=

0.02 and various values of

c,and o.

-

- - -

-Wpti

k

-I

I

I

I

I

r

I

I

I

I