153

6

Radionuclide Transport
Processes and
Modeling

C. M. Vandecasteele

CONTENTS

6.1 Introduction 154
6.2 Transport in the Atmosphere 155
6.2.1 Winds 155
6.2.2 Atmospheric Stability 156
6.2.3 The Gaussian Model 158
6.2.4 The Gaussian Model Applied to Radiological Dispersion
Devices 162
6.2.5 Parameters of the Gaussian Model 163
6.2.6 Important Limitations of the Gaussian Model 163
6.2.7 Long-Range Dispersion Models 165
6.2.8 Plume Depletion 166
6.2.8.1 Radioactive Decay 167
6.2.8.2 Wet Deposition 167
6.2.8.3 Dry Deposition 167
6.3 Transfer in Terrestrial Food Chains 168
6.3.1 Direct Contamination of the Vegetation 169
6.3.1.1 Dry Deposition 169
6.3.1.2 Wet Deposition 170

6.5.2.1 Definition of the Relevant Scenario 197
6.5.2.2 Formulation of the Conceptual Model 197
6.5.2.3 Development of the Mathematical Model 198
6.5.2.4 Estimation of Parameter Values 198
6.5.2.5 Calculation of Model Predictions 199
6.5.3 Uncertainties and Errors Associated with Modeling 199
6.5.4 Model Validation 200
6.5.5 Model Types 200
6.5.5.1 Screening Models 201
6.5.5.2 Emergency Models 201
6.5.5.3 Generic Models 201
6.5.5.4 Experimental Models 201
6.5.5.5 Deterministic and Stochastic Models 202
6.5.5.6 Equilibrium and Dynamic Models 202
6.5.6 Uncertainty Analysis 202
6.5.7 Sensitivity Analysis 204
References 205

6.1 INTRODUCTION

Nuclear electricity production generates large amounts of artificial radionuclides,
which may be concentrated through reprocessing into radioactive wastes. The
many applications of radioactivity in industry, medicine, and research make use
of large quantities of artificial radioisotopes. Finally, some conventional industries
(phosphate mills and oil extraction) concentrate naturally occurring radioactive
materials (NORMs) in their residues. These activities are responsible for routine
and accidental releases of radioactive elements into the environment.
Radionuclides discharged into the atmosphere as gas, aerosols, or fine parti-
cles are transported downwind, dispersed by atmospheric mixing phenomena,
and progressively settled by deposition processes. During the passage of the

15 km at the equator and 10 km in the polar regions, constitutes the common
receptor of routine industrial gaseous discharges and accidental atmospheric
releases. This layer, called the troposphere, is a turbulent zone, saturated in water
vapor and constantly mixed by winds generated by the heat balance at the Earth’s
surface.

6.2.1 W

INDS

Winds are the driving force for the transport of airborne pollutants. They deter-
mine the direction of the plume of pollutants and the speed at which these
pollutants are transported downwind. Winds are caused by the interaction of the
forces created by the pressure gradients between anticyclones and depressions
and the Coriolis forces generated by the Earth’s rotation. When equilibrium is
reached between these forces, air masses move parallel to the isobars. In the
Northern Hemisphere, the flow is clockwise around high pressure areas and
counterclockwise around depressions.
Closer to the Earth’s surface, however, below 650 m, the shearing forces of
contact with the ground modify wind direction and speed. These friction effects
can cause the wind to change direction by about 30 degrees (outward around
anticyclones and inward around high pressure areas) between altitude (650 m)
and the surface. The forces exerted by the roughness of the ground surface due
to natural (mountains, hills, valleys, forests) and man-made (buildings and cities)
obstacles can change wind trajectories and speed. Variations in wind speed and
direction (along the vertical axis) creates turbulence, which increases the disper-
sion of airborne pollutants.

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to the adiabatic lapse rate,
• Stable if its temperature gradient is higher than the adiabatic lapse rate,
possibly positive (inversion), and
• Unstable when its temperature gradient is lower than the adiabatic lapse
rate.

FIGURE 6.1

Illustration of the stability conditions of the atmosphere. The dotted arrows
represent the behavior of
an adiabatic air parcel.
Height
T°
stable
unstable
neutral

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Radionuclide Transport Processes and Modeling

157

The vertical temperature profile in the lower troposphere is directly influence
by
•The thermal fluxes to (insolation in the day time) and from (infrared

Wind Speed at
10 m (m/sec)

In the Daytime; Sunshine

During the Night: Cloudiness
Strong Moderate Slight > 3/8

≤

3/8

<2
2–3
3–5
5–6
>6
A
A–B
B
C
C
A–B
B
B–C
C–D
D
B
C
C

AUSSIAN

M

ODEL

The Gaussian model is an empirical model providing an analytical solution to
the transport and diffusion equations representing short duration (puffs) or con-
tinuous (plumes) releases of atmospheric pollutants. It was developed in the early
1960s by Pasquill [1] and Gifford [9], based on a theoretical description of eddy
diffusion in the atmosphere proposed by Sutton [10]. But despite, and also because
of its relative simplicity and because it can be run with limited, readily obtainable
meteorological information, it is still widely used today.

TABLE 6.2
Rough Correspondence of the Stability Classes Between
Different Classification Systems

Pasquill [1] A B C D E F G

McElroy [3]
McElroy and Pooler [4]
B

2

B

1



The Gaussian model is based on the assumption that diffusion of airborne
pollutants can be equated to a probabilistic phenomenon, which can be described
by a Gaussian equation. In other words, the concentration profiles in the plane
perpendicular to the wind axis (plume model) as well as on the wind axis (puff
model) adopt Gaussian patterns (Figure 6.3). Therefore the maximum of concen-
tration is centered on the plume axis. The diffusion intensities are expressed by
the values taken by the standard deviations, which increase progressively with
the distance from the source.
In theory, the model applies only for sites with very simple topography (flat
lands, without obstacles or discontinuities) and rather homogeneous meteorolog-
ical conditions during the release and on the puff or plume travel path. Concen-
trations observed at some distance from the release point can have extreme
fluctuations, depending on variations in wind direction and turbulence, therefore
the model provides only average concentrations.
In the case of a puff release, the concentration (

C

(

x

,

y

,

z

x

,

y

,

z

,

t

)

= (6.1)

FIGURE 6.3

Coordinate system for dispersion calculations (after Turner [56]).
z
y
x
h
H
(x,0,0)
(x,y,0)
(x,y,z)
(0,0,0)














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160

Radionuclide Concentrations in Food and the Environment

where

Q

= total quantity of pollutants released at the stack (in kg or Bq),

σ

i


effect (due to initial ejection speed or higher temperature of the
gases released at the stack compared to that of the air) (in m).
For a continuous release (plume), the concentration (

C

(

x

,

y

,

z

)

) at a given point
(

x

,

y

,

,

y

,

z

)

=(6.2)
When the puff or the plume strikes the ground, total reflection is assumed
(i.e., deposition on the ground is not included at this stage). Mathematically this
is achieved by considering a virtual source identical to the actual one, but sym-
metrical relative to the ground surface (Figure 6.4). The pollutant concentrations
in air, beyond the contact point, are the sum of the direct contribution from the
source and that resulting from the pollutants reflection on the ground.
The equations become
(6.3)
for a puff release and
for a plume release.
Φ
u
yzH
yz
yz
2
1
2
2

xyz
xyz
xy
(,,)
()
exp
()
= −
−
+
2
1
2
3
2
2
2
2
πσσσ
σσ
22
2
2
1
2




















exp
()1
2
2
2
zH
z
σ
C
u
y
xyz
yz
y
(,,)
exp exp= −

+ −
+











2
2
2
2
1
2
σσ





(6.4)

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h

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162

Radionuclide Concentrations in Food and the Environment

6.2.4 T

HE

G

AUSSIAN

M

ODEL

A

PPLIED
TO

R

(

r

= 0.2

×

height of the cloud top). The model then assumes an initial distribution
of the dispersed radioactive material between five initial puffs positioned on top
of each other at different heights from the ground level up to 0.8 times the cloud
top and attributes to each of them a fraction of the total source term (Table 6.3).
With each puff are associated two virtual point sources located at a height
corresponding to that of the puff and at an upwind distance,

d

y

and

d

z

, such that

σ

y

FIGURE 6.6

Gaussian model adapted to cope with the dispersion of radioactivity after
the explosion of a radiologic dispersion device (RDD). The picture illustrates the coordi-
nate system for two of the five fractions of the total plume considered by the model.
Redrawn from Hotspot 49 [11].
X
Y
Z
d
y
d
z
H4
eff
H2
eff
H0
eff

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Radionuclide Transport Processes and Modeling

163

6.2.5 P

ARAMETERS


, is similar to

σ

y

.
Measuring the wind velocity at the effective release height (

H

eff

) is not
necessarily directly feasible. It is possible, however, to estimate it based on
measurements at another level (typically at 10 m) according to the following
relation:
(6.5)
where m

ranges from 0.03 to 0.64 according to the stability conditions and the
type of ground surface (Table 6.4).

6.2.6 I

MPORTANT

L


40.8 × cloud top20%
3 0.6 × cloud top 35%
2 0.4 × cloud top 25%
1 0.2 × cloud top 16%
0 Ground level 4%
uu
H
H
m
eff
eff
10
=






10
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© 2007 by Taylor & Francis Group, LLC
164 Radionuclide Concentrations in Food and the Environment
FIGURE 6.7 Lateral (
y
) and vertical (
z
) diffusion coefficients as a function of the distance
from the release point. Redrawn based on Pasquill-Gifford approximated equations for a
roughness category 1 [7].

B: moderatly unstable
C: slightly unstable
D: neutral
E: slightly stable
F: moderatly stable
σ
Y
σ
z
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Radionuclide Transport Processes and Modeling 165
These conditions are rarely completely fulfilled in reality, especially over
long distances or long durations. Therefore Gaussian models should only consider
short-range travel. There is wide consensus to consider a range from a few
hundred meters to a few tens of kilometers as valid, however, in practice, modelers
often extend the limit up to 100 km.
The plume occupies a limited space volume, while Gaussian distributions
are, by definition, infinite, therefore estimation is limited to situations where the
calculated concentration values are greater than or equal to one tenth of the
maximal concentration.
6.2.7 LONG-RANGE DISPERSION MODELS
In order to be able to predict plume trajectories over long distances (e.g., the
travel of the Chernobyl clouds over Europe), more complex models have been
developed that call for a much more complete set of meteorological observations
and forecasts from meteorological models (e.g., from the European Centre for
Medium Range Weather Forecasting [ECMWF]), including three-dimensional
wind fields (Figure 6.8).
Eulerian models are based on equations of air mass motion, radionuclide
advection and dispersion, and mass conservation, expressed over a three-dimen-

0.32
0.08
0.25
0.40
0.10
0.35
0.56
0.12
0.40
0.64
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166 Radionuclide Concentrations in Food and the Environment
Lagrangian plume and puff models are less demanding. Unlike Eulerian
models, they do well working with a limited number of different sources and
their variation in time. Because they are based on a mobile grid, they are able to
trace the plume from individual sources. They cannot treat chemical processes
unless they are those that can be approximated by first-order kinetics. When
comparisons are made of observed and simulated frequency distributions for fixed
receptors, Lagrangian models provide good estimates of maximum concentration
values, typically within a factor of two or three of those observed. Many models
combine the Lagrangian approach, which follows the history of the release across
a region, with the Eulerian approach for the simulation of pollutant dispersion
through a three-dimensional grid covering that region.
6.2.8 PLUME DEPLETION
As a first stage, most dispersion models estimate the transport of airborne pol-
lutants from their source without considering the processes that reduce the radio-
activity in the air compartment; for example, models consider the total plume
reflection on the ground surface and neglect deposition.
FIGURE 6.8 Example of a long-range plume trajectory. (Courtesy of Dr. L. Van der

i
(per sec) being the radioactive decay constant of radionuclide i. Of course,
the product of the disintegration might not be a stable isotope, so the source term
must also be adapted to take into account the buildup of radioactive daughters.
6.2.8.2 Wet Deposition
Deposition of airborne material onto the ground by the action of precipitation
can be assumed to remove pollutants uniformly throughout the entire air column
up to the top of the plume with first-order kinetics. As for radioactive decay, a
correction factor f
w
can be applied to the source term, that is,
or , (6.7)
where
α
i
= washout coefficient for a radionuclide i (per mm when t is
given in sec),
r = precipitation rate (mm/sec).
Best estimate values of α are 0.58/mm for particulates and 0.40/mm for
elemental iodine. The α values are much less than 0.4/mm for organic iodine and
insignificant for noble gases [14].
6.2.8.3 Dry Deposition
Airborne contaminants can also be removed from the plume in the absence of
precipitation (see Section 6.3.1.1). A correction factor f
D
can be similarly applied
to the source term:
, (6.8)
where v
g

α
fe
w
r
x
u
i
=
− ××






α
fe
D
v
u
e
z
g
h
z
z
z
=
− ×× ×
∂


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168 Radionuclide Concentrations in Food and the Environment
Best estimate values of v
g
are 0.002 m/sec for particulates (less than 4 µm) and
0.04 m/sec for elemental iodine. The α values are much less than 0.0002 m/sec
for organic iodine and insignificant for noble gases [14].
6.3 TRANSFER IN TERRESTRIAL FOOD CHAINS
Airborne radionuclides are transported downwind and dispersed by the mixing
processes in the atmosphere. They gradually settle on land surfaces as a result
of different deposition mechanisms. Plants are contaminated by two main pro-
cesses: (1) direct deposition on aerial parts of the standing vegetation and
(2) indirect contamination by root uptake when radionuclides deposited onto the
soil are absorbed by plants along with water and nutrients. In a similar way,
radionuclides present in irrigation water reach plants by direct deposition on aerial
parts (sprinkling) or via the soil by root absorption. Gaseous radioelements like
14
C and
3
H (as water vapor or tritiated hydrogen) penetrate the plants through the
stomata and are incorporated into organic constituents by photosynthesis and
other metabolic processes. Contamination of animals and animal products results
from inhalation and ingestion of contaminated soil particles, feed, and water [15].
The most important pathways of radionuclides in agricultural systems are shown
in Figure 6.9.
During passage of the radioactive cloud, people are irradiated externally as
well as internally by inhalation. Thereafter exposure of the population continues
via three main pathways: external irradiation from the radionuclides deposited

larger than 45 µm are generally not retained by the vegetation cover but
bounce off the leaves and fall to the ground; smaller particles are more
easily intercepted by the vegetation [19,20]. For very fine particles
(aerosols) or vapor, sedimentation rates are so low that deposition rates
are determined by diffusion processes. Chamberlain [21] showed that
the deposition of very fine particles is inversely proportional to the
thickness of the laminar boundary layer above the leaf surface. The
thickness of this layer is perturbed at the edges of plane surfaces where
most deposition occurs [22].
The density of the vegetation cover. Chadwick and Chamberlain [23]
proposed that the initial interception in grass could be related to the
herbage density. Such an interception model based on the vegetation
biomass [24] is adequate for plants developing a homogeneous canopy
(like pasture grass and cereals in the vegetative growing period) and for
which a good correlation exists between biomass and the leaf area index
(LAI). However, it cannot properly respond to the situation where the
LAI is not a monotonous function of the vegetation biomass, like for
cereals from “shooting” onward. In such cases, an interception model
based on the LAI gives more reliable predictions [25].
The characteristics of the plants. In grass, most of the particles retained
are found on the shoot base below the animal grazing level. Soluble
radionuclides accumulated there can subsequently be remobilized and
redistributed into plant tissues. The inflorescence of cereals has a shape
that favors the interception of fallout particles, which may explain why
wheat was found to be the major source of
90
Sr from weapons testing
fallout in Western diets [26].
The prevailing climatic conditions. Although difficult to account for, the
presence of dew on leaf surfaces favors the capture of falling particles.

2
),
LAI = leaf area index (in m
2
/kg) characteristic of the plant species and
its development stage.
6.3.1.2 Wet Deposition
Wet deposition is the process by which soluble radionuclides dissolved in hydrom-
eteors or bound to aerosol and particles are trapped by water drops (rain, snow,
fog, or mist) and deposited on surfaces. Aerosol particles are captured by falling
raindrops below the cloud, termed washout, or incorporated in raindrops within
the cloud where they can serve as condensation nuclei, termed rainout. The
contamination of plants by sprinkling irrigation is similar to wet deposition.
The interception efficiency of the vegetation depends on the size of the
droplets and the amount of rainfall, as well as on changes in radionuclide con-
centrations in the rainwater as a function of the length of the rainfall period. The
foliar surfaces are able to retain a certain quantity of water and the excess water
is leached to the ground. Moreover, if rain lasts, contamination in the atmosphere
is progressively washed out and less-contaminated raindrops reach the plants: the
less-contaminated rainfall will also leach part of the already deposited radioac-
tivity down to the soil.
The initial interception (D
wet
) of airborne radionuclides by plants due to wet
deposition processes (in Bq/m
2
) can be assessed by [27]
D
wet
= × (1 – e

−
××
×
e
rt
S
ln2
3
,
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Radionuclide Transport Processes and Modeling 171
S = water storage capacity of the plant surfaces (in mm) characteristic
of the plant type (see Müller and Pröhl [27]),
k = radioelement specific factor,
r = rate of rainfall (in mm/sec).
For high biomass density and low rainfall amounts (drizzle), the water reten-
tion capacity of the plant biomass might not be exceeded and most of the rainwater
will be retained by the vegetation. The expression of the initial interception by
wet deposition becomes
D
wet
= × (1 – e
–Λ × t
). (6.11)
Apart from light rain conditions, wet deposition is likely to be much greater
than dry deposition for aerosols and a few times greater for elemental iodine [28].
Rains are very efficient at driving airborne pollutants toward the ground.
6.3.1.3 Retention of Radionuclides Deposited on Vegetation
Foliar contamination is reduced by radioactive decay, weathering processes (wind,

contaminated parts by harvesting or grazing. The contamination of plants
expressed as the activity concentration may also be approximated in early plant
development stages by
C
v,t
= × e
–((λ + λ
w
+ λ
g
) × t)
, (6.13)
where
Y
0
= vegetation biomass (in kg/m
2
) at the time of the deposit,
λ
g
= dilution coefficient due to plant growth (per sec).
C
t
atm
Q
Y
v
t
,0
Q

6.3.2 INDIRECT CONTAMINATION OF VEGETATION
Indirect contamination includes the mechanisms that rule the behavior of radio-
nuclides in the soil and the geosphere, their interaction with soil components,
and their uptake by plant roots. These mechanisms depend not only on the
element, but also on soil processes and on the physiological properties of the
plant roots.
6.3.2.1 Interaction of Radionuclides with Soil
Soils are heterogeneous systems combining three immiscible phases (solid, liquid,
and gaseous) in different and changing proportions depending on the humidity
level. Each phase is highly complex and variable in composition and physico-
chemical properties. Soil characteristics and thickness are also highly variable in
space. They are often stratified in layers, termed horizons, lying on parent bed-
rock. The top layer, or topsoil, is rich in organic material, while underlying strata
are essentially inorganic. Inorganic compounds are generally categorized on the
basis of their size: clay (less than 2 µm), silt (2 to 20 µm), sands (20 to 2000 µm),
gravels (2 to 20 mm), and stones (greater than 20 mm).
Soils are dynamic systems; the properties are acquired and modified with
time due to the joint actions of natural factors (variations in temperature and
humidity, erosion) and farming practices. Radionuclides deposited on the ground
or dispersed within the soil are first dissolved in soil water. Dissolving proceeds via
kinetics, depending on the speciation of the radioelement: it is quasi-instantaneous
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Radionuclide Transport Processes and Modeling 173
for soluble compounds (e.g., CsI), but can be a longer process when radionuclides
are included in insoluble matrices (e.g., fuel particles or vitrified wastes), as
radionuclides cannot be leached before weathering processes in the soil have
altered the matrix [32]. Once in solution, radionuclides can adsorb on the sorption
complex by exchange processes; (co-)precipitate as hydroxides, sulfides, carbon-
ates, or insoluble oxides; form complexes with organic molecules; or remain in

liq.
= total radioactivity content in the liquid phase (in Bq/l),
M = solid phase mass (in kg),
V = liquid phase volume (in l),
or dividing the measured activity in each phase by the total activity in the system,
K
d
= (6.15)
where
f
sol.
= fraction of the activity associated with the solid phase
(dimensionless),
f
liq.
= residual fraction of the activity dissolved in the liquid phase
(dimensionless).








=
sol
liq
sol
liq

and thawing. Hence the modeling of Cs
+
interactions with clays cannot be fairly
described by a simple K
d
approach and more complex relations must be consid-
ered (see, e.g., Hilton and Comans [35]).
Immobilizing ions by fixation onto the soil solid phase or precipitation delays
or prevents their leaching with percolation water down to below the rooting zone.
However, one should always keep in mind that immobilization might be a tran-
sient process. In other words, if the solid phase can be a sink for radioactivity, it
may become a source according to changes in concentration gradients between
the solid and liquid phases or variations in the soil chemical properties (e.g.,
variation in pH or oxidoreduction potential [E
h
]).
Soluble forms of radionuclides move in surface soils, as in geological layers,
by diffusion and are carried along by the flow of water. A simple way to describe
migration can be derived from the Darcy and Fick laws, taking into account mass
conservation:
+= (6.16)
where
C
s
= activity concentration in the solid phase (in Bq/kg),
C
w
= activity concentration in the solution (Bq/l),
t = time (in sec),
= apparent velocity of the radionuclides along the water flow direction

∂
∂
v
C
x
D
C
x
d
w
app
w
**
,
2
2
v
d
*
D
app
*
v
d
*
v
K
d
d
θρ+×

v
):
B
v
=(6.18)
where
[ ]
plant
= activity concentration in the plant (in Bq/kg dry or fresh weight),
[ ]
soil
= activity concentration in the soil (in Bq/kg dry weight).
By definition, the activity concentration in the soil is averaged over a depth
of 10 cm for pasture grass and over 20 cm for other crop species [36]. The activity
concentration in plants for human consumption is generally related to their fresh
weight, while that in fodders is related to their dry weight.
Immobilizing radionuclides by binding (especially irreversible binding) onto
the soil solid phase or precipitation leads to a progressive reduction of their
biological availability for root uptake and hence a decrease in the soil-plant transfer
factor, which only considers the total activity concentration of the radionuclides
in soil, regardless of whether they are bioavailable or not. This is illustrated by
the set of transfer data obtained in experimental fields artificially contaminated
with
134
CsCl (Figure 6.10). The calculated transfer factors decrease exponentially






λ
H
= removal rate attributable to exportation by harvesting or grazing
(per sec).
Losses by leaching (λ
L
) can be expressed by the ratio
(6.20)
FIGURE 6.10 Changes with time of the transfer factors (B
v
) observed in maize leaves.
01224364860728496108120
Time (months after contamination)
0.0
0.2
0.4
0.6
0.8
1.0
B
v
(kg dw / kg dw)
λ
ρ
θ
L
d
sd
v
dK

where
B
v
= soil-plant transfer factor (dimensionless),
M
H
= weight of biomass removed per unit area at each harvest (in kg/m
2
),
N = number of harvests per unit of time (per sec),
ρ = soil density (in kg/l),
d
s
= depth of the rooting zone (in m).
6.3.2.4 Translocation within Plants
Elements absorbed by plants through the root system (indirect contamination) or
through the aerial organs (direct contamination) may be redistributed within
plants. After direct contamination, radioactive elements absorbed by nonroot
absorption processes are redistributed within the plant depending on their mobil-
ity: alkali ions can readily be remobilized, whereas alkaline-earth ions are gen-
erally not redistributed from leaves [37]. Movement of
90
Sr,
144
Ce, and
106
Ru into
the grain of cereals is minimal if deposition takes place in the early stage of
development, while
65

d
=
××
×
,
DK594X_book.fm Page 177 Tuesday, June 6, 2006 9:53 AM
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