FORCED COOLING OF UNDERGROUND
ELECTRIC POWER TRANSMISSION LINES
PART II OF IV
HEAT CONDUCTION IN THE CABLE
INSULATION OF FORCE-COOLED
UNDERGROUND ELECTRICAL POWER
TRANSMISSION SYSTEMS
by
Joel V. Sanders
Leon R. Glicksman
Warren M. Rohsenow
Energy Laboratory
in association with
Heat Transfer Laboratory,
Department of Mechanical Engineering
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Sponsored by
Consolidated Edison Co. of New York, Inc.
New York, New York
Energy Laboratory Report No. MIT-EL 74-004
Heat Transfer Laboratory Report No. 80619-88
May 1974
I
ACKNOWLEDGEMENTS
The authors express their appreciation to Dr. H. Feibus and
to Mr. M. D. Buckweitz of Consolidated Edison for their kind assistance.
The generous support of Consolidated Edison Co. of New York,
Inc. who sponsored this work is gratefully acknowledged.
C
ABSTRACT
Forced-cooled systems for oil-filled pipe-type cable circuits have
recently been considered.
In such systems the conduction resistance
through the paper insulation of the cables is the limiting thermal resistance.
Assuming bilateral
symetry,
steady-state conditions, and two-dimensional
heat transfer, a FORTRAN IV computer program was written to solve the heat
conduction problem in the cable insulation for arbitrary configurations
of a three-cable system.
For a steel pipe, a cable system is most susceptible to overheating
in the equilateral configuration with the three cables touching.
Proximity effects are very significant in forced cooling, especially
when cable
a
nt provided with a copper tape under the insulation moisture
seal assembly, accounting for as much as 21% of the total oil temperature
rise between refrigeration stations.
This figure, however, is reduced to
8% when 0.005 inch thick copper tape is present.
de
4
TABLE OF CONTENTS
.
.....
* .. .
*
Page
TITLE
PAGE
.
. . . . .
.
ABSTRACT .....
ACKNOWLEDGMENTS
TABLE
. .
LIST OF SYMBOLS
LIST OF FIGURES
.
.
. . . . .
OF CONTENTS
.
. .
. ...
..
LIST OF TABLES .
.
..
CHAPTER 1.
INTRODUCTION
CHAPTER 2.
FORMULATION
The Cable Ins
.ulation
. .
Path
The Inter-Cable Conduction
The Solution
Domain
........
ihe
Conduerosition
Domain
. . . PaMeth
. .
Boundary Cond[itions...
itions
........
Variations
onL the
Statement
Problem
OF. .SOLUTIO.
. . . . . . . . .. ..
.
12
. .
3
. . .
4
. . .
7
.
.
.
9
.
.
.
11
* .
.
12
.
.
.
19
.
.
.
19
.
.
.
19
.od
. . .
...
1
. .
20
. . .
22
. . .
30
roblem
. . . . . . .. ...
,ulatiProblems
Nondimensiona1l
Formulation
. . .. .
. ..
* *
31
I OFSOLUTIONS
. ..
. . .
33
. .
.
33
.
.
34
. . . . . . . . .
.
37
. .
.
41
.*
CHAPTER
3.
SUPERPOSITION
General
. . * .
.
.
The Overall Problem
.
.
.
.
. . . .
The Component tProblems
Validity
.
.
.
.
.
.
of t.he Supoerposition Method
Determination of Conductor
Temperatures
CHAPTER
4.
. . . .
.
.
.
.
.
45
THE FINITE-DIFFERENCE METHOD . .
.
.
.
.
.
49
Discretization
. . . .
of
Domains
..
. .
*
49
Difference Form of Governing Equations
and Boundary Conditions ,
.
.
*
55
5
Page
CHAPTER
5.
THE COMPUTER PROGRAM . . .
.
.
.
.
.
.
.
.
.
.
The Coefficient Matrix . . . .
.
.
Forcing
General
.
. . . . . .
Vectors
Verification
CHAPTER
6.
.
.
.
.
61
.
.
.
.
61
. .
.
.
.
.
.
.
.
.
64
. . . . .
.
.
.
.
.
.
.
.
68
. .
.
RESULTS AND CONCLUSIONS
75
Evaluation
. . . . . .
75
. . . . . . . . . . . .
76
Criteria
Results
Conclusions
. . . . . . . ..
Recommendations for Further Work
.
.
.
.
.
92
.
.
.
.
.
92
.
95
CHAPTER
7.
REFERENCES
APPENDIX
A.
THE RELATIVE MAGNITUDE OF CONDUCTION
AND
APPENDIX
B.
.
C.
. . . . . . . . . .
CONVECTION
RESISTANCES
.
.
.
. . . .
96
INVESTIGATION OF THE CABLE-CONDUIT
BOUNDARY
APPENDIX
61
.
CONDITION
.
. . .
. . . . .
.
.
99
THE SOLUTION FOR MAXIMUM CURRENT . . .
.
.
108
.
.
.
.
108
The Superposition Method for
Solution
2 .
.
.
.. . . . ..
Maximizing Current in Solution 2 .
APPENDIX
D.
E.
.
117
THE DIFFERENCE FORM OF THE CONDUCTOR
BOUNDARY
APPENDIX
.
USER
CONDITION
INSTRUCTIONS
. .
. . . . . . .
Geometry and Mesh Size ...
Input
Variables
Output
Variables
Array
Data
Dimensions
Card
Example
126
.
126
..
.
.
1.
132
. . . . . .
.
.
138
. . . . . . . . . . .
.
.
142
. . . . . .
.
.
143
. . . .
.
.
149
.
155
. . . . . . .
157
. .
. . .
. . .
Assembly
Problem
121
. . . . .
. . .
.
.
. .
Capabilities and Limitations of the
Computer
Program
Program
Modifications
. .
. . . . . . .
. .
6
Page
APPENDIX
F.
LISTING OF THE SOURCE PROGRAM
APPENDIX
G.
ONE-DIMENSIONAL SOLUTIONS FOR
TEMPERATUREAND CURRENT .
The
H.
Solution
. .
. . .
.
220
M
.
2220
.
Genera
.
The
APPENDIX
Temperature
162
Current
Solution
224
. . . . . . .
CONSERVATIVE APPROXIMATE SOLUTIONS FOR
MAXIMUM TEMPERATURE AND CURRENT
.
227
* .*
.
227
* .
.
228
* .
.
229
o
. . . .
. . . . ..
The Temperature Solution
The
Current
The
Effective
Solution
.
Perimeter
..
. . ...
. . . . .
·
.
.
230
7
LIST OF SYMBOLS
2D = height
of D3, measured
along y-axis.
D1 = solution domain consisting of the cable insulation of
Cable
1.
D 2 = solution domain consisting of the cable insulation of
Cable
2.
D 3 = solution domain consisting of the inter-cable
conduction path.
h = local convective film coefficient.
h
r
= dimensionless radial spacing between mesh points
in D 1.
h
= dimensionless azimuthal spacing between mesh points
in D 2 .
h
= dimensionless azimuthal spacing between mesh points
in D1 .
h
= dimensionless radial spacing between mesh points
p
in D2
I = current.
jl = radial
index in D.
j2 = radial
index in D2'
j3 = normal
index in D 3.
k = thermal conductivity.
kl = azimuthal
index in D 1.
k2 = azimuthal
index in D 2.
k3 = tangential
index
in D 3 .
q = heat flow per unit length.
q = heat generation per unit volume.
8
r = radial
r
coordinate
= inner radius
in D 1.
of Cable
1 insulation.
r2 = outer radius of Cable 1 insulation,
r = dimensionless
radial
coordinate
in D 1 .
T = temperature,
Toi = mixed-mean oil temperature outside the convective
Oil
boundary layer.
T
= maximum allowable system temperature.
max
Wd = dielectric loss per unit volume.
W = arbitrary loss per unit length.
W
c
= conductor loss per unit length.
W d = dielectric loss per unit length.
W
s
= sheath loss per unit length.
x = normal
coordinate
x = dimensionless
in D 3 .
normal
coordinate
in D 3
y = tangential coordinate in D3
y = dimensionless
tangential
coordinate
in D 3.
a = azimuthal coordinate in D2.
p = radial
coordinate
in D 2
P1 = inner radius of Cable 2 insulation.
P2 = outer radius of Cable 2 insulation.
p = dimensionless
radial
coordinate
in D 2 .
T- Toi
9 - W/k -
dimensionless temperature.
= azimuthal
coordinate
in D1 .
9
LIST OF FIGURES
Page
Figure 1.1
Cross-Section of Underground Pipe-Type
Cable
Figure 1.2
System
. .
. . . . . . .
.
. . .
. . . . . . . .
. .
System
. . . . . . . . . . . . .
Figure 2.1
The Inter-Cable Conduction Path
Figure
Coordinate
Systems
Figure 2.4
The Cable-Cable and Cable-Conduit
Boundary
-
17
Conditions
. .
Surfaces
of D
.
.
.
. .
. .
.
. . .
Figure 3.1
Curves Ci(x) Comprising the Boundaries
of the
Solution
21
Domain
.
.
24
. . .
Convective
. . .
*
*
*
-
27
*
-
29
*
-
35
A Regular, One-Dimensional Finite
Difference
Mesh
. . . . . . . . .
Figure 4.2
Discretization of the Domain D
Figure 4.3
A Typical Mesh for the Equilateral
Configuration
. . . . . . . .
.
.
.
.
.
.
0
.
.
*
*
.
.
*
50
52
-
54
.
58
.
63
.
67
Nomenclature in the Neighborhood
of Pj2,k2 in D 2
........
.
Points Affected by the Application of a
Governing Equation at Various Locations
in
Figure 5.2
*
23
Figure 2.5
Figure 5.1
16
in D 1 , D 2 , and D 3
Regional Divisions in D1 and D 2
Figure 4.4
-
. . .
Figure 2.3
Figure 4.1
*
Bilateral Symmetry of the Underground
Cable
2.2
15
-
*
Cross-Section of Underground Power
Cable
Figure 1.3
. .
a Discrete
Network
. . . . . . . . .
A Radial Mesh Illustrating the Area
Associated With Each Mesh Point . . .
Page
Figure 5.3
Percent-Error
in Temperature
vs. N2,
the Number of Radial Subdivisions
in D 2
. . . . . . . . .
71
1. .
System
System
1.
Figure 6.1
Nomenclature for Cable Configurations
Figure 6.2
Temperature Distribution for the
. .
77
Equilateral-Pipe Configuration . .
. . . . . .
Figure 6.3
Figure 6.4
85
.
Isothermal and Adiabatic Lines for the
Open Configuration - System 1 . . * .*
88
.
Isothermal and Adiabatic Lines for the
89
Cradled Configuration - System 1 .
Figure 6.5
Isothermal and Adiabatic Lines for the
90
Equilateral Configuration - System
the1
. .
Figure 6.6
.
Isothermal and Adiabatic Lines for the
Equilateral-Pipe Configuration . . .
Figure B.1
. . . . . . . . * .
.
91
Fin Geometry for the Cable-Conduit
Boundary
Condition
. . . . . . . .
Figure B.2
Thermal Model of the Conduit Wall
Figure D.1
Nomenclature for the Discretized
.
Boundary
Condition
Figure E.1
Effective
Surfaces
of
Figure E.2
A Discrete Model of the Equilateral-Pipe
Configuration
D3
.
. . . .
Conductor
Figure E.3
.
. . . . . . . . .
. . . . . . .
. . . . . -
100
102
123
129
151
Computer Solution Printout for
Example
Problem
. .
.
.
.
. . . . . . .
156
11
LIST OF TABLES
Page
Table
1 - Boundary Conditions in D1 and D 2
Table
2 - Values for the Physical Parameters
of
Table
Systems
1 and
2
. . . . . . . .
80
4 - Solution 3 for Four Cable
81
Configurations - System 1
Table
5 - Solution 2 for Four Cable
Configurations - System 2
Table
82
6 - Solution 3 for Four Cable
Configurations - System 2
Table
78
3 - Solution 2 for Four Cable
Configurations - System 1
Table
25
. . . . . . . . .
83
7 - Specification of Subdivisions Throughout
Progra
.
Progra
.
D 1 , D 2 , and D 3 in Terms of the Computer
Variables
Table
N(J)
and
M(J)
. . . . . . . .
8 - Input Variables for the Computer
. . . . . . . . . . . . . .
Table
130
133
9 - Output Variables from the Computer
. . . . . . . . . . . . . .
Table
10 - Array
Table
11 - Data
Dimensions
Card
Assembly
. . . . . . . . . . . .
.
. . . . . . . . .
Table 12 - Input Data for Example Problem . . . . .
139
144
147
152
j_
CHAPTER
1
INTRODUCTION
High-pressure oil-filled pipe-type cable circuits
have been used for underground electrical power transmission
for a number of years.
Such circuits employ a steel conduit
inside which are several cables, each consisting of a copper
conductor wrapped with porous, oil-soaked paper insulation
and a protective outer covering.
The space between the
cables and the pipe is filled with a dielectric oil which is
under high pressure.
The oil, which impregnates the paper
wrapping on the cables, provides electrical insulation for
the cables and also transfers the heat generated by losses
in the cables to the conduit
and the surrounding
soil.
Pressurization of the oil prevents vapor formation in the
paper insulation and ensures proper electrical insulation of
the cables.
In this non-circulating
type of system, heat
which is generated in the cables is transferred from the
insulation to the pipe wall by natural convection through
the oil, and then from the pipe to the atmosphere by
conduction through the soil.
The power-carrying capacity of
underground cables is limited by the maximum allowable cable
temperature, which depends on the rate of heat removal from
the system.
Force-cooled systems for oil-filled pipe-type
cable circuits, which appear to have power capacities significantly larger than those of non-circulating systems,
12
13
have recently been considered.
In force-cooled systems,
chilled oil is circulated through the pipe, and heat is
transferred from the oil to the atmosphere at refrigeration
stations.
Most of the heat is transferred from the cables
to the flowing oil, heat transfer
secondary importance [1].
to the soil being of
The cable-to-oil temperature
difference for a given current and voltage is determined by
the overall cable-to-oil heat transfer resistance, which is
due to two effects:
the resistance to conduction heat
transfer through the cable insulation, and the resistance
to convection heat transfer from the surface of the
insulation to the bulk of the oil.
Based on results of the
natural convection experiments performed by Orchard and
Slutz [2], it is demonstrated in Appendix A that the
conduction resistance for the type of system which was
considered is an order of magnitude larger than the
convection resistance.
Therefore the rate of heat removal
from the system depends primarily on conduction, and an
accurate conduction model of the cable insulation is
required in order to confidently predict the cable
temperature.
Conduction within the insulation is complicated by
the proximity of one cable to another.
When two cables come
into direct contact, their mutual presence causes a large
increase in the resistance to heat transfer near the point
of contact.
Consequently, the cable insulation near the
contact point experiences a sharp increase in temperature,
14
which in turn elevates the conductor temperature, and
thermal failure of the system will ensue unless the oil
temperature is appropriately adjusted.
Given a system with
a maximum allowable cable temperature, it is therefore
desirable to know the maximum oil temperature which should
be allowed in order to avoid thermal failure of the system.
This involves determining the two-dimensional (i.e., radial
and circumferential) steady-state temperature distribution
within the cable insulation for various cable configurations,
especially those which produce the most severe operating
conditions.
This heat conduction problem is too complicated
to be solved analytically.
However, the solution for
arbitrary cable configurations is readily obtained by means
of numerical methods.
The particular system which was studied consists
of three circular conductors inside a circular conduit.
The
dimensions of this system are shown in Figures 1.1 and 1.2.
In addition to the outer moisture seal, the cables are
wrapped with skid wires, which protect the cable coverings
and reduce friction when the cables are pulled into the
conduit.
In order to simplify the geometrical problems
which arise in handling configurations of three cables, it
was assumed that the system possesses bilateral symmetry, as
shown in Figure 1.3.
This assumption reduces the system to
one and one-half cables inside half a conduit, while
permitting arbitrary configurations of the one and one-half
15
w
L1J
L3
-I
L')
(9
w
J
a0
co
-i
-Jw
Wl
W
a
F-
O
-
E
E
Q)
r-
E1
2
U
a)
0
0
*
H
C)
'~
U
.,H
C:
t,)
a)4
04
U)
O
O
FO
4-,U
9
.H
a)
I
En
O
U
To
sq
LULI
LJ
LU
I
_J
-J
U
0Uf)
16
U
C)
r-3
L~) "~
3
w
0*.H
0
z
X r0
0a
"4
0
P~
ro
ir
14
UU
4R
w
0
o~~~~~~~
U>
0
UH
4i
u
17
)le 2
Cable
1
;ymmetry
FIGURE 1.3
Bilateral Symmetry of the Underground Cable System
18
cables.
As Figure 1.3 indicates, the half- and whole cables
are referred to as Cable 1 and Cable 2, respectively.
In Chapter
2 a complete
formulation
of the
conduction problem is presented, followed in Chapter 3 by a
discussion of the superposition methods which were employed
in obtaining final solutions.
Chapters 4 and 5 are
concerned with discretizing the conduction model and with
translating the discretized model into a computer program.
In Chapter 6 the results of several problems are discussed,
and conclusions are stated.
I~
~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~
CHAPTER
2
FORMULATION
The Cable Insulation
In developing a conduction model for the cable
insulation, the following assumptions were made:
any axial
conduction along the length of the cable is negligible, thus
reducing the problem to two dimensions; steady-state conditions prevail in the system; the thermal conductivity
throughout the insulation is taken to be uniform.
Using
these assumptions, an energy balance on an infinitesimal
element in a cylindrical coordinate system yields the
following expression, which is a special form of Poisson's
equation
[3]:
1T
r
r
2T
+
2
1
2
ar
2T
2
-
2 a2
2
r
(2.1)
(2.1)
~$
This equation governs the temperature distribution in the
cable insulation, together with appropriate boundary conditions which operate around the various portions of the
cable surface.
The heat generation term q in Equation 2.1
is due to a dielectric loss which occurs throughout the
insulation.
The Inter-Cable Conduction Path
In order to model the situation which exists when
Cables
and 2 are lying together in direct contact (skid
wires overlapping), a special conduction path was placed
19
20
between the cable and half-cable.
A conduction path was
used because there is a small region between the cables in
which the oil is essentially stagnant.
The thermal conduc-
tivity of the path was taken to be the same as that of the
insulation.
The width of the path is usually taken to be
the thickness of a skid wire, since this is as close as the
cables come to actually touching.
As an estimate of how
large an angle the path should subtend along the cable
surfaces, it was decided to use the angle subtended by the
overlapping skid wires.
For the system which was studied,
this angle is approximately 25 ° .
The inter-cable conduction
path is thus an extension of the cable insulation, joining
Cable 1 to Cable 2, as depicted in Figure 2.1.
Since no
heat sources are present within the conduction path, the
governing equation for its temperature distribution is
Laplace's equation [4:
a 2T
a2T
°
-__ +T a7 2- = 0,
ax
(2.2)
ay
where x and y are the normal and tangential coordinates,
respectively, and where "normal" denotes an axis which joins
the cable centers.
The Solution Domain
Two additional assumptions underlie the conduction
model.
The first is that the oil is assumed
to be well-
mixed, so that the oil temperature outside the convective
21
-i2
FIGURE 2.1
The Inter-Cable Conduction Path (Shaded)
22
boundary layer is uniform at a given cross-section in the
The second
system.
conductivity
is that, because
of the very high
of copper, each conductor
is assumed
to be at a
single, uniform temperature (though the two conductor
These two
temperatures are not, in general, equal).
temperatures are obtained from a knowledge of the losses at
each conductor, and this is discussed in Chapter 3 under the
subject of superposition.
The point to be made here is that
the two conductor temperatures are not unknown quantities in
Therefore the conduction problem has
the temperature field.
as its solution domain only the paper insulation surrounding
the conductors and the inter-cable conduction path.
purposes
of nomenclature,
referred
to as D 1
as D2
the insulation
of Cable 1 is
(Domain 1), that of Cable 2 is referred
(Domain 2), and the region
conduction path is called D
domains D
For
comprising
(Domain 3).
to
the inter-cable
The solution
D2 , and D3 , together with their associated
coordinate systems, are shown in Figure 22.
Boundary Conditions
The solution
regions of varying
condition
which
domains
D 1 and D
size according
are divided
into
to the type of boundary
is acting at the cable surface.
A set of
regional divisions for both cables is illustrated in
Figure 2.3, and the boundary conditions associated with the
various
regions
are listed in Table
1.
Regions
II of D 1 and
D2 are not included in the table, because they join the
inter-cable conduction path and therefore have no surface
23
FIGURE 2.2
Coordinate
Systems
in D 1 , D 2,
and D 3
24
optional
FIGURE 2.3
Regional
Divisions
in D 1 and D 2
25
TABLE
1
BOUNDARY CONDITIONS IN D 1 AND D2
D1
Region
Convection
I
Region III
Convection
Region
IV
Cable-Conduit
I
Convection
D2
Region
Region III
Region IV
Convection
(if used)
Region V
Region VI
Region VII
Cable-Cable
Convection
(if used)
Cable-Conduit
Convection
26
The cable-cable and cable-conduit boundary
boundaries.
are depicted
conditions
in Figure 2.4.
There are several different surface boundary
conditions, but it can be shown that they are all convective
in form.
The convective boundary condition itself is
obtained from an energy balance at the cable surface:
=-k
q
h[T(r2,)
-
Toil]
(2.3)
r2
P21
where r2 is the outer radius of the Cable 1 insulation, P is
the perimeter
of the cable,
and h is the local film coeffi-
cient, which may vary around the periphery of the cable.
The cable-cable boundary condition occurs when the
three cables are in an equilateral configuration.
A small
arc along the surface of Cable 2 then lies immediately
adjacent to the line of symmetry.
The arc length is taken
to be the same as that for the inter-cable
conduction
path.
Since no heat flow crosses a line of symmetry, this boundary
is taken to be an insulated
one, which
is just a convective
boundary with a local film coefficient of zero.
While this
boundary condition differs considerably in form from the
conduction mechanism operating in the inter-cable conduction
path, both mechanisms have the same effect on the temperature distribution.
For in the equilateral configuration,
symmetrical conditions on either side of the conduction path
act to prevent any flow of heat across the tangential axis
27
U-,
Q
)
z
0
F-
-i
D
0
3.) _L
(9
z
LL
0
0
U
3
_)
0
0
04
0
H
rH
U
ro
z
Q
-Q
:D
Qj
UlU
z
0--
cr
L
a
w
ci)
U0
O
28
(y-axis) of D3 .
The trilateral symmetry of an equilateral
configuration is thus preserved by modelling the cable-cable
effect as a convective boundary.
An alternative method
would be to employ an optional conduction path for the
cable-cable effect, but this would introduce unnecessary
complication.
The cable-conduit boundary condition, which exists
when either cable is lying directly against the conduit, is
influenced by the following factors:
convection cooling
near the point of contact; the thermal conductivity of the
conduit, which for steel is large; potentially large AC
losses in the conduit itself; and heat conduction from the
conduit to the adjacent soil.
This situation is examined in
Appendix B, where a portion of the conduit wall is
thermally
through
modelled
as a fin, and the thermal
the fin is compared
to the thermal
resistance
resistance
across the cable insulation for a given set of AC losses.
In the most conservative case, the resistance from the fin
base to the oil is an order of magnitude smaller than the
resistance across the insulation.
Thus the cable-conduit
boundary condition for a steel conduit is essentially a
convective one with a slightly modified film coefficient.
The arc on the cable surface affected by this boundary
condition is again taken to be the same as that for the
inter-cable conduction path.
There are two surface boundary conditions in D3,
each a convective one.
These are depicted in Figure 2.5.
29
CONVE(
FIGURE
Convective
2.5
Surfaces
of D3
30
An energy
at the surface
balance
of D 3 gives
-k a
Ti]
ay sufc
surface = +h[T(surface)ITol
(2.4)
where (+) and (-) apply for positive and negative y,
respectively, and h is a variable film coefficient.
In addition to the surface boundary conditions,
there are two internal boundary conditions.
the line of symmetry
boundary
condition
symmetry
bisects
The symmetry
occurs in Cable 1, where the line of
the cable and forms a portion
insulation boundary.
insulated.
and at the conductors.
These are at
of the
This boundary, of course, is perfectly
The boundary condition at each conductor, as was
stated previously, is one of uniform temperature.
The aforementioned governing equations and
boundary conditions, along with the requirement that the
temperature between D1 and D3 and between D2 and D 3 be
single-valued, constitute a complete formulation of the
conduction problem.
Variations on the Problem Statement
Although the heat conduction problem was
originally posed with the oil temperature as the unknown
quantity,
it is possible
to specify
solve for other quantities.
the oil temperature
and
Assuming that the voltage is
constant for a given system, there are three variables:
maximum allowable oil temperature, maximum cable temperature,
31
and maximum allowable current.
third may be found.
Given any two of these, the
In subsequent
discussions
it will be
necessary to specify which of the three variables is
unknown, and so the following solutions are defined for
reference:
Solution 1 finds the maximum cable temperature,
given the current in the circuit and the oil temperature;
Solution 2 finds the maximum allowable current, given the
oil temperature and the maximum allowable cable temperature;
Solution 3 finds the maximum allowable oil temperature,
given the current in the circuit and the maximum allowable
cable temperature.
Nondimensional Formulation
In preparation for numerical solution, the heat
conduction problem is cast into nondimensional form.
Such
a formulation can be obtained by introducing the following
dimensionless variables:
-r --p -x - oil
-
r
r2
r
-
P =
~~~~~~~~~~T
T .l
X
x=2A
Y
W/k
2D'
(2.5)
where
the minimum width of D 3
2A denotes
2D is the height
of D 3
(at the x-axis),
(along the y-axis),
and W is an
arbitrary loss per unit length (Btu/hr-ft). The form of the
governing
equation
in D 1 and D 2 is then
1 2~~~~2
-
-2 32e
2
r
2
- r
Dr
520
3~2
-
(r2 r) 2 q
(2.6)
3
whereas the governing equation in D 3 becomes
4A 2
D2a
a2e
;:X
2
~-x 2 D2 22 ay
(2.7)
The form of the standard convective boundary condition
becomes
k
2
6
r
= he(1,4)
(2.8)
CHAPTER
3
SUPERPOSITION OF SOLUTIONS
General
In the solution of linear problems, such as this
problem of conduction with uniform thermal conductivity, it
is often convenient to employ the principle of superposition.
This reduces the overall problem to a number of simpler
problems, each having the same geometry as the overall
problem, whose individual solutions may be linearly combined
to form the overall solution.
The required number of
separate solutions is equal to the number of nonhomogeneities, or potentials, in the overall problem.
In the
conduction problem which has been posed, there are three
potentials:
the two conductor temperatures and the
volumetric heating effect,
The overall problem may thus be
decomposed into three component problems.
Solutions to
these component problems need to be generated only once for
a particular cable geometry and voltage (dielectric loss);
the total solution for any arrangement of current-produced
losses can then be achieved by suitably combining the three
component solutions.
In the following sections the superposition
technique for obtaining Solution 1 (which finds the cable
temperature) is presented.
It is then rigorously demon-
strated that the overall governing equation and boundary
conditions are obtained from a linear combination of the
33
34
governing equations and boundary conditions of the three
component problems.
introduced:
For brevity the following notation is
x is a generalized position vector for the
overall solution domain (comprised of D1 , D2 , and D3);
x
C3-1,_
(x) denotes all points in the solution domain which
lie on the curve C (x); n
is an outward normal to the curve
Ci(x); v2 is the Laplacian operator.
The nine curves Ci(x)
which comprise the boundaries of the solution domain are
shown in Figure 3.1.
less:
The nine normals n
1
are all dimension-
normals to curves in D1 are nondimensionalized with
r2, normals to curves in D2 with P2 , and normals the two
curves in D3 with the length 2D.
The Overall Problem
The governing equation for the overall problem is
the following:
V20(x) = f(x) ,
(3.1)
where f(x) describes forcing effects throughout the domain.
0(x) also satisfies boundary conditions on the nine curves
C i (x).
On the curve
C (x) the condition
0(x)IXEC (x) =
1
where
is
01 ,
(3.2)
01 is some uniform (as yet unknown), dimensionless
temperature.
On the remaining curves the boundary
35
C2
C.
FIGURE 3.1
Curves Ci(x) Comprising the Boundaries of
the Solution Domain
36
conditions are
ae
x)
hr 2
(3.3)
e(x)
an2
ECC2
XEC2 (x)
hr 2
2x)
3
k
an3
(3.4)
OC
xEC (x)
~
3~
xEC (x)
~ 3 ~
ae(x)
an 4
=0
(3.5)
=
(3.6)
XEC4 (x)
a
(x)
n5
0
XEC (x)
0(x)
where
x
5 x)
nc
6 W
I
02 is a uniform
02
(3.7)
02 '
(as yet unknown), dimensionless
temperature.
ae (x)
hP2
an7
k
O(x)
xCc 7
ae (x)
an 8
2Dh
-
xeC 8~(x)
~
k
(3.8)
C7(X)
6(x)
(3.9)
xEC8 (x)
37
ao (x)
_
an9
2Dh (x)
k
(3.10)
~xC
9 (x)
xEC9 (x)
The Component Problems
The overall problem is decomposed into three
component problems, each of which has only one potential and
is individually solvable.
OA(x), OB(x), and
The component solutions are
C(x).
eA(x) is the solution
in which
the Cable
1 conductor
for the physical
situation
is hot, the Cable 2 conductor
is cold (at the oil temperature) and in which there is no
dielectric loss.
A(x) satisfies the homogeneous governing
equation
V2OA(x) = 0 ,
(3.11)
and it satisfies the following boundary conditions:
OA(x)
where A
0
(X) =A
(3.12)
'
is some arbitrary dimensionless temperature.
a A(x)
-
n2
xEC
2
(x)
hr2
k
OA(x)
(3.13)
xEC
22 (x)
-
38
3CA(x)
3
hr 2
- kOA
(3.14)
(x)
xcC
(x)
3
xEC3 (x)
a 3n3
A(x)
4
= 0
(3.15)
) = 0
(3.16)
xEC4
4 (x)
aOA(x)
an5
xeCc (x)
= 0
0A(x) [x£C
XE 66 ( x)
OA(x)
(3.17)
hP2
-
an)
an 7
k
GA(x)
xcC7(x)
xEC7 (x)
-
(3.18)
7
SeA(x)
2Dh OA(x)
an8
xeC
(3.19)
xEC
(x)
a OA(x)
8
x)
2Dh
-
n9
x£C
(x)
9
k
GA(x)
(3.20)
x C 9 (x)
OB(x) is the solution for the physical situation
in which the Cable 1 conductor is cold (at the oil
temperature), the Cable 1 conductor is hot, and in which
there is no dielectric loss.
The component solution OB(x)
satisfies the homogeneous governing equation
39
=0
V OB(x)
(3.21)
as well as the following boundary conditions:
OB(x)
=
xc1 (
(3.22)
0
a0B (x)
I
hr
an 2
k
~
2
(3.23)
OB (x)
'xEC
(x)
2
X
hr2
k
xC
3B(x)
(3.24)
xEC3(x)
3 (x)
aOB(x)
39Bn4
(x)
1
an4 I1Ix
=0
(3.25)
= 0
(3.26)
C 4 (x)
DOB (x)
an
xc
C(x)
5
5B(x)
OB(x)
where B
0
CC6(
is an arbitrary
aOB (x)
(3.27)
0
dimensionless
temperature.
hP 2
an7 1
XE
= B
k
7 W
OB(x)
k
(3.28)
xCC 7 (x)
40
3eB(x)
2Dh
aeB (x)
an 9
Finally,
eB(x)[
xEC
k
8xeC (x)
(3.29)
8
(x)
2Dh
(3.30)
2kh 6B(x)
x£C 9 (x)
~c9_
C(x) is the solution for the physical
situation in which both conductors are cold (at the oil
temperature), but in which there is a prescribed dielectric
loss.
The component solution
C(x) satisfies the nonhomo-
geneous governing equation
V2 C(x) = f(x)
(3.31)
and the following boundary conditions:
ec(x)IXEC(X)-
(3.32)
0
aec(x)
hr 2
(3.33)
an2
xcC
XE
2
2 (x)
W
aec (x)
an
3
x£C3 (x)
kr
2 e(x)
~
-
xEC3 (x)
(3.34)
41
a6c (x)
0
(3.35)
= 0
(3.36)
=
4xEC
n
4
Wx
-
OC (x)
5
xEC
x
~5xc
--
ec(x)
5 (x)
6
hP2
6c (x)
k ~C
aOC (x)
an 7
(3.37)
0
EC ()
-
-
(3.38)
xeC
(x)
7-
aec (x)
8
-an8
xeC
Cx
k
~
x£C8
xC 8(
I
2Dh
(3.39)
8(x)
a OC (x)
_ 2Dh 0C (x)I
an 9
xc
9
(3.40)
- xC 9 (x)
(x9)
Validity of the Superposition Method
Having described the overall problem and the three
component problems, it remains to demonstrate the validity
of the superposition method.
The three component solutions
are linearly combined to form the total solution according
to
[5]:
6(x) = a 1 OA(x) + a 2 OB(x) +
C(x) ,
(3.41)
42
where a 1 and a2 are two arbitrary constants, to be
determined from two additional boundary conditions in the
That Equation 3.41 is indeed valid is
overall problem.
proven by substituting it directly into the overall
governing equation and boundary conditions.
The following
results are then obtained:
V 0(x) = V [al0A(x) +a 2 0B(x) +C(x)]
1
2
-_
= f(x)
Check
(3.42)
e (x)I~ xC
(x~~) = [alA
= alA
provided alA
=
01
(x) +a
2 eB(
~
x)
+ C (x) ]jI
I
x -~~
Check,
(3.43)
This presents no problem, since
01 is
unknown, and a1 and A ° are both arbitrary.
XEC
()x)
2
a
[a1 OA(x) +a 2 6B(x) +C(x)]
n
-
~~~~~~~~xC2
-
hr2
-
-- k
[a 1 OA(x)
- + a2 OB(x)
- + OC(x)]
hr
= -
k
e (x)
Check
xeC2 (x)
E (x)
xEC2
(3.44)
43
ae (x)
an 3
a
an 3
xEC
[a1 OA(x) + a2
B(x) + 0C(x)] I
xeC
_ J
(x)
3-~
.. ,
(x)
\,=
~I
hr 2
[a1 6A(x) + a2OB(x) + OC(x)]
(x)
xcC3
hr2
0 (x)
k
xC 3(x)
Check
(3.45)
ae (x)
[a1 A(x) + a2OB(x) +C(x)]
(x)
xcC 4-~
xnC4 ()
= 0
0e (x)
a
-an5
3n5
xEC5 (x)
= 0
6(x)
I
EC 6 (x)
a2Bo = 002'
e02 is unknown,
[a1 0A (x) + a 2 B(x) + ec (x)]
xC
5
x)
(3.47)
Check
= [a1 OA(x) + a2 eB(x) +
a2B o
provided
(3.46)
Check
C(x) ] xC
6(
Check,
This also causes no difficulty,
and a 2 and B
°
are arbitrary,
(3.48)
since
44
a (x)
[a1 OA(x)+ a 2 B(x) + ec (x)]
an7
xEC
-
7-~
(x)
hp 2
[alOA(x) + a 2 OB(x) + C(x) ]
k
xEcC7 (x)
hp 2
a 1 GA(x) + a 2 B(x) +OC(x)]
-n
xcc8 (x)
(3.49)
xcC 7 (x)
ae(x)
an 8
Check
6 (x)
k
a8
xEC
2Dh
8
(x)
[a1 eA(x) + a2 OB(x) + OC(x)]
k
xC C 8 (x)
--
2Dh
k
a (x)
8(x)
an 9
xC 9 (x)
Check
=-2Dh
2kh
2Dh
k
(3.50)
CxC
8 (x)
[a l OA(x)a2x)
- +Bx)
2
an9
Dn 9
8
-
+C(x) ]
xEC9 (x)
[a1 OA(x) + a2 OB(x) + C(x)
6 (x)
W
xcC (x)
XEC 9
Check.
I
XC
9
(X)
(3.51)
45
It is thus established that the superposition technique just
described is indeed valid.
temperatures
However, the two conductor
01 = alA ° and
02 = a2B
have yet to be found.
Determination of Conductor Temperatures
All that has been said of the conductor
temperatures up to this point is that each one is uniform.
These two temperatures, though, are uniquely determined by
two additional conditions in Solution 1:
the specified loss
per unit axial length at each conductor.
Returning
momentarily to dimensional variables, let WCl and WC 2 be the
specified conductor losses per unit axial length in Cables 1
and 2, respectively.
Equating W
to the total heat flow
per unit length transferred from the conductor of Cable 1,
the following result is obtained:
,-
ar
r 1dd=w Cl
(3.52)
0
Likewise, for Cable 2
[
2Tr
=
-k
2~
Pd
ap Plia
= WC 2
C
0
These two expressions are rendered dimensionless and
rearranged to give
(3.53)
46
r2Wc
38
r2 Clw
=
(3.54)
0
2 7T
da = -
P2Wc2
(3.55)
1
l
0
Finally, in terms of the present vector notation,
Equations 3,54 and 3.55 become the following:
a
(x)
an 1
xl
cI(x)
WCl r 2
dC (x) = +Wr
xC
-
1
(x)
dC 6 (x)
n
(x)
xC
6
(3.56)
1-
a0(x)
C
I
=
+
WC 2 P2
P
(3.57)
~~~~~
(x)
0(x) may now be eliminated from these two equations in favor
of OA(x),
B(x), and
C(x) by substituting
into Equations 3.56 and 3.57:
Equation
3.41
47
dC1 (x)
an- [al OA(x) + a2 B(x) + C(x)]
1(x)
c (x)
WCl r2
=+
Wr-
(3.58)
'
1
a
an 6
C
1
C 6
C(x)
6
dC6 (x)
OB(x) + 6C(x)]
[a OA(x). +a 2...
_
WC2 P2
+ .
0Wp
1
(3.59)
Since the component solutions OA(x),
B(x), and 6C(x) are
each known, Equations 3.58 and 3.59 are two simultaneous
equations from which the arbitrary constants a1 and a2 are
determined.
Furthermore, since A0 and B0 are known
quantities (the arbitrary dimensionless temperatures which
were used in solutions
A(x) and
B(x), respectively), the
dimensionless conductor temperatures follow directly from
Equations
q
3.43 and 3.48:
801 = alA
~~~~01 1
and 602 = a2B.
2 o*
and 602
This
then completes a description of the superposition technique
for Solution
1.
The technique for obtaining Solution 2 (which
finds the maximum current) is somewhat more complicated,
owing to the fact that current is then a variable.
This
makes necessary a separation of current-produced losses from
48
voltage-produced losses, as well as a subsequent procedure
for maximizing current with respect to the allowable cable
temperature and the oil temperature.
A full description of
this solution is presented in Appendix C.
Solution 3 (which
finds the oil temperature) is nearly identical to Solution 1,
the former requiring only a minor extension of the latter.
In particular, Solution 3 is obtained by using an arbitrary
oil temperature in Solution 1 and then by equally incrementing all temperatures (including the arbitrary oil
temperature) until the maximum temperature in the field has
reached the prescribed allowable value.
The two temperature
distributions therefore have the same shape, differing only
by a constant.
CHAPTER
4
THE FINITE-DIFFERENCE METHOD
Discretization of Domains
The numerical method used to generate solutions
for the various component problems described in Chapter 3 is
the finite-difference method.
It has as its first basic
step discretizing the solution domain.
Discretization is
the reduction of a continuous system into a system which has
a finite number of degrees of freedom.
The basic approxi-
mation involves the replacement of a continuous domain by a
network of discrete points within the domain.
A one-
dimensional example of this is shown in Figure 4.1.
Instead
of obtaining a continuous solution defined throughout the
domain, approximations to the true solution are found only
at these isolated points
Discretization
of D 1 and D 2 is accomplished
by
defining a network of radial and circumferential mesh points.
Since it is desirable in terms of computational labor for
the mesh to be as regular as possible,
conventions were adopted:
uniformly
the following
points along a radius are
spaced, though the spacing in D 1 nmay be different
from that in D2 ; circumferential spacing of points within a
particular region is uniform; the number of circumferential
subdivisions in both Regions II of D 1 and D2 and the number
of tangential
subdivisions
in D 3 are constrained
to be equal,
thereby avoiding mismatches at the D 1 -D3 and D2-D3
49
50
CONTINUOUS
ONE-DIMENSIONAL
DOMAIN
j-1
0
Pj
Pj+
i..
0
DISCRETE
0
*
*
REPLACEMENT OF ONE-
DIMENSIONAL DOMAIN
FIGURE
A Regular,
4.1
One-Dimensional Finite-Difference Mesh
51
interfaces.
It should be noted that, just as the conductors
were not included in the continuous domains D
and D2 , so
are they not included in the corresponding discrete domains.
The dimensionless radial spacings h
and h
are
p
from
obtained
obtained from
h
r
=
1- r
-
h
N1
=
1.- Pl
1-
(4
f
N2
1)
where N1 and N2 are the number of radial subdivisions in
D1 and D2 , respectively.
.~~
spacings h
and h
The dimensionless circumferential
vary in magnitude depending on which
region is involved, since the regions generally are of
varying
size.
Thus there are four values
seven values of h
of h
in D 1 and
in D2 , or one for each region.
A typical
spacing is calculated according to
(h ) n
_ n-
n-l
(4.2)
M
n
where
4n and 4n-1 are the values of 4 at the bounding
lines of Region n, and M
subdivisions
n
is the number
radial
of circumferential
in Region n.
Discretization of the domain D3 is difficult
because of its irregular siape.
For this reason it was
approximated by the more regular shape shown in Figure 4.2.
The nondimensional normal and tangential spacings hx and h y
52
I
CONTINUOUS MODEL
2A: skid wir-ethickness
c: ftE-c os(2)]
B-2 (c2 ct)
B:
V i(£ 7)
D:
sin()
DISCRETE
FIGURE
Discretization
4.2
of the Domain
D
!DEL
DE
53
are obtained by dividing the nondimensional width and height,
respectively, by the appropriate number of subdivisions.
It
is seen in Figure 4.2 that these spacings are not uniform:
h
h
x
yx
on y and h
depends
are small compared
value
for h
y
y
is a function
to changes in h,
was assumed.
Since changes
of x.
a uniform,
The linear dependence
in
mean
of h
x
on y
was retained in the model.
A subtlety regarding the two convective surfaces
of domain D3 is also mentioned here briefly.
At each corner
of D3 there exists a discontinuity in the area available for
conduction heat transfer.
This discontinuity is most
conveniently accounted for in the following manner:
the
regular form of the governing equation, which itself assumes
no discontinuity in area, is applied at each corner point.
Four effective corner locations, which lie outside the
corner mesh points, are thereby established, and these
corner locations define the two effective surfaces for
convection in D3.
The mesh points along either convective
surface thus lie inside the conduction path, rather than on
the boundary itself.
are discussed
Details of this modelling procedure
in the user's
instructions
in Appendix
E.
The discretized domains are shown with a typical
mesh in Figure
4.3.
The number of points
to be used in a
given problem is dictated by the level of accuracy required
in the solution; as the mesh becomes finer, it more nearly
approaches the original continuous domain.
Also it is
economical to use a coarse network in regions where the
54
FIGURE 4.3
A Typical Mesh for the Equilateral Configuration
55
temperature gradient is small, switching to a finer mesh
where there are rapid variations in temperature.
Note, for
example, how the mesh points in Figure 4.3 are arranged:
since the temperature distribution away from points of
contact should be nearly one-dimensional, most of the points
are concentrated between the cables, where large gradients
are expected.
Difference Form of Governing Equations and Boundary
Conditions
The reduction of a governing equation and boundary
conditions for a continuous domain to those of its discrete
replacement may be accomplished physically or mathematically.
In the mathematical approach, which was used by the author,
the continuous formulation is reduced to a discrete
formulation by simply replacing derivatives with finiteWhen this is done, the original
difference approximations.
system of governing partial differential equations is
reduced to a set of n simultaneous
algebraic
equations,
where n is the number of discrete points in the mesh.
Since
the original continuous system is linear, the algebraic
system will also be linear.
In preparation for replacing differential
equations with finite difference relations, two basic onedimensional finite-difference expressions are listed.
extension
Figure
to two dimensions
4.1 let the points
is straightforward.
... , Pj-l
P
The
In
Pj+ 1 ,..' be
separated by a dimensionless spacing h, and let the value of
56
'(z) at P. be denoted by
derivatives
Then the first and second
..
'1
J
at P. may be ap,,roximated by the following
J
finite-difference expressions [6]:
(d,)
+
(h2 ) ;
(4.3)
1-22
+ O(h
(4. 4)
2j+lJJ
=
'+
2
~~~h2
where O( ) denotes the order of the error.
With the
availability of these computational formulas, the process of
ec1.,,tionand boundary conditions of
replacing the governin
the heat conduction
roblem with approximate algebraic
equations is simple and
irect:
at each internal point the
to
finite-difference aproxii,:.i.rion
:.
d
ential equation p,(v)Msi
the values
sex.'
l_-n.graic
equatlon
connecting
J points.
.. r1iqborn
a.
a typical equation
example,
Region
of u at ti.
he governing dffer-
the poLnt
For
(j2,k2) in D2,
IV is:
2
-
,2
__ j22-liik
L+r 2hJ
2
+
e-
4
:
[i2 2
-22tj,
40+rL
-2
+h-- ej2,k2_
Ot4
+
h2
0~~4
j2,k2+i
2'
Wf'
2
j2+lk2 4
57
where
2
is the outer
radius of Cable 2, ()j2,k
2
is the
local volumetric loss, W is an arbitrary loss per unit
length, and the remaining symbols are explained in
Figure 4.4.
This result was obtained by substituting the
two-dimensional forms of Equations 4.3 and 4.4 into a
typical governing equation, such as Equation 2.6.
Two types of exceptional situations can arise in
applying this equation.
The first is that on the boundaries,
not all the neighboring points of a governing equation will
lie within
the domain.
It is then necessary
to introduce
finite-difference approximations to the given boundary
conditions and thereby to eliminate the need for any point
that lies outside the domain.
For example, the standard
convective boundary condition in D 1 reduces to the following
equation after discretization:
[2h
ejl-l,kl+
where
jlkl
6jl+lkl
hr2.6
k
1jl,kl + ajl+l,kl =
is a temperature
(4.6)
on the surface of D1 .
is then a fictitious temperature outside D1 .
However, when the governing equation is applied at the point
Pjlkl
(whose temperature is
jlkl),
simultaneous equations in the unknown
there will then be two
jl+l,kl' and this
fictitious temperature may be eliminated in favor of real
temperatures within D1.
The same problem occurs at the
conductors, where a finite-difference expression for the
58
J2,k2 +1
1'
I
FIGURE 4.4
Nomenclature
in the Neighborhood
of P j2k2
in D2
59
first derivative is required.
In this case, however, the
situation is less easily resolved.
Since no governing
equation is applied at the conductor, there is no way of
eliminating a fictitious point within the conductor.
This
problem is addressed in Appendix D, where a suitable
approximation to the conductor boundary condition is derived.
The basic method involves satisfying the boundary condition
at a slight distance from the conductor, and then relying on
the fact that the temperature distribution is nearly onedimensional in the immediate vicinity of the conductor.
The remaining exceptional case occurs at mesh
points whose neighboring points on either side have different dimensionless spacings.
This happens for radial
spacings at the D1 -D 3 and D2 -D 3 interfaces, and it happens
for circumferential
adjacent
regions
spacings
at the interfaces
in D 1 and D 2.
of all
In such situations
it is
necessary to have finite-difference approximations which
have been modified to fit an irregular mesh.
The
expressions for the first and second derivatives in a nonuniform mesh which are used in this study are readily
derived, either from a Taylor's series expansion about the
central point or by deduction from the mean value theorem of
differential calculus.
dzj
They are the following [7]:
[h2 (hl+h2 )] j+l+[12 2 ] j -
hlh+h2 j-1 )+ ;h
(4.7)
60
-
dz
=
F
2
[2]
2
1
2
*j+
[lh
+
jp +0(h);
2
]-i
2ih(hlh2
2h
(4.8)
where h 1 is the dimensionless spacing between
and h2 is the spacing dimension between
ij
and
j-1 and
j+l
j,
Also
it is noted that these expressions reduce to the standard
form of Equations 43
and 4.4 when the dimensionless
spacings are uniform (h1 = h 2 ).
CHAPTER
5
THE COMPUTER PROGRAM
General
The result of transforming the continuous
formulation of the conduction problem into the corresponding
finite-difference
formulation
taneous algebraic equations.
is a linear set of simul-
A FORTRAN IV computer program
written by the author generates this system if equations,
performs the matrix inversion and multiplication to obtain
various component solutions, and then combines component
solutions to produce a final solution according to the
superposition principle.
User instructions for the program
are discussed in Appendix E, and a complete listing of the
source program is given in Appendix F.
The Coefficient Matrix
The set of simultaneous equations for the
conduction problem may be written in the form
[A]{e} = {B} ,
where
[A] is an nxn matrix
of coefficients,
(5.1)
{}
is a vector
of n unknown dimensionless temperatures, and the right-handside vector {B} is a vector of n forcing elements.
Referring back to Figure 4.4, a typical algebraic equation
was shown to be of the form
61
62
(ajlk)
+ (ajk)
jl,k
jk +(aj+l,k)
j+l,
k +(ajk) jk 1
+(a.j,k+l
k+l 52 jk+
where
jk
bj,k '(5.2)
is the central point at which the governing
equation was applied, and the an
are the coefficients
m,n
which were given in Equation 4.5. A typical equation thus
involves five points - a central point and its four
neighbors - and a typical row of the coefficient matrix
accordingly has five non-zero elements.
However, the
application of a governing equation at certain mesh points
produces rows with fewer than five non-zero elements.
These
situations are depicted in Figure 5.1, together with a
typical mesh point.
In this figure, points P3 and P4
initially had the full complement of four neighboring points,
but fictitious points outside the domain were eliminated by
incorporating the boundary conditions at those locations
into the governing equations.
It is a simple matter to assemble the various
coefficients a
into a matrix. The only requirement is
m,n
that the rows be arranged so as to place the coefficients of
central points (Pi, P2 ' P3 , or P4 in Figure 5.1) on the main
diagonal of the matrix.
Since a governing equation will
necessarily involve the central point at which it is applied,
it is therefore ensured that only non-zero elements will
appear on the main diagonal, a necessary condition prior to
matrix inversion.
63
-MESH
O
POINT
CONDUCTOR
POI
P.
4
P,TYPICAL
POINT WITH FOUR NEIGHBORING MESH POINTS
P 2 -POINT
WITH THREE NEIGHBORING MESH POINTS
ONE NEIGHBORING CONDUCTOR POINT
P3 -POINT
WITH THREE NEIGHBORING MESH POINTS
P-POINT
WITH
4
TWO NEIGHBORING
MESH
AND
POINTS
FIGURE 5.1
Points Affected by the Application of a Governing
Equation at Various Locations in a Discrete Network
64
The coefficient matrix is inverted by the packaged
subroutine RMINV, which uses the standard Gauss-Jordan
An automatic feature of this subroutine is the
algorithm.
calculation of the determinant of the matrix.
In order to
keep the order of magnitude of this determinant within a
range acceptable to FORTRAN IV, each row of the matrix and
the corresponding elements of the forcing vectors are scaled
so that the largest element of every row is unity.
matrix has been inverted, the temperatures {}
Once the
are available
from
{8} =
[A]
{B} ,(5.3)
where [A]- 1 denotes the inverse of [A].
Forcing Vectors
In the conduction problem the total heat flow is
comprised of the conductor losses, the dielectric loss, and
the sheath loss.
potentials:
heating.
This heat flow is driven by two types of
the conductor temperatures and the dielectric
These potentials are accounted for in the right-
hand-side vectors {B} of Equation 5.1.
reference
the following
vectors
For purposes of
are defined:
{B}1 is the
forcing vector for the component problem in which a
conductor
temperature
is driving
the heat flow; {B} 2 is the
forcing vector for the component problem in which the heat
flow is driven by dielectric heating.
65
The procedure for generating {B}1 is suggested by
point P2
of Figure
all zero.
points
5.1.
The elements
of {B}1 are initially
However, when a governing equation is applied at
adjacent
the neighboring
to the conductor,
points
such as point P 2 , one of
is the conductor
itself.
The
conductor temperature is not an unknown, though, and when
the governing
equation
is written
in the form of
Equation 5.1, the term involving the conductor temperature
is carried over to the right-hand side and becomes a forcing
term.
The non-zero
elements
of {B}1
are therefore
comprised
of conductor temperature-terms which have been referred to
the forcing vector.
The elements of {B}2 appear as volumetric heating
terms in the difference form of Poisson's equation,
Equation 4.5.
It can be shown [8,9] that the dielectric
loss per unit volume
is of the form:
WdW= -C
2'
r
(5.4)
_
where
C is a constant
for a given system,
and r is the
radius at a point in the insulation where the local dielectric loss per unit volume is wd.
the dielectric
loss is known,
Since the distribution of
it is possible
to integrate
Equation 5.4 over any particular area to obtain the total
loss per unit axial length within that area.
A typical
radial mesh and the areas associated with each mesh point
66
The dielectric loss per unit
are shown in Figure 5.2.
length in a typical
(Wd)2
(d)2
=
{dd=
d
=
say A 2 , is given by
area,
~~~~Cr
(A)(2rr)dr
d
W
d
3
= 2C
A
ln
r=
(5.5)
Jr 2
r2
The total dielectric loss per unit length is:
r6
wddA = 2C
Wd =
(5.6)
ln(r)6
1
The loss per unit length in any particular area may be
expressed as a fraction of the total loss per unit length
just from information about the radial mesh.
For example,
ln (__3)
(Wd)2
(
Wd
(5.7)
rl
The computer program distributes the dielectric loss in this
manner.
The fraction of the total loss per unit length
which occurs in each discrete area is calculated from the
shape of the radial mesh.
The dielectric loss per unit
length for each area is then obtained by multiplying the
67
&
FIGURE
MESH POINT
5.2
A Radial Mesh Illustrating the Area Associated
With Each
esh Point
68
various fractions by the total loss per unit length, Wd, a
number which is supplied as input data.
Each mesh point
therefore has an associated dielectric loss and the volumetric loss terms in Equation 4.5 can be generated accordingly.
A question then arises, however, regarding the disposition
of the loss in the innermost discrete area, A1 in Figure 5.2.
It is noticed in this figure that there is no mesh point
associated with A1.
For this reason the loss which occurs
in A
is added to the current-produced heating of the conduc-
tor.
This suitably accounts for the loss, providing a con-
servative approximation to the true dielectric distribution.
The sheath loss is a current-produced loss which
occurs on the surface of the cable.
In the finite-
difference model, this loss is placed in the outermost
discrete area of insulation, which is associated with the
mesh point on the cable surface.
In Figure 5.2, for
example, the sheath loss would be placed in A5 .
treated
as a volumetric
type of heating
dielectric loss for that area.
introduced,
which
is accounted
It is then
in addition
to the
A third potential is thereby
for in a third vector:
{B}3
is the forcing vector for the component problem in which
sheath heating drives the heat flow.
Verification
During the course of developing the computer
program, periodic tests were performed to ensure its
correctness.
One general technique used to verify a
computer program is to solve a problem with it whose
69
solution is already known and then to compare the two
results.
Several such problems, as well as a simpler
checking procedure, were employed in this study.
The first major verification was a check on the
coefficient matrix.
This test was accomplished by printing
out the matrix for a 111-point mesh and then by verifying
each element by hand computation.
The 111-point mesh was of
sufficient size for the matrix-generation portion of the program to pass through all its decision branches.
This test
was repeated for meshes of decreasing size, until the matrix
for the smallest possible mesh had been verified.
In partic-
ular, the coefficient matrices for the following mesh sizes
were verified:
16-point.
111-point, 57-point, 24-point, 17-point, and
Forcing vectors were checked in a similar manner.
The second major verification of the computer pro-
gram was accomplished by solving the following problem:
the
conductor of Cable 2 was maintained at a specified hot temperature, while that of Cable 1 was maintained at the oil
temperature.
Sheath and dielectric losses were not included.
The height of the inter-cable conduction path was chosen so
as to include an angle of 6
in either cable.
This angle
was judged to be sufficiently small so as to minimize the
thermal effect of Cable 1 on Cable 2.
Temperatures in
Cable 2 diametrically opposite the intercable conduction
path (and hence far-removed from the limited two-dimensional
effect) were then compared to corresponding analytical temperatures from the one-dimensional solution.
This comparison
70
was repeated for successively finer meshes in order to
examine the convergence of the solution.
In performing the
test, temperatures at similar radial points were compared,
and a root-mean-square error was defined:
.1/2
N
-1 IN~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
i
[(Tj)computer- (Tj)analytical]
RMS-error
(F)
=
j-l
.
.
N
r
N
(5.8)
where N is the number of mesh points along a radius in
Cable 2.
This error was then expressed as a percentage of
the total temperature drop through the insulation.
The re-
sult for five different mesh sizes is shown in Figure 5.3.
The errors demonstrate the typical (1/h2)-dependence on mesh
size, which is expected, since nearly all the finite difference expressions used in this problem are
tions.
2
(h ) approxima-
It is further seen that the computer solution clearly
converges to the analytical solution, having less than one
percent error in temperature with as few as three radial subdivisions.
A final run was then made with this problem to
check whether a symmetrical cable configuration would produce
a symmetrical temperature distribution.
A coarse mesh of
18 points was used with a cable geometry such that the line
joining the cable centers was a line of symmetry.
In the
resulting temperature distribution, the seven temperatures
71
10
6
0
0o Cr
4
LLJ
o°
2
v
3
2
5
4
z(2)
FIGURE
Percent-Error
5.3
in Temperature
vs.
2,
the
Number of Radial Subdivisions in D22Z
72
above the line of symmetry agreed with their mirror images
to six significant figures.
The third major verification of the program was a
cross-check, using the output from Solution 1 (which finds
temperature) as the input for Solution 2 (which finds
current).
For this test a 79-point mesh was employed, and
all losses in both cables were included.
Using an oil
temperature of 140°F and a current of 942 amperes in each
cable, Solution 1 predicted a maximum temperature of
189.451°F for the system.
This maximum temperature,
together with the oil temperature of 140°F, was then used as
input for Solution 2, which predicted a maximum allowable
current of 942 amperes in each cable.
The solutions
mutually agreed to six significant figures, thereby demonstrating the reciprocal validity of the program.
The final major verification procedure was to
approximate the one-dimensional solution for a single cable
with all losses, by shrinking to a minimum the height of the
inter-cable conduction path.
In the computer program the
conduction path is presumed to have some non-zero height, so
the included angle in either cable was taken to be 2°.
Again temperatures in Cable 2 opposite the inter-cable
conduction path were compared to corresponding analytical
temperatures, and the same error criteria (RMS-error as a
percentage of the total drop through the insulation) were
employed.
The test was made for three cases:
loss only,
conductor
losses
dielectric
only, and then all losses.
73
Using a mesh comprised of four radial subdivisions in the
insulation, the following results were obtained:
dielectric
loss only - 2.6% error in temperature solution; conductor
losses only - 0.4% error in temperature solution, 1.0% error
in current solution; all losses - 0.8% error in temperature
solution, 2.0% error in current solution.
Temperatures in
the dielectric solution are elevated above their analytical
counterparts
because of the referral
of loss near the
conductor into the conductor itself.
However, when the
dielectric loss is considered proportionately with all other
losses, the temperature
small
error is observed
(less than one percent
predicted
to be reasonably
for this mesh).
The currents
in this test are seen to be less accurate
corresponding predicted temperatures.
than the
This circumstance,
though, reflects a limitation of the test itself rather than
one of the model.
is expected
That is, the temperature distribution
to smooth out into a one-dimensional
region far-removed from disturbing effects.
form in a
However, the
current solution depends on the entire temperature distribution.
Since an inter-cable conduction path of any
non-zero size will inevitability produce local distortions
in the temperature
distribution,
it is ultimately
futile to
expect very close agreement with a one-dimensional current
solution.
Agreement
could be demanded
if the size of the
conduction path could be made identically zero, but the
model does not possess this capability, having been designed
for the solution of real two-dimensional problems.
Given
74
this limitation, the current solutions demonstrate excellent
agreement with the one-dimensional results.
The credibility
generated by this test, together with all the evidence
previously stated, suffices to establish the validity of the
computer program.
74.1
CHAPTER 5A
COPPER TAPE EFFECTS
Effective Conductivity
Many cable systems used in the underground power transmission have a
thin copper tape wrapped around the cable insulation directly under the
moisture seal assembly.
The tape is included to circumferentially smooth
out the electric potential and to provide electric ground.
Having a very
high thermal conductivity the copper tape provides a mechanism for transIt therefore causes a
ferring heat away from high temperature regions.
redistribution of the temperature, tending toward the one-dimensional form.
Order of magnitude calculations indicate that the presence of a five mil
(t = 0.005") copper tape (thermal conductivity k
CU
= 220 BTU/hr ft°F) under
the moisture seal assembly have a significant effect on the temperature
distribution of the cable:
Consider a typical system of Fig. 5A.1,
r
= 0.9125
in
r 2 = 2.0675
in
k
= 0.1153 BTU/hr
ft°F
The thermal resistance per unit length in the circumferential direction without
the tape R is approximately
c
R
=
c
1
k(r 2 -rl)
=
90 hr°F/BTU
The circumferential thermal resistance including the copper tape R
c
is
74.2
1
k(r2 -r 1-t) + kt
C
c
.
R
c
Since R
c
=
1
1
9.7 hr°F/BTU
k(r2 -r1-t) + ktht-
and R'
are significantly different, the presence of a highly con-
c
ductive medium under the cable moisture seal assembly must be taken into account.
An order of magnitude calculation also indicates that the copper tape
effect in the radial direction is negligible:
whereas in the circumferential
direction the resistance of copper and insulation are parallel,in the radial
direction the two resistances are connected in series. The radial thermal
resistance per unit length without the tape
R
is
r
R
= in r/r
2
=
1.129002568 hr F/BTU
k
The radial thermal resistance per unit length including the tape, R '
r
-t
i~n
r
R
~~~~~~~~~~~~~r2_
r
in
is
r
+
r
1
-t
= 1.12900432
hr°F/BTU
2
2 rk
2
k
cu
Since the tape is very thin only the mesh points located around the outside
circumference of the cable insulation are affected.
In Fig. 5A.1 the thermal
conduction path between point 0 and 1 consists of the copper tape (thickness t)
and a layer of insulation (thickness L -t).
The path length is L
r
thermal resistance of the two conducting media
L
- = k(Lr-t)
____
+ k
R rCU
eff
r
t = k
L
R
is
.
The
74.3
A
Oil
Circle A
angle
4
angle
_- I
th Lo
l
Fig. 5A.1. Development of the expression for the combined
effective conductivity of a layer of the cable insulation and
the thin copper
tape.
74.4
where k ff is the effective conductivity of the entire thermal conduction
path betwee 0 and 1.
In the computer
program
L
= (r -rl)/N;
r
where
21
N is the number
of radial
divisions.
Hence,
t+k(
rk
- t)
2 r
k
eff
-
cu
N
(5A.1)
r 2 - r1
N
Equation (5A.1) applies only for points on the insulation outer circumference
and in the circumference direction only,
al other mesh poinst are unaffected.
Finite Difference Equations
For the mesh points located on the outside circumference some of
h -J
z4t
_O
4
.
r
i
of the governing equation (2.6) and the boundary condition must be adjusted
to account for the higher conductivity in the circumferential direction.
Thus for the point 0 in Figure 5A.1 the finite difference approximations
in non-dimensional
form
(L
r
a20
40- 200 + 03
r2
2
r
04
4 - 03
3
2h
r
= hrr2,
r 2
= h r2)
L
a
a 2
74.5
a20
a
_=
0 2 - 20% +
2
0
~-ct-2
h
0
2
1
(5A.2)
2
0
2
h 0 (hcl+h
2
)
halh o9
+
2
0
ho (ha+h a2)
Twhere 0 is given by (2.5) and O
Toi
W/k
W/keff
=
Then using (5A.2), the equation (2.6) about a regular boundary point 0
(h
= h
-2
) becomes,
0
-200
[4
r
+r[ 4]+I['1
_-0
+
2
03
2h
r
02 - 20 +
4 - 200 + 03
r
h .
4 -03 +h
r [ 2h ]
II
h2
L
L
J
h
(5A.3)
2
For a boundary point at a regional interface (ho
-2
le
(r2r) 2
r
#
ho ):
2
2
(hDi+h02)
ocio
I
0
r
-2
(r 2 r)
2
h
(h +h
1
)
W
(5A.4)
The boundary condition (2.8) is unaffected for all boundary points;
k
r
04 - 3
~=
2h
k2~
2
Eliminating
r
h2r
0
(5A.5)
04 from (5A.3) and (5A.4) using (5A.5) and noticing that from
0= k AT
e AT
0 = keff W
obtain
0'-=
keff 0
k
(5A.6)
74.6
Hence
for h
= h
2hhr
k
= h
2 (-2
rk2(
k
-
(h2
-2
eff
)
2h
+
k
r
k
k
2(r
I
eff
k
2r)
2
h
h
cx
and for h
r
a
+
[1
eff
k
+
2r
2
q
(5A.7)
(5A.7)
r]
h
hhrcx
r
h2
2h hr
-2
r 2(
+
+-[ ff
k
k
khh
k
+eff_ _ ~ rr
k
)
2h r
2
r
+
+ 2r
h2+
ft
k
k
+
)1®
2
+
h ah
lc,
3
r
k_
eff
( 22r
+h
h )®1+
h, 1 (h
eff
h
k
_
_
2_
]9
,o
-
+[2 ] - 0
2]2
2 ]
r
-2 .
_(2
)
q
(5A.8)
W
Computer Program Modification
From
(5A.7) and
(5A.8) it is apparent
that to account
for the presence
of a thin high-conductivity tape under moisture seal of the cable insulation
it is only necessary to multiply the original coefficients (corresponding to
k~~~
!
the homogeneous insulation material) of ®1's and
eff
2's by
k
and to add
k
2
ftihft
(1
follows
thatt
followsth
can be eased
eff)
k
to both
2
h
f
(5A.7) and
(5A.8) for if
= hh
e
=h
c2
, it
2
2.
-hc
a
The simple complementation of this procedure
by the following manipulation.
74.7
k
From
of 0
(5A.8) the coefficients
's and O's for
1
2
h
h=
0l'S and 02 s in (5A.7).
= 1 (original
coefficient
k
which are the original coefficients of
2
reduces to
ci h~~~~~~~~h2
hc
for h
eff
It is possible
therefore
to use
(5A.8) for all
circumferential boundary points.
Further,
let the original
coefficients
of
's and 0 's be
1
2
h Ul(hc+h
=0
)
(5A.9)
1
2
=0
h o2(h
2
+h o)
Also, let the new coefficients of 0l's and
1
2 f
=P
(h D+h a )
h
2
2's be
2
=o.
P1
01
(5A. 10)
2
h2
P
f
(h l+h
where f-
)
0
2
2
f
keff
k
k
2
Let Q
h
=
h
(1 - f
= factor
to be added
to the original
coefficient
alo02
02's.
2
Then from
(5A.9) and(5A.10)
0
or
h
(01
-
P1)
4
(-
+
1)
= P
+
hCa (h o2+h
1
h 2(l -t)
h
Q
)
1
(5A. 11)
h
and similarly (2
- P2) (
)
(12
=
Q
(5A.12)
of
74.8
Eliminating h
and h
Q = 01 + 02 -P1
between (5A.ll) and (5A.12) obtain,
(5A.13)
- P2
Thus to modify the original matrix of coefficients of O's it is necessary
I!I
keff
to multiply the original coefficients of Ol's and 02 s by f = k
add the difference (1
0
1
and 0
2
of 0 1'sand
1
- P 1 ) and (2
- P 2 ) between the original coefficient
0 's and the new coefficients
P
2
to the original coefficients of 0
s.
and to
1
0 f and P
1
= 0 2f
2
2
An important property of the matrix
0
Awas used to find the locations of
's and 0
s,
s:
the numbering
system is such that all coefficients of 0 's lie on the main diagonal and
0
the coefficients of 0 's immediately precede 0's
in each row and the
~~~~~~0
1
coefficients of 02 's immediately follow O 's in each row with the exception
2
0
or tne row corresponding to the governing equation written about the point
in cable 2 at an angle
= 0.
Also all
's are circumferential boundary
0
points.
Verifications
The first major verification was a check on the coefficient matrix.
This test was, as in Chapter 5, accomplished by printing out the matrix for
a 42 and 17-point mesh and then by verifying each element by hand computations.
Tha second major verification was an energy balance performed on
selected mesh elements, again by hand computation.
The final major verification
was accomplished by solving the following problem:
the intercable conduction
path was reduced to a very small size so that approximately uniform convective
boundary conditions existed at every circumferential point.
The problem
was run with and without the tape and both solutions were converging to the
1-D solution as the intercable conduction path was being reduced.
CHAPTER
6
RESULTS AND CONCLUSIONS
Evaluation Criteria
As a first criterion for evaluating the severity
of a given set of system operating con.ditions, the numerical
solutions produced by the computer program are compared to
the corresponding analytical solutions for a single, undisturbed cable.
The one-dimensional temperature and current
solutions for a single cable are presented in Appendix G.
Since the undisturbed cable represents an optimum operating
condition, the one-dimensional solution provides an upper
bnini
fnr system performance.
A second criterion for evaluation may be obtained
from a modification
of the one-dimensional
solutions,
in
which a conservative allowance for two-dimensional effects
is made.
manner:
This modification is effected in the following
it is reasoned that an effect on any portion of the
cable surface which disturbs the one-dimensional temperature
distribution is less severe than the effect of insulating
that portion of the surface.
The conservative approximation
is then made that such disturbances do indeed effectively
insulate appropriate portions of the surface, and that the
entire sector defined by such an insulated arc is likewise
insulated.
All losses which would have occurred in the
insulated sector are then placed into the undisturbed fraction of the cable, and the one-dimensional solutions are
r7 P.
76
employed with appropriately scaled-up losses.
This proce-
dure is explained fully in Appendix H, where the conservative approximations for maximum temperature and current are
derived.
Since by physical argument the disturbed portions
of the cable surface cannot have a more stringent condition
imposed than that of being insulated, the approximate
formulas of Appendix H provide a lower bound for system
performance.
Results
While the primary product of this study is the
computer program itself, a total of 16 cable problems were
solved by the author in order to provide preliminary inforThese
mation about some typical operating conditions.
16 problems break down into the following:
Solution 2 (for
maximum current) and Solution 3 (for maximum oil temperature)
were employed for four configurations of cables, and two
cable systems were considered.
The four cable configura-
tions were equilateral, cradled, open, and equilateral-pipe.
These are depicted in Figure 6.1.
The systems considered
were a 2500 MCM system (System 1) and a 2000MCM system
(System 2).
Values for the physical parameters associated
with these two systems are listed in Table 2.
The thermal
parameters were taken to have the following values for all
16 problems:
0.1153 Btu/hr-ft-°F for the thermal conduc-
tivity of the insulation, and 5.0 Btu/hr-ft2-°F for the
thermal film coefficient in convective regions.
Also the
conservative assumption of a thermally nonconducting conduit
77
EQUIL ATERAL
CRADLED
OPEN
EQUILAT ERAL PIPE
FIGURE 6.1
Nomenclature for Cable Configurations
78
TABLE
2
Values for The Physical Parameters
of Systems
1 and 2
System
1
System
Inner Radius
of Cable Insulation (in)
0.9125
0.8155
Outer Radius
of Cable Insulation (in)
2.0675
1.9675
Skid Wire Thickness (in)
0.10
0.10
5.36
6.63
at Conductor
1.19
1.13
at Sheath
1.24
1.18
Typical Dielectric
Loss (watts/conductor-ft)
3.18
3.18
942
888
Thermal Conductivity of
Insulation (BTU/hr-ft°F)
0.1153
0.1153
Thermal Conductivity of
Copper Tape (BTU/hr-ft°F)
220.0
220.0
Copper Tape Thickness (in)
0.005
0.005
DC Resistance
of Conductor
(/ft)
AC/DC Ratio
Typical
Current
(amps)
2
79
was made.
This latter assumption is significant only for
the cradled and equilateral-pipe configurations, where the
.
t
cables actually touch the conduit wall.
A nonconducting
wall substantially increases the thermal resistance in the
vicinity of the contact point, thereby producing a local
region of high temperature and hindering the removal of heat
from the cable.
Results for the 16 problems are given in
Tables 3 -6.
In these tables the first column of percent-
ages is a comparison of computer solutions to corresponding
one-dimensional solutions.
These negative percentages are a
measure of how much worse the given operating condition is
than the best pssihl
nondi;on. The second column of
percentages is a comparison of computer solutions to the
conservative approximate solutions from Appendix H.
These
percentages, which are positive, provide a measure of how
much better the given operating condition is than the
estimated worst condition.
Upon examining the four tables, the equilateralpipe configuration is immediately identified as the most
severe operating configuration.
This is expected, since the
greatest obstruction of cable surface area occurs in this
configuration.
sequence:
The other configurations follow in logical
equilateral, cradled, and open.
It is shown in
Appendix B and also in the heat transfer report [2] that a
steel pipe is very effective in conducting heat away from
the cables to the bulk of the oil.
So had the problems
80
TABLE
3
Solution 2 For Four Cable Configurations System 1
1-D
I-I
I (amps)
1-D
1075.4
-5.0%
+0.2%
Open -- tape
1105.0
-2.4%
+0.3%
Cradled
(Nonconducting Pipe) -- no tape
1030.2
-9.0%
+2.0%
Cradled
(Nonconducting Pipe) -- tape
1086.0
-4.1%
+7.5%
978.2
-13.6%
+3.7%
1071.3
-5.4%
+13.5%
946.4
-16.4%
1057.0
-6.6%
(1132.1)
-
Open
-- no tape
Equilateral -- no tape
Equilateral -- tape
Equilateral-Pipe
(Nonconducting Pipe)
--
no
ape
Equilateral-Pipe
(Nonconducting-Pipe) -- tape
One-Dimensional
T
T
oil
max
= 140 ° F
= 185°F
I - current from computer solution
I
- current from one-dimensional solution
I* - current from conservative approximate solution
I,- current from conservative approximate solution
*
To . 6
+21.3%
81
TABLE
4
Solution 3 For Four Cable ConfigurationsSystem
1
T .- (T il)l
Uj-j-
Toil
(F)
oil
-DD
To (Toil)l-D
Toil- (Toil)*
T-
o (Toil)*
0.0%
Open -- no tape
148.3
-8.9%
Open -- tape
149.9
-4.2%
Cradled
(Nonconducting Pipe) -- no tape
145.7
-16.6%
Cradled
(Nonconducting Pipe) -- tape
148.9
-7.1%
Equilateral -- no tape
142.4
-26.4%
Eqi-I1teral
-- tape
168.
Equilateral-Pipe
(Nonconducting Pipe) -- no tape
140.3
-32.6%
+11.5%
Equilateral-Pipe
(Nonconducting Pipe) -- tape
147.3
-11.9%
+25.4%
One-Dimensional
I = 942
T
max
Toi
oil
(151.3)'
amps
- oil temperature from computer solutions
(Toil)*
O
r
= 185°F
(Toil)l
T
-9
D
-
oil temperature from one-dimensional solution
oil temperature from conservative approximate solution
- conductor temperature
+4.4%
+2.8%
+10.7%
+4.8%
+17.8%
82
TABLE
5
Solution 2 For Four Cable Configurations System
2
I-ID
1
-D
Ii_ D
I (amps)
I-I*
1I*
Open -- no tape
950. 2
-5.0%
+0.3%
Open -- tape
975.3
-2.5%
+3.0%
Cradled
(Nonconducting Pipe) -- no tape
911.9
-8.8%
+2.4%
Cradled
(Nonconducting Pipe) -- tape
959.2
-4.1%
+7.7%
Equilateral -- no tape
864. 6
-13.6%
+4.0%
Equilateral -- tape
946.6
-5.4%
+13.9%
837.2
-16.3%
938.8
-6.7%
Equilateral-Pipe
(Nnn-onductin_
Pine) -- no taDe
Equilateral-Pipe
(Nonconducting Pipe) -- tape
One-Dimensional
T
T
oil
max
(1000.4)
= 140 F
= 185°F
I - current from computer solution
I1-D - current from one-dimensional solution
I* - current from conservative approximate solution
(0.0%)
+9.3%
+21.9%
83
TABLE
6
Solution 3 For Four Cable Configurations System
2
_ T
(Toi)
Toioil
ilD-D(Toi
TOil-(T
°
Toil ( F)oil
D
il)ll-D
To-(T oil~~T
-(
l)*
-( oil
T
(Toil),
Open -- no tape
144.4
-8.8%
+0.3%
Open -- tape
146.1
-4.3%
+4.5%
Cradled
(Nonconducting Pipe) -- no tape
141.8
-15.8%
+3.4%
Cradled
(Nonconducting Pipe) -- tape
145.1
-7.0%
+10.8%
Equilateral -- no tape
138.2
-25.5%
+5.8%
Equilateral -- tape
144.2
-9.4%
+17.9%
Equilateral-Pipe
(Nonconducting Pipe) -- no tape
136.0
-31.4%
+12.4%
Equilateral-Pipe
(Nonconducting Pipe) -- tape
143.3
-11.8%
+25.4%
One-Dimensional
(147.7)
-
I = 888 amps
T
max
= 185°F
T il = oil temperature from computer solution
oil
(T il)l
D
- oil temperature from one-dimensional solution
(T il)* -oil
T
temperature from conservative approximate solution
- conductor temperature
O
84
been solved using a thermally conducting conduit material such as steel,
then the results for the equilateral-pipe configuration would have been
essentially the same as those for the equilateral case, and the latter
would have been the most severe configuration.
A complete temperature
distribution for the equilateral-pipe configuration without the high
tape under the cable moisture seal assembly of System 1
conductivity
is displayed
in Figure
6.2.
Two observations are made regarding the conservatke
solutions.
approximate
The first is that in cases without tape they are reasonably
accurate, being conservative by 9.3% in the least accurate case (Solution
2, System 2, Equilateral-Pipe).
They are therefore useful whenever a
highly refined solution is not required.
The second observation is
that the conservative solutions become less accurate as the configurations
become more severe.
This tendency is readily exnlained. for as the sur-
face area of a cable is increasingly obstructed, two-dimensional effects
grow stronger.
Since the conservative approximations are based on the
one-dimensional solutions, they become increasingly deviant with the
severity of the configuration.
So despite the conservative nature of
the approximate solutions, they are not recommended for design purposes.
Also it is noticed that there is a large discrepancy between corresponding percentages in Solution 2 and Solution 3:
temperature
deviations are somewhat larger than current deviations.
crepancy, however, should not be a surprising one.
solutions are applied to a given configuration.
Such a dis-
Consider that both
In Solution 3 the heat
flow is constant, and the temperature distribution must be linearly
adjusted
so as to account
for the two-dimensional
constraints
imposed
85
~A:
A
47.8
18C
18C
18
18.
'ES IN
I,= = 942
AMPS
the
n -
1473
14 7 3
F.
86
by the configuration.
In Solution 2, though, the heat flow is the adjustable
quantity of the total heat flow, only the current-produced fraction (usually
about 2/3) is variable, and thus the current-produced losses must be disproportionately adjusted so as to align the overall heat flow according
to the two-dimensional constraints of the configuration.
Furthermore, the
current itself varies as the second root of the variable heat flow.
Therefore
the relationship between the two solutions is a complicated one, and there
is no reason to expect any similarity between their respective deviations.
Finally, it is evident from the tables that cable proximity effects
are very significant in forced cooling especially for cables without the
high thermal conductivity tapes.
In System 2 with no tape, for example,
the maximum allowable oil temperature is 11.7°F lower for the equilateralpipe configuaLiolL
haff ot the one-dimensional
Cat.
Since force-cooled systems are typically designed for an axial oil
temperature rise of about 45°F between refrigeration stations, the 11.7°F
difference itself would accout for 26% of the axial oil temperature rise.
On the other hand, in the same system with the tape present the maximum
allowable oil temperature is only 4.4°F lower than in ID case, or about
10% of the axial oil temperature rise.
For the more realistic equilateral
configuration the same figures are 9.5°F
or 8% with
the tape.
This means
r 21.6% without the tape and 3.5°F
that depending
on whether
the tape is or is
not present under the cable moisture seal assembly System 2 in the equilateral
configuration would require either an 8% or 21% higher flow rate, or
either an 8% or 21% shorter axial distance between refrigeration stations
then the same system in a completely free (one-dimensional) configuration.
87
Approximate allowance for cable proximity effects must therefore be made
in the overall design of force-cooled systems.
Since there is a sig-
nificant improvement in the results when the copper tape is wrapped
around the cable insulation, such cables are from thermal considerations,
the more suitable for force-cooled power transmission work.
Isothermal lines for the four oil temperature solutions of System 1
are shown in Figures 6.3 - 6.7.
One-dimensional portions of the various
solutions may readily be identified in these figures by isotherms which
are circular arcs.
As expected, all the distributions smooth out into
one-dimensional form away from points of cable contact and conduit
contact.
Regions of high temperature within the insulation are identi-
fied by isotherms which depart significantly from the circular shape,
protruding outward from the cable centers.
This effect is observed to
be most prevalent in the equilateral-pipe configuration, decreasing in
strength in the equilateral, cradled, and open configurations, respectively.
Thus the isothermal lines in themselves provide a vivid illustration of
the severity of the various configurations.
adiabatic lines.
Shown with the isotherms are
These lines are everywhere normal to the isotherms,
and they represent curves along which the heat flow travels.
that not all adiabatic lines originate at the conductors.
It is noted
This circum-
stance is attributable to the dielectric heating, which occurs within
the insulation itself.
161
183.5
I = 942 amps
TOIL = 148.3
---
ISOTHERM
ADIABATIC LINE
17
N
(NO COPPER TAPE)
16
16
FIGURE 6.3
Isothermal and Adiabatic Lines for the
Open Configuration
- System I
89
i7t
178
3
183.4
163
i3
169
I = 942 amps
185.0
TOIL
146.5
-
ISOTHERM
---
ADIABATIC LINE
ALL TEMPERATURES
178
IN
F
173
HOMOGENEOUS INSULATION
(NO COPPER TAPE)
161
FIGURE 6.4
Is(thermal and Adiabatic Lines for the
Cradled Configuration - System 1
90
4
I
INE
ATURES
SULATION
E)
FIGURE
Isothermal
6.5
and Adiabatic Lines fr
the
Equilateral Configuration - System 1
9i
URES II
LATIC N
FIGURE 6.6
Isothermal and Adiabatic Lines for the
Equlilateral-Pipe
System
Configuration
1
-
92
156
160
15 6
160
160
180
174
168
162
156
Figure
6.7
Isothermal and Adiabiatic Lines for the Equilateral-Pipe Configuration
with copper tape in the moisture seal assembly - System 1.
93
Conclusions
On the basis of the results discussed, the following conclusions
are drawn:
1. For a thermally nonconductiong conduit, a cable system is most
susceptible to thermal failure in the equilateral-pipe configuration.
For a thermally conducting conduit, the equilateral-
pipe and equilateral configurations are equally severe, and the
latter represents the worst operating configuration.
2. The conservative approximate solutions developed in Appendix H
are useful for obtaining good estimates of maximum temperature
and current.
However, recourse should be made to the computer
solutions whenever design information is required.
.able
..iguration,
...
_r
. , ...
o21
configu
3. Cable proximity effects are important in forced cooling.
The
BV
with no copper tape)
to 26% (equilateral-pipe configuration with
no tape) of the total oil temperature rise between refrigeration
stations.
4. The presence of a thin copper tape in the cable insulation moisture
seal assembly significantly smooths out the temperature distribution
in the cable insulation and thus higher maximum allowable oil
temperature and higher currents are permitted than if a homogeneous
cable insulation is used.
Numerically the improvement is from
about 4.3% for the oil temperature and approximately 9.5% for the
current.
If this figure is unacceptable, thicker copper tapes would
smooth out temperatures even more.
94
Recommendations for Further Work
In order to fully exploit the convenience of having a separate region
associated with each boundary condition, a provision should be written
which would allow Regions IV and VI of Domain 2 to be used simultaneously.
Such a provision does not presently exist, because there is only one
configuration for which it would be desirable.
However, the exceptional
configuration is the equilateral-pipe case, in which both cable-cable
and cable-conduit boundary conditions act simultaneously.
Since this is
the most severe condition for a thermally nonconducting conduit, it is
expected that this configuration will be frequently used, and the change
is probably warranted.
It is noted, however, that in the present program
all boundary conditions are independently specified by means of a variable
film coefficient.
The equilateral-pipe configuration can therefore be
modelled as accurately as the user desires, and the proposed modification
represents only a convenience in laying out the system geometry and mesh
size.
Instructions for implementing the change are given in Appendix E.
Finally, some improvement could be effected in the input-output
formats of the computer program.
Throughout the development of the program,
attention was continually given to simplicity of I/O procedures and to
ease of user operation.
Yet certain aspects of the final I/O formats are
less convenient than is desirable.
are offered, again in Appendix E.
Suggestions for their improvement
CHAPTER
7
REFERENCES
1.
Long, H. M., Notaro,
J., and Webster,
D. J., "Economic
Analysis of a Generalized Design for a Forced Cooled
Cable," IEEE Trans., Vol. PAS-90, No. 3, May-June,
1971, pp. 1225 - 1231.
2.
Glicksman, L. R., Orchard, W. P., Rohsenow, W. M., and
Slutz, R. A., "Cooling of Underground Transmission
Lines: Heat Transfer Measurements," Unpublished
Report for Consolidated Edison, Energy Laboratory and
Heat Transfer Laboratory, Department of Mechanical
Engineering, MIT, January, 1974.
3.
Kreith, F., Principles of Heat Transfer, International
Textbook Company, Scranton, Pennsylvania, 1965, p. 80.
4.
Ibid.,
5.
Arpaci, V. S., Conduction Heat Transfer, Addison-Wesley
Publishing Company, Inc., Reading, Massachusetts, 1966,
pp. 126 -129.
6.
Crandall, S. H., Engineering Analysis, McGraw-Hill Book
Company, New York, 1956, p. 246.
7.
Forsythe, G. E., and Wasow, W. R., Finite-Difference
Methods for Partial Differential Equations, John
Wiley & Sons, Inc., New York, 1960, pp. 186, 188.
8.
Von Hippel, A. R., Editor, Dielectric Materials and
Applications, The Technology Press of MIT and John
Wiley & Sons, Inc., New York, 1954, pp. 1- 10.
9.
Pugh, E. M., and Pugh, E. W., Principles of Electricity
and Magnetism, Addison-Wesley Publishing Company, Inc.,
p.
Reading,
10.
79.
Massachusetts,
1970, p. 39.
Holman, J. P., Heat Transfer, McGraw-Hill Book Company,
New York,
1968, p. 28.
11.
Ibid., p. 29.
12.
Feibus, H., Consolidated Edison Company of New York,
Inc., New York, Personal Source, January, 1974.
95
APPENDIX A
THE RELATIVE MAGNITUDE OF CONDUCTION
AND CONVECTION RESISTANCES
The equation which defines the resistance per unit
length to heat transfer is the following:
AT
AT
q
(A.1)
where q is the total heat flow per unit axial length, AT is
the temperature difference driving the heat flow, and R is
the resistance per unit length to heat transfer.
The heat flow per unit length due to convection
from the outer surface of the insulation to the oil is [10]
q = 2fr h(AT)
0
(A.2)
,
where h is the convective film coefficient, and r0 is the
outer radius of the insulation.
The convective resistance
per unit length is therefore
-h
1
2ffr0 h'
h
2rr~h
(A.3)
The heat flow per unit length due to conduction in
the cable insulation, which is assumed to be one-dimensional
in this type of calculation, depends not only on AT, but
96
97
also on the distributed dielectric loss.
Because of this
additional dependence on the dielectric loss, it is not analytically possible to model the conduction path as a simple
resistance.
However, a conduction resistance may be ob-
tained numerically, by substituting appropriate values into
Equation
A.1.
The
(AT) for a given
set of losses
(q) is
available from the one-dimensional solution presented in
Appendix G.
A typical set of values for (q) and (AT) yields
R
AT
hr-ft-°F
(q) = 0.93
Btu
'
(A.4)
where AT is the temperature drop across the insulation, and
(q) is the heat flow per unit length emanating from within
the insulation (the conductor and dielectric losses).
The corresponding value for Rh from Equation A.3
is
1
Rh
2r
hr-ft-°F
0
h
0.18
Btu
(A.3)
where the most conservative value for the natural convection
film coefficient (5.0 Btu/hr-ft -°F) was used [2].
relative magnitude of the two resistances is
Rh _
Rk
0.18
0.93
(hr-ft-°F/Btu)
(hr-ft-°F/Btu)
The
therefore
_
19
A.5
(hr-ft-01F/Btu)-*(.5)
0.93
98
Thus the natural convection resistance, which is always
larger than the combined forced and natural convection resistance, is small when compared to the conduction resistance, and the latter is the limiting resistance to heat
transfer.
APPENDIX
B
INVESTIGATION OF THE CABLE-CONDUIT
BOUNDARY CONDITION
In order to examine the cable-conduit boundary
condition described in Chapter 2, a portion of the conduit
wall is thermally modelled as a fin.
which
to determine
The geometry from
is shown in Figure B.1.
a fin length
It
is based on a cradled configuration of cables, since that
configuration produces the largest conduit loss.
possible
lengths L
and L 2 , the latter
is chosen
maximize the temperature drop through the fin.
Of the two
so as to
L2 is found
from the following relations:
sin
r2 +
R
1
2
-r
p
0.660 ,
(B.l)
3
from which
1
= sin
-l
(0.660) = 41.3 ° .
(B.2)
Then
2 =
180-
2Bl
99
= 97.4
(B.3)
100
FIGURE B.1
Fin Geometry for the
Cable-Conduit Boundary
Condition
101
and
L2 =
(Rp +
= 8.92"
(B.4)
One end of this fin is insulated by symmetry, and as a
conservative approximation, the side of the fin which is
adjacent to the earth is likewise taken to be insulated.
Furthermore, the pipe loss is taken to be concentrated at
the end of the fin which touches the cable (again to maximize the temperature drop through the fin).
model is illustrated in Figure B.2.
This thermal
Considering the heat
flow to be one-dimensional, the fin temperature is governed
according
to [11]
d2T
dz
h P
-2kA
dz(T-T
p
oil
) =0
(B.5)
where P is the convective perimeter, and A is the crosssectional area.
Using the dimensionless variables
z
n = '
2'
e
=
T-T
T oil
W/k
(B.6)
p p
Equation B.5 becomes the following:
d2 _2
-9-d2
d--
P eh
0= 0
2 kp A(
--
.
B
7
(B.7)
102
W,
"
Wp
kp
PIPE LOSS/I UNIT LENGTH
CONDUCTIVITY OF PIPE
- THERMAL
FIGURE
B. 2
Thermal Model of the Conduit Wall
103
Making the substitution
m
2
2
2
~h
hpP
= L2 k'
(B.8)
P
p
the governing
equation
is just
2
d20
---
2
-
m 0 = 0
dn
The boundary conditions
.
(B.9)
are
-k
t dT
P
dz
=
(B. 10)
W
1 z=O0
p
and
dT
dz
= 0 .
z=L
(B.il)
2
These are rendered dimensionless to give
L2
da
a -TIIr
(B.i2)
and
dO!
7,1
-= 0 .
n=1
(B. 13)
104
The general solution of Equation B.9 is
a () = Cle-m +m C2 e
(B.14)
The two arbitrary constants C 1 and C2 are found from the
boundary conditions to be
L2
L2
tm(l-e-2m) '
C 2 =-
(B.15)
tm(e2m-1)
When these are substituted into the general solution,
Equation B.14 can be manipulated into the following form:
(
=L2
cosh[m(1-n)
tm
]
(B.16)
sinh (m)
The following values are then taken:
= 3.3 Btu/hr-ft 2-F
P = 1 ft
h
P
=
1 ft
=
25 Btu/hr-ft-°F
(conservative,
based on
[2])
(B.17)
k
(1% C steel)
A = 0.0208 ftP
2
A
=
0.0208
ft
Equation B.16 finally becomes
6(n) = 5.73 cosh[l.92(1-n)]
.
(B.18)
105
It is desired to know the temperature drop from the fin base
to the oil, so it is now convenient
to return
to the form
W
T(rl)-Ti
oil
=
5.73 -
(B.19)
cosh[1.92(l-ri)
k
p
~~~~~~~(B.19)
p
The desired quantity is obtained by evaluating Equation B.19
at
= 0 and by substituting
W
(8.85 watts/system-ft).
The quantity in parentheses is a typical loss value, which
is divided by 6 to account
splitting
of the heat flow to either
cable contact.
oil
5.73
=
and for the
side at the poin
of
The temperature drop is then
1
T(0)-T
for the 3 cables
1 (8.85 watts/ft)
25.413
(25 Btu/hr-ft-
= 3.9°F
F)
(3.413
tu)
Btu) cosh(.92)
(watt-hr)
.
(B.20)
It is now desired to compare this temperature drop
to the drop across the cable insulation.
Using once again
the one-dimensional solution for a caole with all losses
(Appendix G), the temperature distribution in the insulation
is given by
W
(W
d
O~)d
d(~
4fW(ln
(ln E)221
+W
(+2
d
c
2irWc
(W
in
~
4±
+W
+W
Wdc++W
c
d
s
2-7Whr2
)
k
(G.16)
106
where Wd, Wc, and W
s
are the dielectric, conductor, and
sheath losses per unit length, respectively, and W is an
arbitrary loss per unit length.
The dimensionless
temperature drop across the insulation is given by
(~1)
-86(1)=
61
ln
-
Wd\
1
2W
2TrW
(2
+W c)(.1.
(B.21)
Returning to the dimensional temperature by means of
Equation G.6, this result becomes:
T 1)
( -T(1) = --
ln ~
1 /W d
W
))
(B.22)
The dielectric and conductor losses corresponding to the W
P
used previously are 3.18 and 5.66 (watts/conductor-ft),
respectively.
These values, together with
k = 0.1153 (Btu/hr-ft-°F) give
T(Ci) - T(1) = 31.1°F .
(B.23)
The relative magnitude of the two temperature
drops is therefore
(AT)
fin390
(Afin = 3.9 0 F =
0F
(AT)
insulation
31.-1
12.5%
.
(B.24)
107
It may be inferred from this result, which represents the
most conservative comparison of the two effects, that
contact between the cables and the steel conduit does not
significantly alter the overall temperature distribution.
The cable conduit boundary should thus be modelled as a
convective one, with at most a slightly modified film
coefficient.
_
_
_
_
APPENDIX
C
THE SOLUTION FOR MAXIMUM CURRENT
The Superposition Method for Solution 2
In Solution
2 it is necessary
that current-
produced losses be treated separately from voltage-produced
losses.
This makes it possible to distinguish the variable
component of the temperature distribution from the
stationary component, and subsequently to adjust the
variable component so as to maximize the current.
When the
losses are so separated, there are two complete problems,
each one having three components and resembling the problem
presented
for
in Chapter
3.
In fact, the solution
in Chapter
3
(x) may be taken as one component of Solution 2,
provided that the forcing term f(x) is clearly identified,
either with stationary losses or with variable losses.
Accordingly, let f(x) describe all forcing effects in the
domain which are attributable to current.
The solution
(x)
then denotes that part of the total temperature distribution
which is current-dependent.
It is now necessary to determine the stationary
portion of the temperature, that portion which depends on
voltage.
called
Let this part of the total temperature solution be
D(x).
The solution
D(x) satisfies the governing
equation
V2 OD(x) = g(x) ,
108
(C.1)
109
where g(x) describes all forcing effects in the domain which
are attributable to voltage.
D(x) also satisfies the
following boundary conditions (using the notation introduced
in Chapter
3):
D(X)l xEC (x)
1 -
where
=
(C.2)
D1
D01 is some unknown dimensionless temperature.
hr2
OD (x)
k
-
OD(x)
xcC 2 (x)
a ED(x)
an2
hr2
=
xEC
-
a an4
D(x)
43n
(C.3)
x eC 2 (x)
--
(x)
3-
(C.4)
OD(x)
k
xcC
-
(x)
3
I
xC
-CC
(x)
4
= 0
(C.5)
= 0
(C.6)
= 8D 0 2
(C.7)
W
DOD(x)
an
5
xcc5
(x)
8D(x)xeC6(x)
-
6
-
_
_
110
where OD02 is some unknown dimensionless temperature.
hp2
3aOD(x)
= - --
an7
(C.8)
OD(x)
x C7(x)
7
a OD(x)
an8
(C.9)
2Dh OD(x)i
D
xkCc8(x)
xC8
C8(x)
aOD(x)
-mx
Dn
-
9
xEC 9 (x)
2Dh
(X [D C(
2kh OD(x)
I xC
(x )
9 -
(C.10)
As before, OD(x) may be decomposed into three
component problems.
However, it is not necessary to
introduce three new components, for the solutions
A(x) and
eB(x) of Chapter 3 already describe the homogeneous
components required.
So only one additional component is
needed, and let it be referred to as
E(x).
This component
satisfies the nonhomogeneous governing equation
V2OE(x) = g(x)
(C.11)
The boundary conditions satisfied by OE(x) are identical to
those satisfied by OC(x), Equations 3.32- 3.40.
111
Once again the three component solutions are
linearly combined according to
OD(x) = b OA(x) + b 2 OB(x) + OE(x) ,
1
where b
2
(C.12)
_
and b2 are two new arbitrary constants.
The
validity of Equation C.12 is readily established by direct
substitution into the appropriate governing equation and
boundary conditions, Equations C.1- C.10.
Since this
procedure is identical to that followed in Chapter 3 (see
Equations
3.42 -3.51),
constants b
it is not repeated
The two
and b2 are again determined from a knowledge of
the losses at the conductors.
Equations
here.
By analogy with
3.56 and 3.57,
a3OD(x)
dC(x)
3n 1
C
xEC
-
1 (x
1
= 0 ,
(C.13)
(x)
_
and
OD(x)
I
an6
C6 X
()
XE
dC
6
(x) =
(C.14)
0 .
6 W
The zero right-hand sides of these equations reflect that
there are no voltage-produced conductor losses.
When
112
Equation C.12 is substituted into Equations C.13 and C.14,
two simultaneous algebraic equations result which uniquely
determine b
and b2.
The stationary part of the total
temperature solution, OD(x), is therefore available.
Attention is now turned to the total solution,
0I(x), which includes both stationary and variable losses.
0I(x) satisfies the nonhomogeneous governing equation
V2 I(x) = f(x) +g(x) ,
(C.15)
as well as the following boundary conditions:
I(x)IxeC1
where
01
=
(x)
01
(C.16)
'
is some unknown dimensionless temperature.
aoI (X I
an2(x)
hr
=
IxEC
I
2
k
( x)
-
aeI (x)
(C.17)
OI (x)I
0~x
IxfC
(x)
2
hr 2
=
an3
(C.18)
ei(x)
xC 3(x)
xC3()
=0
XEC4(
aeix
(x) cC
k
(C. 19)
113
3aI(x)
n5
C(x)
= 0
(C.20)
xc55
I(x)
xcC 6 (x)
-
6 02
=
,01~~~
~(C.21)
where eI02 is some unknown dimensionless temperature.
TOI (x)
hp 2
an 7
-
k
61(x)
xEC77 (x)
M6I x)
2Dh
an 8
k
xEC
-,
8
2Dh
an 9
k
xcC
(C.23)
eI(x)[
XEC8 (x)
(x)
aeI (x)
(C.22)
xcC 7 (x)
(C.24)
eO (x)
x
(x)
C 9 (x)
The total solution eI(x) is obtained
as a simple
sum of the particular solutions 6(x) and 6D(x):
eI(x) =
(x) +
D(x)
(C.25)
The validity of C.25 is established by direct substitution
into Equations C.15 -C.24:
V
2
6I(x)
= V
2
[0(x) + 6D(x)]
=
f W +
W
Check
(C.26)
114
I(x) Ix(c (X) = [(x)+(x)x) ]lX C(x)
=
1 + 0Dl
Check,
(C.27)
01 + OD0
provided 01
directly
=(a
from the linearity
1 +b1 )A0.
This result follows
of the problem:
al and b
were
determined by the variable and stationary components,
respectively,
of the loss at the conductors.
Since the
variable and stationary losses may be added to give the
total loss at the conductors, the temperatures alA
and
l~~~~~~~~o
blA° may likewise
be added to give the true conductor
temperature.
36I(x)
3
an2I=)
x¢
2 (
2
2E
[0(x) +
2
-n
D(x)]
x'C2 (x)
-
hr 2
[6(x) +
2x
D(x)]
'x~C2 (X)
hr
2 6I(x)i
k
Check
x (EC
()
x£ 22
(x
(C.28)
115
30I (x)
[ (x) + OD(x)]
an
~C
xcC3 (x)
hr2
2
k
3
x)
[0(x) + ED(x) I
xC 3(x)
hr2
OI (x)
k
aOI (x)
a
3n4
xC 4
Check
[0(x) + D(x)]
xEC4 (x)
(x)
= 0
M6I (x)
Check
_a
9n 5
(C.30)
[8(x) + D(x)]
xEC
5 (x)
x
XEC5 (x)
=
01(x)j xE C
=
(C.29)
xeC 3 (x)
[0(x) +
0
(C.31)
Check
D(x)]
XC
6 (x)
6
=
02 + 0D02
Check,
(C.32)
This result is
provided e102
= (a2 +b 2 )Bo.
= 002 + D02
analogous to Equation C.27, again following from linearity.
116
aeI (x)
a
dn 7
xEC 7 (x)
-
[0(x) + OD(x)]
an7
xCC(x)
-
hp.
[6 (x) + OD(x)]
C7 (x)
h p2 I
I
- I (x)i
I
Check
C
(C. 33)
(x)
7-
aOI (x)
an 8
xC 8 (x)
an 8
I 0(x)
+ eD(x) ]
Ix,C
(x)
-.. ,,,.,
2Dh
[6 (x) + 6D(x)]
XEC 8(X)
2Dh
k
6I (x)
Check
ix C 88 (x)
(C. 34)
aeI (x)
an 9
XEC 9 (x)
n_9
3n9
[6(X) + D(x) ]
xcc
2Dh
k
2Dh
k
9
(X)
[6(x) + D(x)]i
xC
6I (x)
9 (x)
Check
XtC
9
9
(X)
-
(C. 35)
117
It is thus established that the overall solution OI(x) is
available from its stationary and variable components
according to Equation C.25.
variable component
It now remains to adjust the
(x) so as to maximize current with
respect to the allowable cable temperature and the oil
temperature.
Maximizing Current in Solution 2
The current I is introduced into the temperature
solution through the relation
O(x) =
1 (x)I
2
(C.36)
where yl(x) is a constant of proportionality whose magnitude
depend on position x.
Equation C.36 follows directly from
two elementary facts:
1. Because of linearity, the cable
temperature is directly proportional to the cable loss.
2
2. Current-produced losses are directly proportional to I.
(x) is available,
It is now recalled that the solution
provided that the current-produced losses (and hence I) have
been specified.
be an arbitrary current
Accordingly, let I
0
from which a temperature distribution e0o(x)is determined.
Then from Equation C.36,
y (W)I
e0
0
0oX == ¥1(x
I2
,
(C.37)
118
and hence
e (x)
l
=
(C.38)
I 20
Equations C.38, C.36, and C.25 may be combined to give the
result
6 I(
2
2 +
(x)
=
(x)
,
(C.39)
0
where
(x), I,
and
D(x) are all known.
It is desired
allowable value, say
C.39,
I(x) take on some maximum
to have
ma x
.
Inserting this value into
maxEquation
Equation C.39,
mx=
o (x)
2
+
D(x) .
(C.40)
0
Upon rearrangement this relation yields
m
max- D) (x)
I
o
where the dimensionless scalar
brevity.
(C. 41)
0-
(x) has been introduced for
6(x), which is known throughout the domain,
119
determines the ratio I /I2 which will produce a temperature
0
of
It is now necessary to choose
max at the location x.
max~
the particular ratio I2/I2 (and hence the particular value
~
of 6(x)) which will yield a maximum temperature
This is accomplished simply by taking
distribution C.39.
2 2
the smallest possible ratio I/I
0
Let
denote the minimum
The desired temperature distribution is
over all x of 6(x).
~
in the
max
.~
then
0I(x) = 60 (x) + D(x)
Proof is as follows:
let x
o
(C.42)
be the location at which the
minimum value of 6(x) occurs:
~
= 6(x o )
(C.43)
It then follows from Equation C.41 that
eI(xo ) =
max
(C.44)
.
Now consider any other location in the domain, say x1 .
It
is known from the definition C.41 of 6(x) that
6(x 1 )0o (x1 ) +D(x
1)
=
max
(C.45)
120
It is also known that
< 6(x
1
),
(C.46)
m
since 6 is the minimum
over the entire
domain of 6(x).
It
then follows directly from the relations C.45 and C.46 that
6o6 (x 1) + D(x 1) =
I(x1) < 6max
(C.47)
The distribution C.42 is therefore proven to be the correct
one, and the maximum allowable current is determined from
Equation
C.41 with 6(x) = 6:
I = i /
0
.
(C.48)
I-
APPENDIX
D
THE DIFFERENCE FORM OF THE
CONDUCTOR BOUNDARY CONDITION
In Chapter 3 the heat flow emanating from the
conductor of Cable 1 was given as
Tar
=~~~
-kf
q=
rirlrd
=
wCl
.(3.52)
0
It is now convenient
to express
this in the dimensionless
form
7r
I_~8
-
0
e
ld~
rldP
WC
-
w
(D.1)
rl1
where r1 denotes the dimensionless inner radius of the
As the discussion
insulation.
of Chapter
4 indicated,
a
problem is incurred in the discretization of this boundary
condition.
mation
For if the standard central difference approxi-
4.3 is substituted
for the derivative
at the
conductor, a fictitious temperature within the conductor is
introduced.
Since no governing equation is applied at the
conductor, there is no way to eliminate such a fictitious
temperature.
In approximating the boundary condition D.1,
121
122
it is therefore necessary to have a difference expression
which involves only real temperatures.
There are a number of methods for approximating
this boundary condition.
As a reasonable compromise between
accuracy and simplicity, the following method is chosen,
where reference is made to Figure D.l:
location
r
in between
r
and r2
some central
is sought,
at which
location a good approximation to the derivative can be
achieved.
The boundary condition D.1 is then satisfied at
the location r,
[l|
WCl
(D
rather than at the conductor:
n r~d%
8
~ WC1
_
0 r,
The difference form of the derivative is constructed
according to
a|
ar
where
~
l,kl
r,,~ r
2
01
-r
(D.3)
1
01 denotes the conductor temperature.
to ascertain
the location
r,
attention
corresponding one-dimensional problem.
In order
is turned
to the
The analytical
temperature distribution for the half-cable with prescribed
conductor loss W
is readily found to be
I_
123
1
)k1
FIGURE D.1
Nomenclature for the Discretized
Conductor Boundary Condition
124
=
-
0(r)
is available:
r
n
kn r
-
)
D.4, a criterion
With the distribution
location
W (k
Cl
r* is chosen
(D.4)
(.
4
.
for determining
the
such that, as the
true temperature distribution approaches the one-dimensional
solution
(which it does in the vicinity
of the conductor),
the difference form of Equation D.2 becomes exact.
This is
accomplished by replacing the discrete temperatures of
Equation D.3 with their analytical expressions and by then
The difference
substituting the result into Equation D.2.
form of Equation D.2 is
(kl
kl
\
(D.5)
r*(Al)kl = -W
2-r1
Replacing the discrete temperatures with analytical ones
gives
... ()r*)|
(
kl
A
W
(D.6)
r2-rl
Upon expanding through Equation D.4, this becomes
CWE
1
kl
kl
L\
ln -
rW-r1
KrA)ll1-l
1
r=
-
W.
(D.7)
r2_ r 1
-
125
does not depend on kl, the
Since the term in brackets
summation
can be carried out.
D.7 then yields,
Equation
upon rearrangement,
-
r2
i
r=
in r
r1
(D.8)
- in r1
When this result is substituted back into Equation D.5, the
difference form of the conductor boundary condition becomes
6 lkl-
01
wmm~N m
kl
The procedure
in
-
2
1
1
Akl]
0
alk2
ln
- --
for Cable 2:
analogous
is of course
E[(1'k
k2
r1
r2-n
WCl
~~~
(Aa)k2
J02
=-
WC2
W
(D.10)
J
-ln P
The results D.9 and D.10 need not be weakened by
the assumption that the temperature distribution becomes
one-dimensional near the conductor.
The distribution is
always one-dimensional right at the conductor, since it has
a uniform temperature.
So the only requirement is to choose
the radial mesh size so as to place r* out of the range of
strong two-dimensional effects.
This choice is a matter of
judgment, and it depends on the given problem.
_ _· I_ I
APPENDIX
E
USER INSTRUCTIONS
Geometry and Mesh Size
In setting up a problem for computer solution, it
is necessary to provide information about the region size
within the cables and about the number and distribution of
mesh points.
This section discusses specification of region
size, subdivision of regions, special considerations for D3 ,
and weighting
of the mesh so as to have a good expression
for the gradient at each conductor.
Reference is now made back to Figure 2.3, where a
set of regional
divisions
is depicted,
and to Figure
2.2,
which shows the origin of cylindrical coordinates for both
cables.
The domain D1 always has four regional divisions.
The domain D2 employs only six regional divisions, since
Regions IV and VI are never used simultaneously.
The angles
included by the various regions are determined from the
azimuthal coordinates of their bounding radial lines.
For
example, the angle ~1 specifies the location of the boundary
between Regions I and II in D1 , and it thus determines the
size of Region I.
The angle ~2 likewise specifies the
location of the boundary between Regions II and III of D.
The angle included by Region II is then (c2 -~1 ).
The sizes
of all the regional divisions are therefore specified by
three angles
in D
and by five angles a in D2.
However,
since D1 and D 2 share a single inter-cable conduction path,
126
127
only six of the above eight angles are independent.
The
orientation of the two cables is determined by al and
by ~1 and ~2)
(1/2)(a1 +a2).
.
2 (or
Specifically, the orientation angle is
The convention of establishing regional
divisions within the cables actually has two purposes.
It
first of all provides a way to clearly identify a given
portion of the cable surface with a given boundary condition.
The second purpose of the divisional convention is to
provide a mechanism for varying the azimuthal distribution
of mesh points, so that they may be concentrated where the
largest gradients are expected.
Mesh points inherently exist at all regional
boundaries.
They are placed inside a given region by
specifying the number of subdivisions within that region,
both in the radial and in the azimuthal direction.
Here the
term "subdivision" denotes the smallest element of the
region, rather than the act of subdividing.
Thus if a
region has three azimuthal subdivisions, it is uniformly
divided into three sectors by two radial boundaries, and two
azimuthal mesh locations within the region are thereby
introduced.
The number of radial subdivisions does not vary
from region to region; it is uniform within a particular
domain.
Thus a choice of four radial subdivisions in D2
places four uniformly spaced radially points at every
azimuthal location in D2 .
Placement of mesh points inside
D3 proceeds in similar fashion, by specifying the number of
normal and tangential subdivisions within the domain.
In
-~~~~__11
128
the computer program the various numbers of subdivisions
throughout the solution domain are denoted by the variables
These are described in Table 7, together
N(J) and M(J).
with a column containing the minimum allowable value of each
variable.
It is noted that the variables M(2) and M(11)
cannot be chosen
to be less than two.
This is necessary
in
order to preserve the basic trapezoidal structure of D 3.
Attention is now returned to the discretized model
of D 3.
The width
of D 3 is taken to be equal to the skid
wire thickness, a number supplied directly as input data.
The height of D 3 is determined from the outer cable radius
and the angle
(a2- 1), as was shown in Figure
as the discussion
of Chapter
4 indicated,
4.2.
However,
the height
so
designated is only an apparent height and not the effective
height.
For when the regular form of governing equation is
applied at the four corner points of D3, four effective
corner locations are produced which lie outside the corner
mesh points.
These effective corner locations then define
two effective surfaces, as shown in Figure El.
The
effective upper and lower surfaces of D 3 extend halfway to
the neighboring mesh points above and below the domain, as
suggested by the figure.
A conduction resistance based on
this extended length is implicitly added in series with the
convection resistance for boundary points in the numerical
model.
Of primary concern to the user is that an appro-
priate allowance must be made for this extension in
specifying the height of D3
Say, for example, that each
129
rr-rrr--
I\ /r-
UPPER SURFACE
FF.
LOWER SURFACE
H
= APPARENT
HEFF
= EFFECTIVE
HEIGHT
HEIGHT
FIGURE
E. 1
Effective Surfaces of D3
130
TABLE
7
SPECIFICATION OF SUBDIVISIONS THROUGHOUT D
D2
AND D 3
IN TERMS OF THE COMPUTER VARIABLES N(J) AND M(J)
Type and Location
of Subdivision
Number
of
Subdivisions
Minimum
Allowable Value
Radial N(1)
1
N(2)
1
M (1)
1
Region II
M(2)
2
Region
M(3)
1
Region IV
M (4)
1
Region
I
M(5)
1
Region
II
M(2)
2
Region
III
M(6)
1
Region
IV
M(7)
1
Region V
M(8)
1
Region VI
Region VII
M(9)
1
M(10)
1
Normal - D3
M(11)
2
Tangential - D3
M(2)
2
D1
Azimuthal D1:
Region
I
III
D2:
131
azimuthal subdivision in Figure E.1 happens to be 10
size.
in
The effective height of D 3 is then based on an
included angle of 50° , whereas the apparent included angle
is only 40° .
So in order to achieve this true included
angle of 50 ° , the apparent angle (a2 -al) = 40° would have
been specified, and the mesh points above and below D3 would
have been chosen so as to place neighboring points at an
azimuthal distance of 10° .
An additional consideration is
that the four surface mesh points in D 1 and in D2 which are
adjacent to the corner mesh points of D3 should be
reasonably symmetrical about the y-axis of D3.
This is
necessary so that the effective surfaces of D 3 remain
parallel or nearly parallel to the normal axis.
Some degree
of foresight is therefore required in laying out the
regional divisions and in choosing appropriate numbers of
subdivisions.
The final topic of this section concerns the
conductor boundary condition.
It is recalled that the
discrete form of this boundary condition involves a
summation of temperatures around the innermost discrete ring
of mesh points.
In the summation each temperature is
weighted according to the azimuthal sector associated with
the given mesh point.
Attention is now called to the
physical circumstance that Regions II of D1 and D 2 are
regions of elevated temperature, owing to the presence of
the inter-cable conduction path.
The temperature trails off
rapidly on either side of these regions, tending toward the
-
:
1.32
one-dimensional distribution.
Since the gradient at the
conductor is constructed numerically by means of summing
discrete temperature differences around the cable, it is
essential that a good sampling of temperatures near
Regions II of D1 and D2 be taken.
This ensures that the
elevated temperatures in those locations will not be unduly
weighted.
Based on comparisons with one-dimensional
solutions, the following convention for weighting mesh
points has been found to produce a sufficiently accurate
numerical expression for the gradient at the conductor:
number of azimuthal
subdivisions
in Regions
the
I and III is
chosen so as to place a minimum of two radial mesh locations
adjacent to Regions II, each at an azimuthal spacing equal
(or nearly equal) to the azimuthal spacing of points within
Regions II.
In Figure E.1, for example, this means that
there should be a minimum of two 10°-sectors on both sides
of both Regions II.
This convention should also be followed
for all regions whose surfaces are insulated, for the same
argument then applies.
Input Variables
The input variables used by the computer program
are listed in Table 8, together with a brief description of
each variable.
The five angles ALPHA(J) of D2 are specified
sequentially, skipping over any region not present.
So if
Region IV is used, ALPHA(3) denotes the III- IV boundary
133
TABLE
8
INPUT VARIABLES FOR THE COMPUTER PROGRAM
Variable Name
Description and Units ( )
ALPHA (J) = the five angles a which specify the boundaries
of the six regions of D 2.
(degrees)
FILMP (J) = the variable film coefficients for the surface
(Btu/hr-ft
2-F)
mesh points of D2.
FILMR(J) = the variable film coefficients for the surface
(Btu/hr-ft2-F)
mesh points of D 1.
FILMX3
(J)
= the variable film coefficients for the mesh
points on the upper surface (+y) of D 3.
(Btu/hr-ft2-OF)
FILMX4(J) = the variable film coefficients for the mesh
points on the lower surface (-y) of D 3.
(Btu/hr-ft2-°F)
IPARAM = 1 or 2:
1 denotes
that Region VI of D 2 is
present; 2 denotes that Region IV of D 2 is
present.
(unitless)
IPROB = 1, 2, or 3, corresponding
to Solution
1 (for
maximum cable temperature), Solution 2 (for
maximum current), or Solution 3 (for maximum
oil temperature).
(unitless)
M(J) = various numbers of subdivisions, as per
Table
7.
(unitless)
N(J) = various numbers of subdivisions, as per
Table
7.
(unitless)
PHI(3) = the angle in D 1 which specifies the boundary
between Regions III and IV. (degrees)
RAD1 = the inner radius of the insulation
of Cable 1.
(inches)
111111111111--1
------
134
TABLE
8
(Continued)
Description and Units
Variable Name
( )
RAD2 = the outer radius of the insulation of Cable 1.
(inches)
REST1 = the DC resistance of the conductor of Cable 1.
(Q/ft)
REST2 = the DC resistance of the conductor of Cable 2.
(pa/ft)
RHO1 = the inner radius of the cable insulation of
Cable 2.
(inches)
RHO2 = the outer radius of the cable insulation of
Cable
2.
(inches)
SKID = the skid wire thickness.
(inches)
TMAX = the maximum allowable temperature in the cable
system.
(F)
TOIL = the oil temperature outside the convective
boundary layer.
(°F)
WD1 = the total dielectric loss per unit length in
Cable
1.
(watts/ft)
WD2 = the total dielectric loss per unit length in
Cable
2.
(watts/ft)
XHFILM = the thermal film coefficient for the
one-dimensional solution. (Btu/hr-ft2-oF)
XI
= the current
in Cable 1.
XI2 = the current in Cable 2.
XI20I1
= the ratio of the current
current
in Cable 1.
(k-amps)
(k-amps)
in Cable
2 to the
(unitless)
XK = the thermal conductivity of the insulation.
(Btu/hr-ft-°F)
-
135
TABLE
8
(Continued)
Variable Name
YC1
Description and Units ( )
=
the AC/DC
ratio
at the conductor
of Cable
1.
(unitless)
YC2
=
the AC/DC ratio at the conductor of Cable 2.
(unitless)
YS1
=
the AC/DC ratio at the sheath of Cable 1.
(unitless)
YS2
=
the AC/DC ratio
(unitless)
ICUTAP
=
0 denotes that no tape is present; any other integer
at the sheath of Cable
2.
indicates that tape is wrapped around either Cable (unitless)
THICK1
=
thickness of tape wrapped around Cable 1 (in)
THICK2
=
thickness of tape wrapped around Cable 2 (in)
XKCU1
=
conductivity of tape wrapped around Cable 1 (BTU/hr-ft°F)
XKCU2
=
conductivity of tape wrapped around Cable 2 (BTU/hr-ft°F)
______l_·___rl
________
136
line.
All angles ALPHA(J) are measured from the vertical,
as was shown in Figure 2.2.
For all the variable film coefficients FILMP(J),
FILMR(J), FILMX3(J), and FILMX4(J), a separate value is
specified for each convective boundary point.
The total
number of values specified in each case is thus determined
by the total number of mesh points on the respective
surfaces.
The sequence for specifying the various coef-
ficients is as follows:
FILMR(J) starts at
= 0 and
proceeds clockwise around D1; FILMP(J) starts at
= 0 and
proceeds clockwise around D2; FILMX3(J) starts at (-A,D) and
ends at (+A,D); FILMX4(J) starts at (-A,-D) and ends at
(+A,-D)
The inter-cable conduction path is merely skipped
over in specifying FILMR(J) and FILMP(J).
The variable IPARAM specifies whether Region IV or
It is convenient to use
Region VI of D2 is present.
IPARAM = 1 for the cradled
configuration
and IPARAM
= 2 for
the equilateral configuration, even though those boundary
conditions have not been implicitly programmed.
For the
open and equilateral-pipe configurations the choice is
arbitrary, because the strict correspondence between regions
and boundary conditions then no longer applies.
Of the 11 variables M(J), only ten are specified
as input.
The omitted region of D2 is skipped over, and the
program subsequently assigns a value of zero subdivisions
for the region not present.
-
137
Of the three regional
necessary to specify PHI(3).
angles
in D 1 , it is only
PHI(l) and PHI(2) are deter-
mined automatically from ALPHA(l), ALPHA(2), and the outer
radii of the two cables.
It is noted that the variables REST1 and WD1
describe only half a cable.
So if Cable 1 and Cable 2 had
identical properties and losses, REST1 and WD1 would be
exactly half of REST2 and WD2, respectively,
The variables
XI1, YC1, and YSl are not affected by this distinction.
Should it be desired to compute the total
dielectric loss per unit length W d rather than to specify
it directly, the following integrated-out form is available
[12]:
Wd =
2
-12
(SIC)(df)
watts
conductor-ft'
log1 0 (D)
t(7-354)(101)
3
(E 1)
where
VZk = line-to-line voltage (volts)
w = 2f
= frequency
(Hz)
(SIC) = specific inductive capacitance, or relative
dielectric constant
(df) = dissipation factor
d = inner radius of insulation
D = outer radius of insulation.
_
_I___~
138
The conductor and sheath losses are computed according to
W
I2RY
c
(E.2)
c
and
Ws =I2R(YY
),
c
$
(E.3)
respectively, where R is the DC resistance per unit length
of the conductor,
and Y
Y
is the AC/DC
c
is the AC/DC ratio at the conductor,
ratio at the sheath.
Output Variables
A listing and brief description of the output
variables from the computer program are presented in Table 9.
In the computer printout six values of ALPHA(J)
are written
rather than five.
However,
two of the six are
always equal, reflecting that one of the regions in D2
(either Region
IV or Region VI) has an
ncluded
angle of
zero degrees.
Because of the matrix scaling method employed in
the program,
that the determinant
it is expected
o
the
coefficient matrix will never attain an unwieldy order of
magnitude.
The variable DETEP4 is nevertheless
so that its magnitude may be
rinted out
onitored for each problem.
The user need not be concerned with this variable so long as
it lies in the general range 10
-50
50
+r50
to 10 5 .
However,
f
if it
takes on values significantly outside this range, another
matrix scaling procedure may be required in order to avoid
an underflow or overflow.
If the value of DETERM is ever
139
TABLE
9
OUTPUT VARIABLES FROM THE COMPUTER PROGRAM
Variable Name
ALPHA(J)
Description and Units ( )
= the regional angles of D2 , as per Table 8.
(degrees)
DETERM
= the determinant of the coefficient matrix.
(unitless)
IERR = 0, 1, or 2
This is a condition code from the
matrix inversion subroutine.
IERR = 0 denotes
no difficulties encountered in the inversion.
IERR = 1 denotes that the matrix is not
dimensioned correctly or that the subroutine
is not called correctly.
singular matrix.
IERR = 2 denotes a
(unitless)
M(J) = various numbers of subdivisions, as per
Table
7.
(unitless)
N(J) = various numbers of subdivisions, as per
Table
PHI(J)
7.
(unitless)
= the regional
angles of D1 , as per Table 8.
(degrees)
TAMAX1
= the maximum (conductor) temperature from the
one-dimensional solution, based on the
properties of Cable 1 - Solutions 1 and 3
only.
(F)
TAMAX2 = the maximum (conductor) temperature from the
one-dimensional solution, based on the
properties of Cable 2 - Solutions 1 and 3
only.
TANALl(Jl)
(F)
= the analytical temperature distribution from
the one-dimensional solution, based on the
properties of Cable 1. (F)
--
140
TABLE
9
(Continued)
Variable Name
Description and Units ( )
-
TANAL2(J2)
= the analytical temperature distribution from
the one-dimensional solution, based on the
properties of Cable 2.
(F)
TCON1I = the conductor temperature of Cable 1 Solution
2 only.
(F)
TCON2I = the conductor temperature of Cable 2 Solution
2 only.
(F)
TCOND1 = the conductor temperature of Cable 1 -
Solutions 1 and 3 only.
()
TCOND2 = the conductor temperature of Cable 2 Solutions
1 and 3 only.
(F)
THETA(J) = the temperature distribution for the entire
solution domain - Solutions 1 and 3 only.
(°F)
THETAI(J) = the temperature distribution for the entire
solution domain - Solution 2 only.
(F)
TOIL = the maximum allowable oil temperature Solution
3 only.
(°F)
XANAV1 = the maximum allowable current from the
one-dimensional solution, based on the
properties of Cable 1 - Solution 2 only.
(k-amps)
XANAV2
= the maximum allowable current from the
one-dimensional solution, based on the
properties of Cable 2 - Solution 2 only.
(k-amps)
141
TABLE
9
(Continued)
Variable Name
Description and Units ( )
XIlMAX = the maximum allowable current in Cable 1 Solution 2 only. (k-amps)
XI2MAX = the maximum allowable current in Cable 2 Solution 2 only. (k-amps)
.
~~ ~ ~
"~---I--
142
identically zero, the matrix is then singular.
Provided no
underflow has occurred, the probable cause is either that
the matrix has been dimensioned incorrectly, or that the
calling statement for RMINV is not corrects
The printout of the M(J) includes all 11 values,
with a null value inserted for the region not present.
All three regional angles PHI(J) of D
are printed
out.
The analytical temperatures TANALl(Jl) and
TANAL2(J2) are printed out for each radial mesh point in D
and D2 , respectively,
They are written sequentially, moving
radially outward; the first temperature in each sequence is
the conductor temperature.
The complete temperature distributions THETA(J)
and THETAI(J) are printed out in the following sequence:
starting with the mesh point nearest to the origin of
coordinates, all temperatures in D1 are written, the
azimuthal index moving through its entire range for each
increment of the radial index; the identical procedure is
then followed for all temperatures in D2 ; finally, all
temperatures in D3 are printed, beginning in the lower lefthand corner of the domain (-x,-y) and moving through the
entire range of the normal index for each increment of the
tangential index.
Array Dimensions
A number of subscripted variables, or arrays, are
used in the computer program.
_
__
These arrays and the
143
variables which determine the size are listed in Table 10,
together with their dimensions in the present program.
For
brevity the following computer variables have been used in
the table:
M14 = M(1) + M(2) +M(3) +M(4) ;
(E.4)
M5210 = M(5) +M(2) +M(6) +M(7) +M(8) +M(9) +M(10) ;
NM3 = N(l)x[M14l4+ 1] +N(2)xM5210+
(E.5)
[M(2) + l]x[M(11) - 1
o
(E.6)
Arrays or portions of arrays which have no variable
dimension listed in the table have been permanently
dimensioned at their present size.
Data Card Assembly
Instructions for assembling data cards for the
computer program are listed in Table
llo
While most of this
table is self-explanatory, a few additional remarks are
offered here.
Attention is called to the integer variables N(J)
and M(J), which employ the I-format for their input.
It is
necessary that all these entries be right-justified to their
respective columns.
Values of the variable film coefficients FILMR(J),
FILMP(J), FILMX3(J), and FILMX4(J) begin on card 9, as the
table indicates.
The total number of cards required for
these variables depends on the mesh size chosen for the
I
144
TABLE 10
ARRAY DIMENSIONS
Name of Array
Variable Dimension (s)
Present Dimension(s)
ALFNT (J)
J = 7
ALFSQ (J)
J = 7
ALFTN (J)
J =
7
ALPHA (J)
J=
6
ClFRAC (J)
J = M14 +2
J = 20
C2FRAC(J)
J = M5210
J = 36
COEFF (J,J)
J = NM3
J = 168
FACTOR(J)
J = NM3
J = 168
FILMP (J)
FILMR (J)
J = M5210-M(2)
J = M14-M(2)
- 1
J = 33
J = 16
FILMX3(J)
= M(11) - 1
J =1
FILMX4(J)
= M(il) - 1
J =1
IWORK(J,K)
J = NM3
J =
168
K=
2
M(J)
J = 11
N(J)
J = 2
P(J)
J = N(2) +1
J = 5
P1(J)
J = N(2) +1
J = 5
P2(J)
J = N(2) +1
J = 5
P1HX (J)
J = M(2) +1
J = 3
P2HX (J)
J = M(2) +1
J = 3
P3HX(J)
J = M(2) +1
J = 3
P4HX (J)
J = M(2) +1
J = 3
PALF (J,K)
J = N(2) +1
J =
5
K =
7
PALFNT(J,K)
J = N(2) + 1
J = 5
K = 7
PHI (J)
J = 3
PHINT (J)
J = 3
PHISQ (J)
J = 4
PHITN (J)
J = 3
145
TABLE 10
(Continued)
Name of Array
Variable Dimension(s)
PMID (J)
J = N(2)
R(J)
J = N(1) + 1
J= 4
J= 5
R1(J)
J = N(1) + 1
J=
R2 (J)
J = N(1) + 1
RlFRAC (J)
J = N(1) + 1
J= 5
J= 5
R2FRAC(J)
J = N(2) + 1
R1HX (J)
J = M(2) + 1
R2HX (J)
J = M(2) + 1
R3HX (J)
J = M(2) + 1
R4HX (J)
J = M(2) + 1
J= 3
J= 3
J= 3
RATIO1 (J)
J = N(1) + 1
J = 5
RATIO2 (J)
J = N(2) + 1
J= 5
RMID (J)
J = N(1)
J = 4
RPHI (J,K)
J = N(1) + 1
J= 5
RPHINT(J,K)
J = N(1) + 1
Present Dimension(s)
5
J= 5
J= 3
K=
4
J=
5
K=
3
(J)
J = N(1) + 1
J=
5
TANAL2 (J)
J = N(2) + 1
J=
5
THET1 (J)
J = NM3
J = 168
THET2 (J)
J = NM3
J = 168
THET3 (J)
J = NM3
J = 168
THET4 (J)
J = NM3
J = 168
THETA (J)
J = NM3
J = 168
THETAD (J)
J = NM3
J = 168
THETAI (J)
J = NM3
J = 168
VECTR1 (J)
J = NM3
J = 168
VECTR2(J)
VECTR3(J)
VECTR4(J)
J = NM3
J = 168
J = NM3
J = 168
J = NM3
J = 168
TANALl
XHALF (J)
J= 7
XHPHI (J)
J= 4
------
146
TABLE 10
(Continued)
Name of Array
Variable Dimension(s)
XHX (J)
J = M(2) + 1
J = 3
XHXHY (J)
J = M(2) + 1
J
XHXSQ (J)
J = M(2) + 1
XIVAR (J)
_
.-
-
-
-
J = N
3
Present
Dirrmension (s)
=
3
tj = 1 ,
XM (J)
=
11
XN (J)
=
2
147
TABLE
11
DATA CARD ASSEMBLY
Column
Card (s)
1
2
(s)
1
IPROB
2
IPARAM
1-
10
SKID
11-
20
RAD1
21-
30
RAD2
31 - 40
RHO1
41-
50
RH02
51-
60
XK
61 - 70
REST1
80
REST2
71-
Format
Variable
I
I
FF
F
F
F
F
F
F
F
3
1-
10
YC1
11-
20
YC2
21-
30
YS1
31-
40
41-
50
YS2
WD1
51 -60
61 - 70
4,
4,
Solution
Solution
1
2
Solution
3
F
F
F
F
F
F
WD2
XHFILM
1 - 10
XI1
F
11-
20
XI2
F
21-
30
TOIL
F
1-
10
F
F
F
21-
30
TMAX
XI20I1
TOIL
1-
10
XI1
11-
20
XI2
F
TMAX
F
11 - 20
4,
F
21 - 30
_
F
_
_
^1·----^---1------·
11(1118
148
TABLE 11
(Continued)
Card(s)
5
Column(s)
1 -
10
ALPHA
(1)
F
11 - 20
ALPHA
(2)
F
21-
30
ALPHA
(3)
F
31 - 40
ALPHA
(4)
F
41 - 50
ALPHA
(5)
F
PHI
(3)
F
1 -
7
1 -5
N (1)
I
6 - 10
N (2)
I
1-5
M
(1)
I
6 - 10
M
(2)
I
15
M
(3)
I
20
isi(4)
I
21 - 25
M
(5)
I
26 - 30
M (6)
I
31 - 35
M (7)or(8)
I
36 - 40
M (8)or (9)
I
41 - 45
M (10)
I
46-
M (11)
I
11 6lb-
10
50
9- (10)
FILR
(J)
F
FLIMP
(J)
F
(14)
FLIMX3(J)
F
(15)
FLIMX4(J)
F
ICUT AP
I
11 - 20
XKCU1
F
21 -
XKCU2
F
31 - 40
THICK1
F
41 - 50
THICK2
F
(11-13)
(16)
I
I
__
_
_
Format
6
8
__C
Variable
__
1-
10
30
149
particular problem.
The individual coefficients are entered
sequentially across each data card, with each value
occupying ten columns.
When all the values of a particular
variable have been specified, the next variable begins on a
new data card.
The numbers in parentheses in the Card(s)-
column are typical for common mesh sizes.
If it is desired
to run more than one problem
time, data decks may be assembled in series.
at a
Each separate
deck should be arranged according to Table 11.
A final data card having a zero in columns one and
two must always be placed at the end of the overall data
deck.
This double-zero card tells the program that there is
no more data to be transmitted.
Example Problem
This section illustrates the solution of a
particular cable problem using the computer program.
For
the example problem it is desired to know the maximum
allowable oil temperature for an equilateral-pipe configuration of System 1 cables, given their current and the
maximum allowable system temperature.
In particular let the
current in each cable be 942 amperes, and say that the
maximum allowable system temperature is 185°F.
The thermal
conductivity of the insulation is taken to be
0.1153 Btu/hr-ft-°F, and the film coefficient on convective
surfaces is taken as 5.0 Btu/hr-ft2 - ° F.
Also the con-
servative assumption of a thermally nonconducting conduit is
made.
A complete set of input data for this problem is
1_1
150
listed in Table 12, and the resulting discrete model is
depicted in Figure E.2.
A number of observations are made about the
discrete model used in this problem.
The effective included
angle in either cable associated with the inter-cable conduction path is seen to be 30° , a slightly conservative
angle.
An insulated arc of this size is centered about the
point of cable-conduit contact which, from elementary
geometrical calculations, is found to occur at a = 220 °.
Also it is seen that there is no particular association
between regions and boundary conditions:
Region III of D2
is primarily concerned with the cable-cable effect, while
Region V is involved with the cable-conduit interaction.
This breakdown of convention is necessary in modelling the
equilateral-pipe configuration, because in that configuration there are too many separate effects operating around
D 2 for the available number of regions.
(It is noted,
however, that the thermal model is equally effective.)
The
angle PHI(3) = 170° is likewise arbitrary in this problem,
since there is no cable-conduit contact on the surface of
D1.
The radial mesh size of four subdivisions has been
found from experience to be sufficiently fine to produce an
accurate solution; use of a finer radial mesh does not
significantly alter the temperature distribution.
It is
finally noted that the azimuthal distribution of mesh points
adjacent to insulated regions follows the convention outlined in the first section of this appendix; such a
151
ONAL DIVISION
_ATED
FIGURE E2
A Discrete Model of the Equilateral-Pipe
Configuration
152
TABLE 12
INPUT DATA FOR EXAMPLE PROBLEM
Card
1
2
3
4
5
6
Column(s)
Data
1
3
2
2
1 - 10
0.10
11- 20
0.9125
21- 30
2.0675
31- 40
0.9125
41- 50
2.0675
51- 60
0.1153
61 - 70
2.68
71 - 80
5.36
1 - 10
119
11 - 20
1.19
21 - 30
1.24
31 - 40
1.24
41 - 50
1.59
51 - 60
3.18
61 - 70
5.0
1 - 10
0.942
11 - 20
0.942
21 -30
185.0
1 - 10
20.0
11 - 20
40.0
21 - 30
120.0
31 - 40
190.0
41- 50
250.0
1 - 10
170.0
153
TABLE 12
(Continued)
Card
Column(s)
Data
5
4
10
4
7
8
5
2
10
2
14,15
9
10
13
20
1
25
2
30
8
35
1
40
6
45
3
50
2
1-10
0.0
11- 20
0.0
21 - 30
5.0
31 - 40
5.0
41 - 50
5.0
51 - 60
5.0
61- 70
5.0
71 - 80
5.0
1-
10
5.0
11 -20
5.0
21-
30
5 0
31- 40
5.0
41 - 50
5.0
51- 60
5.0
61-
70
5 0
71 - 80
5.0
154
TABLE 12
(Continued)
Card
Data
Column(s)
5.0
1- 10
11
11 -20
5.0
21- 30
0.0
40
31-
12
41- 50
0.0
51 -60
0.0
61-70
0.0
71- 80
0.0
1- 10
5.0
11 - 20
5.0
21- 30
5.0
31- 40
5.0
41-
13
-
50
US
51- 60
0.0
61-70
0.0
71- 80
5.0
1- 10
5.0
11-20
5.0
21- 30
5.0
14
1- 10
0.0
15
1- 10
0O0
16
5
17
1,2
0
00
155
distribution should produce an accurate numerical expression
for the gradient at each conductor.
The computer solution for this problem is shown in
Figure E.3, where the desired oil temperature 140.3°F is
printed.
The present version of the program requires
approximately
220 K of core memory
for execution.
It
requires 310 K of core memory for compilation on the
FORTRAN IV Gl-compiler; it is too large to permit
optimization on the FORTRAN IV H-compiler.
Capabilities and Limitations of the Computer Program
The present computer program has two notable
capabilities which have not yet been specifically mentioned.
The first is that Cables 1 and 2 need not be the same size.
In the case of unequal cable radii, the included angle
associated with the inter-cable conduction path is still
designated by ( 2 -X 1 ); the angle ( 2-
) is then auto-
matically adjusted so as to equalize the lengths of the
D 1 - D3 and D2 - D3 interfaces.
The second capability not yet
mentioned is that there is no restriction to alternating
current; either one or both of the two cables may carry
direct current.
This is handled by merely setting to zero
the appropriate AC/DC ratios and dielectric losses.
In
addition to these two features, it is noted that, while only
certain orientations of the two cables are physically
realizable, the computer program permits arbitrary configurations.
Finally, attention is called to two automatic
tests which will facilitate the location of certain input
156
FIGURE E.3
Computer Solution Printout for Example Problem
An
NW
Wo0 o
7 0
UN - " Om
.-O (01
U
f
Q 0?7-MO
m 10 IA W
N
7 -
I
*0
0
4w
'I
m
N 7-LOM lo
g*e
0
*
1 In4'N 7t-
0*
*
U'IAO 4
2 '0 U Q
IW
IQ W 7- UW
-4
-
-_4-4
ol
Li
-
-*
0
'_
Q
WU'% m
%d 4
-
0 0
U)4t
U)
W -'0
4r
u1n
I
-
U
U)
W
f.
ff
0 -
Sf U)
4-4-_44-
WIRI 4
4)
I
N NU_
§
.2 Mf W Q 7r-
- QV
-
0
U-, rW 4
M a
N1N
7@
a a, W _ M M a,
-
11I . W
la
'
0
N
-
a, 0 N
N
4
r- r-rQ
N
0of
L
N
0 IQ MN MoIAC)
co
IA
IAU)
N
OM
Uwi
N
N WM
-_
U
o o
-
r- N
N
CU)
0'?7-
0
_
_ W i_ _ _0 -0 _)I
_-7-7
o0
V
N
'4N f0 ICU f
-ffO
O) O'
0
N
0vW
7--47- LA I0 tD . Li
M'N
U 4'W N - U U
*
IA7F 40U
IA NN
INO
0 N tf)
- 4f 10 tD
U) 00 U' N 40%
tD
U)
U)40 NoN
RI N u' I 'C
"4
4 4
1-1-'
1-,
r0
0 00
.
0
0
N-
M
0N
_-
-4_
N
0
*
UN
U)U
)
O Q
U)r
W r? UIr
M
w W
Q
- W
'o W 0
V
U
RI r- N
4
0
f
IA - N 0
D ) -'
1 - en 0W
0
0
00
*
SO *
U' N _-'
0
P-'-
CDO
0n
('
-4
0
M V
_
_ 1U
RI)
o
N
0
*O.
'4 ) 7- 7- q' .) 0v 7- U' 0 Nx7--_)
D '0 IA7-
-
-
T
N
n
0
7
' 70 '05le
W O4
4
0
_
0
o '
0'
4''U
X) 1- 7- 0
''
N
a,
* S
0
ff1
0
0
0
*
*
00
00
0
W
'0 N a,
'0
Q V I
IA.
0' 0 U)7-
A in
IA4 4 000.
N Oi 4n U' N1 U ff N 40NI- A f' 'CN N-
~4
0*
5
*
O
0
0
I_
N
IA-
-
-4
p'0
00
0
O0 .
0
c-
0
0
N
t- U)ff1
4
*
I 0 a, I W M It - -40 LUi U N' Q
- 0
' 7-
U A)
N
0
0s
o
-4 N N'
0
f
7- IT N N
0' 7a 0f
0
1
*
*00
*
WI U) N
N eo N W
g
0
O 7- 7-4" 4 4-
*
N
N
i
r
*
U' N
N
IA
7 f]
*
A
'
'
*
7IA IA
*
S
4' '0 N U)
M
00
000O
0
O
0
'3.4
1. 8
Li
Li
RI1
O
O
0
0
0
~~~
- or r.0W
v WI
VI
P3 IA
W '10
- t N
0
I cr
ff1o
U)
N 7o
0 N .0ff f r1- Li U''
0
-U
'U
A7
7 U .0Lg7-'00.
V
O
O
tS
4)
M M
S
_A
t-
U
.)
Li
0
W
I-
U-
04
It
:
"
a.
L3
_
0
L
W
o
-4
-4
U.
u.
..
LD
,)
0-
_
-4
_
71
Li)
I
M
_X
n
"-4---"_ _
0
U
II
0
H:If
-- M
7
W
0N
U.
M
7-
le
N N
U..4
-.._4
*
vtD 0
_
0;
-
0O
7- 7- '07-IA U)
r N LA
N IA IA 7-U
0 v) 0 .3 rN 7CD PU
o
N
U'U 7-A U)
U' N ff '1 - IA U4 U' Nb ff'*
'C
V~~
o~~~~~~~~~~~
o"NU
f
)5'4
)7 NU'
U) -'
'7-IAU'
3 ?U
7
_0
'44
C)
-
l-
U)
L3
LD
Li, U
Li
(a
-
'-
,U)
N
Li Lic
Z
-7
4
-X
-IJ
zr
z
I-
I--
-4,-
t.,,-
-
^it
N
-4
X
X
4
_
A.
-,..
'.-
157
errors.
Should the input specify that the maximum allowable system
temperature is less than or equal to the oil temperature, an appropriate
error statement is then printed, and the program passes to the next
problem.
temperature
A similar procedure is followed if the maximum allowable system
is too small for the given
dielectric
loss.
Two limitations of the computer program are also called to the
attention of the user.
The first is that there are constraints on the
admissible size of D3 .
In particular, the thickness of D 3 must be non-
zero, and its included angle in D1 , (
2-
)' must be less than two radians.
Secondly, the included angles of the various regions must all be non-zero.
have been used successfully by the author, but
Included angles of 2
regions smaller than this are not recommended.
Program Modifications
briez suggestions for effecting modifications in the present computer
program are offered in this section.
are the following:
The modifications to be considered
provision for simultaneous use of Regions IV and VI
or D 2, and simplification of I/O procedures.
A modification which would permit simultaneous use of both Regions IV
and VI of D2 could be effected without much difficulty.
However, before
making such a change, consideration should be given to the handling of
boundary conditions.
Li the present program all boundary conditions are
specified by means of a variable film coefficient.
This was done in order
to retain the greatest amount of flexibility in treating boundaries.
If it
is desired to preserve this feature, then there is no substantial advantage
in performing the above modification.
For when boundary conditions are
specified separately by means of the variable coefficient, regional divisions
158
are important only in varying the azimuthal distribution of mesh points;
there need be no correlation between the various regions and specific
If, on the
boundary conditions (as the example problem illustrated).
other hand, it is desired to treat some or all of the boundary conditions
implicitly,
then the modification under consideration may be necessary.
Boundary conditions may be implicitly written into the program by inserting
the appropriate values of the various film coefficients directly into the
matrix-generation portion of the program.
This would be convenient for
boundary conditions which never change from problem to problem.
For example,
an insulated surface might be identified with a particular region (such as
D2, Region IV).
Then upon specifying the size and location of that region,
The number of such
the appropriate boundary would be inherently insulated.
permanent kinds of boundaries depends on the particular problems the user
elects to solve with the program.
This method for handling boundaries,
though, would make it necessary to have seven available regions in D2 for
the equilateral-pipe configuration.
The modification required for this
involves the input format for ALPHA(J) and the matrix-generation statements
for Region III through VII of D2 .
Provisions for generating the variables
associated with all seven regions of D2 presently exist; certain statements
are merely bypassed at IPARAM-type decision branches.
It would probably be
convenient to introduce a third category, IPARAM = 3, for a new branching
criterion.
This
criterion could then be used in the matrix generator to
choose such branches from the (IMARAM =
)-type and (IPARAM = 2)-type tests
so as to move sequentially through all the regions of D2 .
The (IPARAM =3)-test
could likewise signal a special input format for ALPHA(J), indicating that
six rather than five angles are to be read.
The program so modified would
159
be most convenient, provided that boundary conditons were treated implicitly.
Finally, some brief suggestions are given for simplifying the I/O
procedures of the computer program.
could be improved:
Concerning input, two primary areas
specification of the variable film coefficients and
specification of the effective size of D3 .
The former is bothersome because
so many values must be entered, and because of the need to keep track of
all the individual surface mesh points.
Implicit treatment of boundary
conditions would completely eliminate this inconvenience.
specification is retained, a provision might at least
the input.
If explicit
be written to simplify
For example, it might be desirable to merely specify a single
coefficient and direct that it apply for all the mesh points of a given
surface (when appropriate).
Or since most of the surface points are con-
vective, it might be more convenient to read in just the non-standard
coefficients.
angle (-
Specification of the height of D 3 by means of the apparent
) is akward; much foresight is required in order to end up with
the desired effective height.
with (
-
It would be much more convenient to work
) as the effective included angle, referring the associated
aximuthal adjustments in D1 and D2 to the computer.
three suggested improvements are mentioned.
Concerning output,
First of all, in its present ver-
sion the program prints out very little of the input data.
Such procedures
as solution identification, error location, and output analysis would be
facilitated if more of the original data were written.
Secondly, it would
be a simple matter to include the conservative approximate solutions in the
program.
These could be written alongside the one-dimensional solutions,
thereby making the upper and lower bounds for system performance immediately
available.
-
Lastly, the overall temperature distribution is not very des-
160
criptive in its present format.
Separating the temperatures in the print-
23
out according to the three domains D1 , D2 , and D
would present no problem,
and this would help considerably in identifying features of the distribution.
Also effective use could be made of plotter routines for illustrating the
temperature distribution graphically.
I___·__
APPENDIX F
LISTING
OF THE SOURCE
PROGRAM
162
163
U
-0
O
°
V O
O tDO O ° U.
O C O - O
U ' Q 0 - I' O
(- U----O -. - O,C) C.;ON
N
U'
F
U U O 0" '
(,
j UDO
O-
N )O(
J
On U
n
Ln O
mm m
. .
Ux
-JZ
-- 4
C
CL
N9.
I-
C
4..
LV)
* U-' U. .
--
i,_
0..
,m[ C
F L
o.
- 11.
4')
)¢
L ',.
x )
0..
I"
1-.
O-4
.4 9.
OC
_*?'
"-" -
V)
cY
W~
t x .-.
-
I-J
r¥ ,Q...* U . 1.Q I--
X ?~
0
ur*
.._N
-L
_-...
.
- J
.
.-s
0
.Z.
'JJ
=L 9 4-*,I
w
>Wi0~ -= Z~ 0' 9 LU
-C
L
U1
ti,
LI¢.J
'0 i_ j -,.~
CL
--
Xx.
' - 0.t
XCL
U~,X N1 '-Io: -
* Q')
<
--Z
Z.
* C(U
f
-
-
9.
CL
4~
* JJ(/
_Z-I -
N
I
X
-
I - I-C
J 4~~
a>
co %
- -
e,* e=.,
..
;-
_lS
< "J
dN
I.- ;"'
I - V.I. CL)0.
< C u.
-"'
,-.,
~' 0X _'-
'""' .-.. X
xL.- x .. L
v0.
=to _ to
,
C..
t
q a
Y
-AZ
W,
U_
u CU
XF CILN
-,.
ZC
_4'vf
r
Z -~,,,
C
-.4
-4z
c)c
L)
L U b."
3
U.
0
-.
I-)
Ig
r.JCJ'
:. -3u
Oex__
44
-f
cD
O
X
N
X C
Q
,..
_-
- U.N
Z -... _
u
4I-N
I
M
U i.
-
CJ)-
*9
etn
%Q.L O
-~l... C..'
CK
-w
CZ'""
L/~- rE
0- I.v
- C:
IT
U
N N
XO
j
X
I
-4
o2
.L)
_00
. N 4
* 1/) %F ~-- I1 II II
. CD
U N". I
I f
.
(3""~(-JX .:
-4
a
O (_
(N
*
_
__
X
ow
.
0 q
' -'_ .)
_IL:
..
U.
aD
-U 0.
-J
U,
cc
U Ct
S
-4
U)
_
0f. U.. T ,...
a;
- ?
-J
m
.LL
.,.J
e) LL
'q C
-0.Z
9.-.
X
.-.,-
-99 -
U-)- Q* -
0uQ
0.
CC
.4UI '-4 L
a
-4
,..¢
-
i
_
OZ~
_-U .).)
-0
0(
NI..) U)
II-
-4
a)
0
X9.-J
en
_·
CD
a O I-- I.-
~LJ
N
N L LiLA. C U
b Q-cr
1
C U
-
-
_
-
UJ
-J
x
.- P.O c-
4
-40
N
i'
ON
II UU
,.4
it 11 IX
D _-
Cl
0 0
" '*-0
'X
Xowz
F-
Q
..
*
* O UJ
I
D
11 -
_ Z -11
-J
I.
-
":_"-Z
Z
U. O0
FU
w
W
n
164
O
,
¢
',
O
4o Co 4 4oo o 4ot o " I(Q o
00o~<)o00000
oo o
-;
O O O
'
J
O ' Ct
'o
uw
unm>Q
ro
o
o
o
%oQ'
AO CL oAoV %o
'
go a, rOO
o o-r- oo~
ooo~o
-O
N%1o4
_ _
0
Q
4,4
fn
NN
II
00
'-4
U-
13+0
I 'I'
It
·
0
"3
4
;
O.
3. -
_ <
J
_
_
X
Ur
-J
:1
X
WN
ff
N
S-,
-
_
Q
. *%%% ,
. wU_
C)
0
.
%,
*
:'
c-4
tOL.)C
,0 ..
I C1t4Li Cc
O!* -L>1
* -II NII O
'0
* 4
*SeS
-4
-^
-4
II
IC
-
-
nI II -
I
^C
__
!U
SD
z<
_L o-1
ii
O-4
co
- I
I- _-III "
lb
.
o -
_
C
_
,_
,
· ,-k,, LlJ
Ci _
LHA
. UU
X
S
CI_ U
J
O<CL
NO A CS
11 . _
* a:Q4 _ Ii)1 1tLA
-,4
4O=-Oa4i
1
.
_
_- _
4
*
U-
;C-- J4
_ 0.0.
1 _~ U.
If 11I
" '
C%,-4 Q.
U-O OUl Y11': 1 ie-J
co
!"-
-
_
_
, iJ
Z~
CJC3
_,
{3.
X
O
C
*
_
C; Oc
_
0.
0.
X
CL
U
4:
A
"O
.
,-
-F
C
- C - L-J
!_
I'' IY4: 0.
CC
0 I. CC
_^
-
4
J0u". NNi
4.I 4
,,..
0. ::o
.L,
0N
to
-
-N
N
N
5I
165
O ) OunD W O
M
o o
o
00-4
A.
W. :
N
I
" I
-_
uN
N
-
- -
- '
rQ
Or
r onO
o
.
H--UOZI
,-d
t-
,._
-X
0z
CI 11 - 11 ^11 )
N EON
D C;
o
*O O
-
N -*
y1 2: -
%o
N
-
"
11
Z
i
~.
It.
* a4+x
r- co
In'.0
CN NJ m
m M
,_
I4 --_N .4 _- N M
+ I I + + + +
. I- co
;L
_
;
I _ N N N1- . _.
fo
IIZZ zzzzxa.x
x xz
Z Z ZZ 3E 3E ;
-I U-Nkn o
II
.,+4 _ 4 N_
,N
W1
3E
3[;
II11
oX Z
U Z
of
1 U Ia.Q
2:
u -- r X
= a-E
c.Z- { L
M
. .4,,,/ -,j C,
E
in
zI
c x
x'4
Ii H I
Z CM
II X JLt Co
ox
V
U% U U
,)C) X
; : ::
u-I-.-
. o
.4
.)_
-
^
O
_.e
Co
C; o o o o _ _ _ O _ _ _ _ _
If- N -
O
-
o o
0-
r'. ,to
-
urD
N
C'
I} 11 0
_(N
O
o o Co
11
4
H1It t
_V
,. 3E
a;G.
C x
LO
-
%
ZZ
-
_
II 11I
N
M
0. 0. 0C.
IzzCNNN-.4
zzxx
166
_
Q
-
- _ --
·-4
4-4
-4
C.
ro-W P C - N
- N CV 4 TW a
LT
-
4
--
-
-4 -4 -4 -4 -4 -
4 -4
N M}%r
+
4 U
()
N N N N N N N N N
4 -4
--4
.
N
.
-4 -4
_ N - N +
1N
.4 '-4 4 -4,-4
+
I
.
li
II
II
I
II
11
I I X
=4N n ,, - N T_N,4 rNII
L,QQ
CI Q. Q. C4 04 N N
2N NN
C X
It
4 -
--
L' . . .
M
' I
_
4 -4
-4 -4 -4 -4 -4
-4
a o 4 N
(P
4
M
4 4s 4 4
flf m Mr%
4 -4 -4 -4 -4 -4
I
0
I
+ + + + + + .4 -+ -4-.
_ -4
_4
- _ -
NNNNNNNNNL('s\_--+
,
- P- cc OD
fcorN
I
I
I
I
I
I
N N~o,, cN eN N' N eq c, cv4 u', .- ,- -- -. -. --
-N LA U- U'
i
It
. . 4.-"N
-44
4 -4
4 u 'o I
. 000C)00C;00C;00C)C~
-
rNr-j
A4
-
-4
P 0o -_ N
N MM
LM CL
_ _ _ _ _ _ _-UN
r\L
n LI LO I-N UT' X N
l("I,=~I icqt. ,,IX 11 11
~ 2
,.. I11 11 N x
Q ?- PL M Mf CP
i n
Ul U'\
t1\
lx> 0\ cr UN _~
_ ,
X
i-t
X-"
M' VN 'D r-
I
N
I
o
-
I
.
-- =~=,
M (P
-I - L -X -X -X
X
X1
11 aI
tt
11 11 11I I of I
- N
X 2X_-4
a
M ;, QL f
;
O4
167
0 -4N rf4 4 LanQ F a O
N
- -O -- wWQ,
T-n
Un n
4O-n u'- - -44
L
1-4
o
4.- -4
-4
-4
4
_
-4-4
.4_.
-4
.-.,4
-.
_
.4
%
o.
,o
- "'4v
4-A)Cy,0- -r---Nr-m'r~
--r
%
.W
_
.4
. _ _
-
_
r-4
- w -4.00o
4
-
_
4
_
4
000o0o0(-00o0o00o0(-00o00o0o0000o000o0oo000o0o(2o;0o000o0
_
N
Z
X
:E--
X
xl,.·--
4r
-
_
%-4
-4~
--
-4
-,
C·
II
_4
-
3L
un_ __
JlL
I.. ~" C r,,
t t 1it
.4N
0.
-4 N tQN
-J J
i
_E
ZZ
a Z ZZ U.L.
a x
xExa :Z
-4 01 W1-
N
.l
-J
-J1
C)
-4
-
If
, CI
I jWW
0
C
0
LD
l.
.
N
--'
I
X.'
IiII
I - I ,LZZZ
, N000
^t0
cc
f
-I
cx - N
O LL
..a
-v..tZ
3L
~-E+
9,· ..,,
'~~Y
,*a + I 0.
"I_ O'1-"
- 9L0
N N * : N ¢N
(N
4Q.
0U
_ -a.
..
CX
. -LXU
X :IC_
N.JW i
eL2
_ C L
3LI
~ Z
~ ~
(.'IC ve:E:
11 c'~11
0
J rW_ -.
_-. , ~ I,. r..O
N
J.
9
X XZ.
C
I
CZ ~
-CC
re
<
II rJ
-uJ
Nli.q -.
_
9 .L.
o4+
%0 -
X
2;
U
LL OC; UU.-- 4
,1
0
II
%
W=-,
-4I.I.
Z , l.9
-4 m U.
xIt
11
-
I,-
'--L; I
.J
II CL-
1
CL
j
-
'0. Ui
a C
iLI
91 -J
_ X X
;;
9l,I
C
+ J
-h
11 11 I
tt X-
I-
%-.3
O UN
LJ.
X-
N
I+J
_-
C)
_
C,
If It -L
0
LI a
4
nlcX
168
_
m
c
O
' O f- a) oa 0 - CN en Q.
m o w m 0, as r C701 0" (
QQ-
-
-
4
N O
co
-
-
C
0
-4
,4
-4
-4
-4
C
QQQ
)C
-4
-
n
; N. (D
;- o( 0, o -C.
a,
-- 4
UN Cy
-
-
Q
o
N
N
O
N
N
o
C) o
N
N
Nrf O
o-4
I- C a,
L;
Q
Q o
'C
?
4 _4 _. _
_
-_
-_
N N
C0 oCQo
N N N N N
N
N
U0 oC0o
( 4
-
o
II
*
_
0
C)
0
-4
X
-.
-
+
rs
4,
a:
-
i-
-.
*
X
C..-
I
.L4-J
'
-0
-L
X
4
0
~. X "- _.
N QL
_
XN
It
"X
O
*[:3.
n
X
---4)c_>X - ZFC -"
It
·'
. I
1
T O
I
X X X O X.
VA
tn
_C -4
-
-'4
CK
.
II
-
_ . C
~ .il~
IIII U
t
0 '"} - r,1
_,
,
,
,' , _
.
N
~ .~
tt t. * t
q0
N
U-
g-_
II C
,
II h-
: If
I.. ;:- (J,"
O11:
C)1.:
,O
_
9.
XI
it
F
LA
-
.-
O
_
-4
~ "'Z
'-'",-
-
N
LA
X 4". N ja.,
..
. JX- JX
1 .- ' _4
Y - _f
Uez
- *I
O:~ + 6tL. 6~,~IL-
-
0,Q
_.
O~ Z"')
ZO
. -. II
- , ~,
·1
w--4
_
tL "'
11 a:L
U-
a._3_
~ +O .
.
.J-,d
IL
-
,..
U.
'~
X" -. 3
3"
1t
(X X
cz
C3
z~~~i
~I
-4
X4. Lm
- CI
XC Ul
X It I -X
1
O+
X
: 1(
Q0 C
X
'j
IO.L-U*OII
;X:.
,0~
;;]
N
_tl]
- -1f+X CX *.
.
:Z . -X W
I--3
X O
X
w
-L
O
"-t X
L.
N,4-N
-. *
-4
IN
-
A
I-
O 0r~
-9*
-K
169
-o4
-- -oN
-
re1o
, un 'o
W.ao0 i.Q
Nc.Q
fn ,r
LO,m e-o(Q
C0 C Or
o O (i;O 0O,
-4 ~DN
N O N'rO N N
(j f
M
c m O
Cj
r
O ';
N N N N N N N N av r N (N rN N NC" N
tN{\1r N N N N J NJ J 'N NJAJ N
N
N
Q
1
J.
a.
N
*
*
-
C*~~~~~~
N
A
.1
-
-_N
*
4-
IN
L.
-
I
..
:.
~L
-
-
L
0
)
NCL
+
4. .s
*
*
-N
_4
_
ZN
*
'.
I
C
_-
C
*
l.L
U-
z
%
-4
CL
z
.. 4
,4'N.
I1L
0.X
-4
I
-
-,
n
Z-
-
N
II 11
I,
tJ)
CLJ)ON
C) C
Ln O
GO
C
00.t
C
nI
---44
w
_,
I11 u'
I
i :)
C - 0
i-rN
Z
CL
U
x
I L
0..
X
C)
N
N*
Z
1,-
-,Z
,=4
II
,-
O-i.L
QL
C)
_4
'N
_
N
O Q*"
-4" .
(N
0
I1
* _* I14
. I!
itll
0.
7
C
r- o0.
*4
.II
tN ·; N-$Tif ,;
Z
*N
N D
NC) N
U
N _ NLLI-FN
JZ
J
U.
CL Q
C::
a.
-
-
_4
N
ZN
z N
C)
c*
*N
Zl
O.
CN
C
X -_ nZc,_-
i '
Ln
-
UJ
z-II Z tN i.t-
(J
CN
x
*
CL
a
L)
.- o
tJ
c~
CL*
N
6
- o I_,,,
x
N
N
*
- .1--
ZNC
0
tJ
.L
X
x
_
o
H
-4
0
-4V)
_r
II
-
.
-
00 -
t
-4
U .
N
I
0
I
_-
CJ
14
"- ,.= ?
*
N
N
CK
X
-i
I
+
*XT
.LOC)Q
CL
*
-
-4
X
O.
N
CL
4-.
N
I
N
C
L
*
*N
J
O
U
0
X
U.
.J
-.L
c
NN
0'
4
N
N
1
_
0C"_
U
N
(N
I-
*
u
n -PU
N
.4
b
0
- "L Z
w
JZ
L) <
w
CLL.
4 N
N N
NN
170
(~
4
0
'
e" r
4
(X) 0 4 r M u4 n O v W
U)
w
IO
o -0
COW
O cO
C
r - c
r
rC
rC
r
rO
F.
OO
U
O
'
(
-.
U
.O
o
O
k uO Z
n
O u~
N N N ONN N N N N N
.N N N N N N N (N N N N N N N)ooQC
N N N N N N N N N N N N;C
;C
QooC
m ,T Un,Or--c
o0
-
u.
-J
J
Xl
*
L.J-
1.
I
-,
X
S
.N
J*
S
-4
0
+
*
*
0
,1
X
I
N
*N
*2.
0
-4
c)
M,
N
t0
0.
X..
N
-4
O0
*
*
*
N *
(.20
*
.#
-
N
0 Pt-d
·
4*
~*
. NN
-4
CL
L-)
~.J LL
N)X * 0
NO
Z * Z~
W UJ.
1-Y
* a
·
.Q A
. .
...4N
-i
QS
0
O
OCN
_
n 17
NNN
·
o
U
N,_
LP
.
D4N
Z.-
-'C* H}
II
i
N...
ItNU
n
-
W ;2
V
^jZ
U- Z
0 U.-0
'-.l
0ZJ00U.-Z
_
N
n
<
N
N
N
(N
II
*
kn
II
LJ u..J
U
~
00
-
P
Z -
,U.
I-ILcx
oH
I
%
-(C3 -
N}
NN-Jz--.)N
_
CX -W_ C~ c~
Li
o.*
C
C
ffi
-U
-~
r,~D*
C.=Z * 0D---'
n
-
X
OX
ft-
e~
4.IC
C-JC
-~~~
I!
'*
-- C .
U..).
U.
w.
<.Uo
S .-J O-_
#N -. . .SWr
SW
-0 ~
IN L
L
s_
£
II
n
0
-4
N
SWOSW-- _ oJ
V 0-
410 Z.tl
DC
0rZ
fn
0
N1
N
NN
m
N
r
N
N.-
I N
i-N
-J
,,
z.
_
N4
171
a,
u
a
_ N m
C.~O C.) Q-.N
m O O
_ - _4 _4- N N N C N
Q o - -4 - -4
n n
m
r c)
n
n m m n
LA
C - Q.UO C O
O O (D
_.OCo.T
W a, a, U a, Ol cr T al 9r cr a cJ C.. o cU o
N N NN N
( N N N N
m M
nn
f
;QoQQooQooQQoQoQQ
o
JooC
U-
-J
_j
L.
*
-4
.
US.
w
LL
-J
_J
I
x
U.
_
-
-L
A-
-
Ir.
.L
X
-
+
-J
X
UN.
.I.
z
**
U.
U
-rX
.Jx
*
X
I
C.D
-N
x
-4
x_j
*
I-L.,
.*__
N
*
N-
r-
-M-
J
X
$
*
·(~N
w
(%
Ic
L--,U0i-
9.494
4
xL-L
Qa
N
W
-
04 L11 CL
Z IeU-
z
L.
-
cx
--
J4
z. Z
-
Z
N
f
N
A)
-4
U-
I
f0
A
+
x
-4
_4.
_~~~
a-a-
O
U-
U-L~Z
_-
'-
< z
,,~U
0)
O*,
F
-.
UJo
Uo
,U .
U), l _
~
-I'-
,4-4
U-
0
0
<-cc UJU
UIs
CZ
-'r .j
LO
I. I<t
-
O
u1n
x
_
O O, [ I,L. U
_-.
V) C. CL,_
z
-4_t -
9
-4
x
*
-_
L U < 0<
UN2.
i
_ +Z
4J
IQ
9
N C.
N - U. <
-9
I-
CNi
-4 L
:r.-
N.
_-4-
_l
<-U
1-94~
0
N-<LA.
NN
-
L.
_
II 1,i
N -4%r'44
LL
-
W _ - 4U
.
X~ LL.C
Mt.
CN
c
-
-
*
1
1- -_
+.9
*
I-
-X
-
I-
-
_1w'
~ C-
LLO LA- >:
0O
-4~~~~~~~~~~~~~~
N
N
O
N
Z
_ LL Z LA
-4
N
LAF
U.'
LN
UN
Lp
LU
172
O U rU 0
{
C
N
N C'J N NN 'M t
0 M. rq)M0Q
'
U.
' r- 0
n
M
Q0C"
rn
F,0)
o -4 N , 4 r r
4 %M
? -T T 4
eC n M
M
M
r
Q U _-N
C
O4UO Q
Mc
O
U t
SUN UNL Ur C
n C)
m r rn
f . )0n
t
*
*
I-
-4
a-
U-
rU.
J
X
=J
-q
..
u.
u.
-.
.L
+
k.
uLL
.
-_
*
-
aZ.
w
cc
Ck
2
-A.
')
.
X
L.
X
-
z
2
r
U.
0.
*
-
-4
,+
U.
N
*
U.
_
+-
*
a.
N
x
*X
X
.
x
I
*
4
-T
-J
-J
-J
a_
-J
<
I I
X
CN
N
0.
'-Nr
X
In~~~-
U.
_A- U_
J- U
I eW
N
x
.
-J
.1 d_
Il Ct
N(. )~
NON
7
rI- I Cx1
.,J
0
I ON
Ni11
_ _ %
!~;,
N
I~
U
M
U)
1_ 4 _
_
_LZ
_
*9''
-
~t
X>-,E,
*
_-*N
X(.
V)'N V*r4'
11N
O..
C ^0
. X
INo4N
_
. *
11
N
j^ N8
tIN
i
S)
y.
I t
~"
41Xj
II(
L/
(,Q N
Yt 1% -
_ .L N
IL
-
* Y
1 o)
I
;L
M
-~N>.f CO
CL
I*
I 'NOQrX
_a.
w
. -_,1 L
It
--
-.
r-
,,.,
3
a.
rv
-
r-
U.
.:L
N
-4
*
._4 o
U..
r-IA
U)
o
U
0
N
r
r
z
0. ').C..)
x C3 *
*
I
o
;v
a.l
i"
1i
I i
]E
I
'
X
cr
X
at C
_ii
C
-
Y
OC
0
Cr:
X~C
_
mN
-a N
a
1 rC
H
I
- II le
X
x ~ -,..:s ,.. f ,,..,
401
,,
%o
-
CL
.
0.
0
N
N
173
U'm
r W LT,0 -4
-
N M .r
,4
fr
, [, O 'O w 'a w NU,7 r f rM
n e r ;OOC
M t
n0
n 0)OOC
m
O
V 0o
(-
M4
r- r
en
4 N M rt
0P
r- X
)
0-
N
N (-'ln
w cX w w ~
0 (
F-r
M M
C f0n 0n 0n
M
0
0 0
0
M 0
)OC
c
M)C
ur %Q
a
M M 0
N
*
*
*
*
LcrJ
0U
4N4t
t
0.
*CC
N
I-
43;
C0
X
*
X~~
C4- k
.-
-
N
N
"-
N
CL
*^
·- .
)LJ
eL).
,-.4) 3k
~1
X ~-
:II
_< CD
N
X
aEN
M:
Z,.,
m
x- II " -'0
ff)
1 1
,IN
I
I 1-- Y I Z-OL
O u.u Y
I - C. 1
1 .j
CJ
tI
I 0.
I
_II rna -I4
I F - 1I
I
-I
X y C4-%
M X: u 2-M
X
<'
L
-N-4 tt r - r~ t -
.X
4m
c
X0.
CL
Im
NccX C
-
0
rn
'N
N
.
11,,
'
M "
I
3F
;T
C)
rr
II A.
X
IN
YXN11
f
O
NI
IE-Cl
=X
t.W,
NI
e C
ZII.L
1 rnX
-,S
_e
,-
Ql. .
0-
_E ;
)(
Ch O
w
_ rj
t4
~~~~~~~~~,--~.
r
- ~~~~CL.
X Li
4_
i
M
t
-.
CL
yx
;I·,-U1,,
I , II~..
ct .. II ?&J
~
YC. f
t
(P
,)1,
..rnr~~~~~~~
II~N I_
~-I4.~.
I r
?:
j
N
1
"
,.
_!t-
, t-
*
;tl.
~~rnm
-
X
*;l
S
:L4
1
u-
z
I
-4
0
N
Mn
N
0
O
174
N (0 U
0 OQ
U
0
r a (U
a, VS all
onnm
U
Q
rT Ttr
'
- W o
J0 -
p0
U N0
cr
-
¢T ¢rt
0
0
(
-
-
-
t
0
-- N
-U 0A r W
- U -.0
W;(
-4 _4 N CVN I N N N N N N m ) re)
ITr T xrt
t
*
*
xCJ
-4
v
lN
X
X
.
*
x
a-
I
-
-x
.L
X
I-
+
"I
X*
CL
-X
N
4-
LL~
:
J
x
Q.
N
CY:
*
CyJ
Z-
N
.-
T-
x
: C
N
(n
CL
I0.1
N
^
-
--
(
-..
'4*
"3
..
I xx x
_4
u~. a..-
~-1-
Nt' Li
*
N -X
N. I
CN
.,..i.L
_;~ -rrT
*,J
>
.-
)::..7.
0z
-N 0-~fr
)
* Ii
-7
I
U. I
U.I.
r. cDv
_..*.,(._-7
, -,
'
,C
, Q ¢ UOC t 4
2L90ttn
v'
II -, II X. II. tI * X -,II A. rII IX
X. XJ.
X
X?
_ X
~ I U.-4 . ti %U : xovcwu u.
I ,_
II X
I IX
t
-" _,?
L Ino.-x JXO
II rn -,,1 (..
ror'N:
-:
(V -4 ,X X
L
0.0-U.KX
-J .
.QO¢¢L
! (C N
z L- Cl
b7
-4)x~)I
_f
CX
hO s U. W
.
N
U'
'4.
O
Ln
U'1
0
U)
'
61A
tv
- --
z
U.
W
U.
U'1
C;
60
V)
Q
.
-4
t
aj
-
- U.
J.-- a.IaXa.*x
_
U
) U
ItX
..
, U.ci
x-J
^
'O 'o
-
-J
ao
ifN
;LJ U
-
a.
Lm.
I.
&~
)C
I -)
:~ -
-
tI
x 'r * -
-,
-c
It
ff +5-;
Y.YUU
'
cc
-
_--
II
- 4. Y
X
~L
,,-
W - %LW
,a,.
mm
X
T ...IX
X
C)
*-I,,
__I --,4
X
X ,L<5
L*_ *_1,e~
-
I
N
- J.II U
Na.
24.1
-4
- X . .- x:X
14
;L
II c-4
CK
-;
_L N
a.
-
4
1
x
X
-
a.
-
_4.
-J$
'-
D
-co
4
175
r ' uLn
Fn
r cc c U
N m 'f) 'r
O OMf M (fYmn 4r 4 O .
-
U r'OT
O 0
N
LA O
L U LO
N
Li~~~f
(3
V)o
a-
-
-4...
I-4
-4
CL
_
-
ct~~~C
-a.)
CJ
U
0
0
I
.
* - .CU
J-.
-4
-4I
~ ~II ~'~~~I
O* -
Zin
II if 1 LI,,,, . I I* IILI -
0
0 U) U"A
O
cr\
LANU
0
0
Ln
O
V
L..
_-4
W
I~
O'.
I
I U
-~~~~~~~~~~,.- I
-
cr
oCK:,
-
-
-L
D
-
L
..
0
t
-
_ W
0 UQ
+ ,X, _
N
~T .=.
CY -rI
LU
-
U)
D
.
-4
A
u0 L
I
,"
-4_
A. CL
-4
* Z -4Q U- - - O N *z *
N~- * _WI.- W O. a-J N Ln.4XI* *
-4
0''"" _Li{...3
N
*3-4
*-4.*
!lN
O_U
* ,.
11N
3K
H-_ t10
110-4 [ It-4-4of N
---) I
r~
~. a
: c
' ,- -j
I
_" ._',- %
.-
.
~t
-t"'-
-4
_4I11 III b-
uti W.-;
uiw
umJULiLJ
0- CLjUUI..0~0U
3C
'Z
.'44'
J. O.~'~ .- U,
r-
t ~,
Y.. CU
* + 0
CL.~
*
N
3
cr
-4
CN
CD
-4
r **,
"J"_
4 .. '0O
O O U Ca O
-
-
L
- O4
O 07 0
N .,)
O <
'0 O O
to LA
W
ct
-4
-
'
-,
aw
-
(.L
-
r-
LAJ3XLL -
0
o ouJZa O'"=II
001 2 :L0
is
0t
LLU...r-J
IItl 1)
-,, Y: ,-".V""~,, ,,
O ) '0
-4 -4 -4
LAL A U,
-4
n ""
- U'G
Zr,~
N
P.. L.)
I I LL
~(1
LA _J V IS C- VA
o r
N (A N
L
U
It II U.
~lAL
0
-I
UCLO
00 0
176
Q%Q -U
r
J
0
(.
4" U
N
o0o
r
Jau N
N '
r- r
a cc co
0 0I0
0
r-- fl rw- r r
QI O T t
-
o
o
Q
o
o
Qo
o
- 'V
; Cx - N f 4 Lr U I- Ord
4 N (
a: a: W a: a: m O
c
QS 0' 0' U 00
0D
C
0 T0
¢
)
Qo
o
0r 0
t
Cso
0
o
00
Q
o
0 0 0 Ut0 n0 a0 n0
o
Q
o
U
o
o
o
o
Q
Q.
u
1'.
U-
U.
,LU
0
-4
+
-4
+t..J
1
QU
a-
- Li
.1
-
k-
0
-5
..J
cr
7
4.
-
ULJ
I II-
--
-1 L
U,
U
·
-
_
r ,
U
0
_
LU
W
t.7J
-1
J
':
4.
CX
_t
£- I(.N
4
'-
1.*+
o .-U
_I
O.a
--
.
L 4
. +
- 4
~
4
L.
q_;
.L
L
C
JI
-CZ
-
L JJ
_
~UL
C
I
U-
N -.U
"LU
_9".
CL
'
-
..
(
_
._J
CL
C'
-3
(N,
--
,. UL
-I
M
-
Of
LL
LL
7
C
02
N ,, ,L
cL L z
-
U' tf
_ ~~~U
C
U -
6Z:L
_
_I
U-
-
LS' U-)
_
C
_II
I.
CL -
U'
rlQ
Z
_ Z uX
U"
L(,
Z .
i
-I
_ ~.,
_ It
_ ,,. .
-,
4 C' LIn
I
-
-
- Z-
LU Ucf -
(-
_
_
a0
r' L
,--j0-L
w
I
C_
* **
I.
-4
_
5
=:CLU-aL
UL
L.
i' 2
a
_.
'(
w
"
_T
U
CD
XL
z
L<'.,_
_ -Ii -U--U
. .,
-'X
-
177
uO C r-
o
M c
'0
CIn
-
Q rCO m
-a
C.; N M C
U) 00 - -4 'O - - - -4 N
o L L fn .'NTNtfo
Lo oLo LC)ir Uo r
rO
N NN
-
Ur
%)a:
CO (M
a. - N
0
o U
NN N N N M M
U
L oA
A u-~
V
o LA t4
)
r cr
O Oo
(.
M
i
M
Lno n o u
-
0
C
Lf
1U.
LL)
UuAJ
C
I
4-
-4
cL
r
I
-4
-..
-J
x
CD
U
cj
I
I LL
I
YY
N + + .)
aa
-4
i
W
4.
ZUJ
LLu
U.
aC CL-
eC
__LL
I U.
-4++
-4 -4
_
U.
N , . .)
,-- ,- -iI
-. 4
Z " -- r '-Z
LU
M[ a. (~ ~'. '"
_
Uj
. y(L
Q.L
C.
.-o..
_ ~.
4 Ui
--
Z
"N
,C
W-
ZI.-,AU
Neq
-U7 U?.)
I .. I4,
·oJ
.,-,-,- L)
N~-_-* {;;-
u _0
--U
c,'%t
,-.,
_'
N$ { - _ N
,- r~
L
r III -4-
(W
UCD
3 C.C--* CD
* )
_9.-_ _ INI
L C.
CL
LC il
n
C
C
,Z u'
-
U,'
L-J
'-
U.,
_U
O)
.J
I U
U .
Cu .-,.-' .
IL
L
0CX
-
-4
Z -. -,
L)
_ LL
I--
I;Lc
U
C3
U
U-
0
7.,,
-
.. i
U
iC
c--
.4;
,-J
-J
C)
_4
U-
J.
.s
UC
I-
O
I I
O -
I
-S
=I
.:
-
t
tt
_-
LL
,Ot.~u,1)
'- =4--U
Z
4-) L) C: QC
°
(D Ca u
'.OC--
Z
00 W
FZ0
0CCQuL)LJos-w
.
U U
,0
U
,oL
L U,
U
U
%,
;0
.- -
I !UCUW~~C
I=-, LD U4;,.
ll
U
LA. -
_
4.D
a- C.)UW
-J.D ZU
UJ
11 A- I"J
.O ¶ON-1- NX
Z
-4
0 U 0
ral tV cs
178
_ N f<
'4 '
U., b' a'
<r4 7' T
'V
0
<. Lu_U.)V
tC ) )
U oU
(l Ir U'0 Q aoM
0 o- N
'
'
-
'
) %nU
Ln J u
U
UL0n n -;t-;)
tru
U) o Oo
o
O
'0 0 '
,
0A
LA
'P
0
0
l
o0 o
o o
o-:)<
)
) "4 N M T tn,
'V
f
Jr
4 N M
r' X c7,
r-I- r-~ I- T' U O
%
p0 u' U'P u'P
0 p ''
'0rc
C U o o C) o o U U o U
Lnc 'P
QU
-4
2)
U)
sx
Uk--
U
4-
N
-4
N-
cN
z
"4
-4
z0..
0.
-+
-
.
c1
0
0
,.J
O
L)
._4
-b
*
'N
. ZL.
9. '-"
N
_
A
i-
0
O.O.(L,.L
'%_'
rv
m.XC
L.-.
.
+ -+4U).. / I
.4..
U)U
GU C
.
CL
:EL.;
:r
I,-4 ".. U)
,-
a
*
_1 V*4-4* 'I
J/ h-
_j
UJI· ~., rvL
"
p
JO
*4
4w
,~
,,-
_j M,
* *
_
I -I!
.I C13
-4*It f"
I
*
I I
-
U
t 11 itU-
~r
CC CCO Z Z
CD
-
U. 0.
HU
A"0
tv
. * 4.t-
. * *
--
"
I
.c
,it.
._. Il.
_. a:,Oni- .11
--tILL- .r
,_3
I Li
W
JO
(;.U)UJ
o fiU)0--U):
2-
*
O cM
U
U
>s
U)U)W
)0U U
3 II NJ , 6U)
I I
u-'
II O
_
-)- -)-- -
-I-
H U-
U
L,OO
U
'.0
I-4.
-Q
'.0
V.4
.
UO
ul
__U
."-"4
y
*
CL Z
z
if
0. 0.
W
yZ 11
. _
_ It
*+ *
OL
Z* * 0ZU' JL
* LA
11
U
IU
".
E:
Z
Li
LLD L) 0
U
L)
0U,_ Q _ C O O Q O U'P
,iI
L
VJ
t I4
M
U.
x C
,9u% 1IIU
LA. k.
--ILZ
U.,)
~.
t~
C
_,, , _-.t. IJ 21*._
u_- _. ..
~ -~. ~.
(_k-
U'
U)
- O - .
Ou),_-L
'u ILZZ
N
ou)1
U'
U U-
-
M
0
I-00d
_NU.
S
tr
WZ
-.I.l.
N
I
-
..J
UJ
II
I
.*
,U-
*
"4
£w_4
cw U)
0
LU
c"
z
. 0
A"X
-4
0-4
LU
CD
[
-
' MU U,Q c~c
LU
+4-44 +.k.
+4 +- V
)
,I,.+.
LL
,,'-4"-4
'
O-1
W
aU
.. N
J ZJ,",4
E:Z. L. ;
"t
U~
N"
-
O.U.
4r ~iL
O-
U
C.
" ,."-
"4
-4~~~~~~-
Li
A
_4
O
N
CU
UL.J[D
-U1
L)
L)
c
CC
U0
-J
C)
~.k
O
,'P'P
-4
'0
)
-4
.)
-O
U
_ _J
LU L) -
A
' -'.
1 79
.
c) m
O
X
Cy, 7
I r rf-a. ONO
00X 0 m =U)
N aO 0O 0 Q
0C0
C oW
r_m 0 cOO0 0U
N f a 0, C
cr cD _
Qr o
r U Tn u Lr >T u -UN
0o
0 a CO aO0
n
n
0
'A
1n n LnjoL in Li, Vq
oo
l OUUQO O UDOOUUQO nO U-Qooc.ooQ
0
uO
'O
'o
'o
)
'O
A)
O
O°
'O
UOWU-o
o Q uouucQ oo'0 Q0
w uo'
-4
0r
-J
U.
*
-4
x
CD
tU
-.
I-
Z.
~x
-
L.
U.
zN
0
CL
L4
C+
Z
-
U
-4
w
I.U
LI
J
NJ
cc
+
CA.
.
rs
~~~-4
-4
.
C)
_
-4
U
LL
U
-J
U
CD
31t
**
C
-
-4
--t u
U-u
U-
-,
Q-y4y4UJ
*-r
- -
+
_
* -4
Q
o
i
C
J~
-
-, ^~4
t
I
- N
X e,
-..
4.-'
N
11
4
CI
_
z
-
_i _ Z
xo
Z_U.
CCOI L)Q s;L
u
CD
A
-
O
O
C-
U¢
cii
O
Z~rfY.
-0.
-4 0.
1+ + U.
4
_
II ) _
CoJw 1
C
_I C
U W
.U._
Z 0 Y
O40L
t
U
a
CV) L
O
*Y.
* jr
i
-
jLI- .
-4_
:
*4
.,I
L- L
Ur L
*'-4
._4 0
t,.
rW
ULJ
1
Iu 1I-
**
_
c
1
_
S.f
f-
¢ ¢ JU
)
c
.
Z
j .I---1
i1~
L I.
I-.
.J
(- Q)
-LL
0 U u.JZ
-4
S
-
Ci
-4,0
U"
Qe
Z' O
le
I
C Q
4
S
0t Wr_a
0,%2
I
ce
UX
"O
+ +
U.
. CL~~r.L
4
I
aaU~~U
_
4-4.- eu
Jz l
*.C
_4 C4
----
C..O
++
*'+ Q.(LD
C
I
Uw
_-*
LA ItL I C_~
0L.~
-4
gt fI11LL -w,ZO
CU
LU
-.. 4
-4
D-
-4
Y 0w a
LU
X O0
_4C
-.4
C.
4 L
_4
J-Q
U
L)
U-
0. -~LU -C
J
._-
CJ
_
C00
O .
CZ
N C
N
- O u
'O 10
-44
,7I
O
Q
L:C U U
i)cl'
_
1
*
*_
IU.-O
c
4
Q
I
UO
;cC,.Ctt
LL
P"
180
.
O
. O O (; O
NN
N I
-4 - ---4 4) 'Q O) O C O O
r
r (O
o C)
_ N MI u
N N (N N N M) 1 n fn ff) M1
~00w
'O O ZJi
U 0Ua
i0O
00U
CQ CU O O
c
O C)OQ
O
U -4 N
0
O r- W
.
4 4 4 4 4 4 4 4r
M
' o CU WO O Q O O OQQ
OUC-;0
C Q OUO
C)
-
-,4
A
-'
.-X
C
-4
a-q
mi.
U-4
"C
ztIN
1-4
Ac
-
-4
-4
U.
-
.J
-.
-
-
z
i-
a4J
-O
u
c4
4-
iOD
U
IO
CD
Z
C:
NON
Q
G
-s 0
0:K
_
_0
'j3
a-
I
a-N
U-
it
If
I
-4_ -4
AU.
L a
"'U4 CLf
L UU
0A
I _4_1
U4
.L 4
_
_
I4
-4_
OI ULI. ' Z- _
U-C
I_
_CLJ0
X
*
L..
0_CL O
-
14
LL
t
r-
O
W
U
LAJ
CODU
0- L)
',IT4 .
N
U
L(14
MLl
L
rv If i
U. *
uu
A
ifU-
Z 4-
LUL
"-4
; ----
U,
44N 4JjU .N 4Q
J U4 I0t.-'-LU
I:C~
*+ *N_ La
4_
H..,*_ *
A
CILLL U.
-B
',1..1e
O
U.
O
-
U
..
-
-
t-
U
W
a
-
QQLLJ
> ~ ~z
Z ·- CD _
-4-4
O
0 ,-~~
ICO IIN
_- ,_,.
.
ICG i
-I
II
U. UJ
-42.
-4-
-
4, --,- ~)
V) -
OU
-A. Q-3-W
z
i
All ~
**£l
cCII
UN w:a:cu
-~~~~~i
CL W
4
-)
>
-
0
XU
U.
fX
_ _ J
r1-
-4
-
OWr
-4
I,I
_-
N
-4*_
Uj.
.- u
n
, 44-.-
: 00¢XCII
00
4s
* -
LLC
WCC.
_-l_ 101
ZI E: N:
->* 0-,-_-4a
I4U4.0IC
U.
-4
II L
LA. 444
Qt -4Li
.--
-0-XU- ,
.- M% '"-
U.
I
M Q.
CL a -
CoC
- -LU6U
~ ~ ~ ~ ~
*
II
U
+LA.
Uj
lc.
CL
N
-
2.¢
- -
U
OL)LU*
..U- A
LJ_CLAJe
ON
NN
r* I-!
O
'.0
,O
p.__
-r
181
-4 n
N
Cy% c
UL
t Jr V i VN m
.o
U
C)-- M f-
aWO
O r
0
r
W
"
V M -Un
-
r
CJ O o o Q o
, Lfl U. v s ' O '
'Ai O
O o CQ 0' '0'O o C;
'
Z '
'
'
?-
r
1Q r-
WxO"
r'- CJ
4- Is-
-
04 M
)X a) Z
'o
'0 ) OQ 'a'
O C.)o'o Q
-J
-
0.
U.
v
N
"-I
C.
1-
z
N
-
-:
1-
- 4.4
,4
I
-14
m
.3C
0Y
+
-J
-J
/,m
9.
-
+,
-,
,I c- a
U..
-~hLhj
r-
C
L
-t4U
a---U.zi-,1
OC
IINNA~~~~II
44
L
c~~~ _ - _. -*.
-
0
A
++ C
.J
LI
I
aJ - Iit
-J -x LJ
e-
r-
cN - . .- J ..
CS)
L
D-
.. ,+ +I'"
IU.-.
,]NI,,
_+
'L)
Q
w
-L -
z. r
0
9'
* 44 o U..
CD lJ
C O
. C0a.0
C0)a.Q0tJ
_CS
-
,4r
I II IL.Z
_~
J1I
,C
ti
F-.- ... j - uO0QS¢ ¢
Q: UJ
.C.
1 C)..
0LLO
W U Yt
nc.)-,a. -
004 n 0
-
-J
JC
-J
-.
1:
a
I I.L
n0~JCL -ILJU)
t.
l L.- C
U-
_*..)1.1.
_ - w:.
._
.
9
1,4 ,-~II
,"~0CL
;]E:.4--~
_::J
V
9'-
z
0 t- -
-V
fO
r- f- I-
I4 C
z~0
-.--
Y:
-
O
1
O LU..
.4
..
U -.
--4
.
14L)
P_ NA
_ L
rsH
+V
-Q.0,.) ,.,. CL
-.
r.
CL
0. "CL. ..
C./) C~
-4.E.-..--4
00
-
-,T9'
'49,'
f
0
J
-
-":..
, L
,L
C)
A
5
_
L-
W
I
a.
O
-J
cc
_
CD
6o
-)
LJ
4J
U
Li
U
Li
CJ
.L -
-.3
Q'
~4
w
-a UJ
N
-Ir - z cr
II
U1.-
C) O
wU C
II
?-~C) ,,
-_
,.
,- ?~. L. C3
182
,: Cj
XO'n u w
O cCUa
un
, a
U
NM
C)-A
V c M
,
u ' O rO
ON
rU
U M aM U U
U o U Q o o Q cj
M
U
o- ~ NO M
-4
- _.
rW
--
-
L"
c o
N
'
r r
r
r
r
F
I r
r r
@ 0 Z Q z Z Q 0 O Z U o 'O r r
0
o
-4
0
V-
M
_J
.
U-
-
z
-4
-4
CN
UJ
CL
U
Li
CL
UJ
-4
LaL
-U
[%.
-J
Nz I-I
*
_
~-4~
~-4
CL
N
4
.-
-J
J
LiW
C)
~L
eY
+ .lC -6
U Y_
.e ,
4
-,
::
I
L.)"NIN-J~
+
J
U
4
-JI
3-e,. , (3.'--
-.-
-
II tl.
. ,-4 CD
U:
44
r-& -J-4U1
~j_ LLz
9.
U-rn-n
°.
aj *
eZ
a
F -4 S
11
It
2-'
W0
; - _9 U. z-
CD ui <J Q
.CKOU*
U
_
CC
L) a UJ d
0
~
W.
0
a,
14
CO*
Cl
N
ZxaD
z
0
U- sNU. 1
_U U ~ -_-4IJ -
-
. U.
QC
"
O0
NLNC
"O
3U
Li
L -4
C',IIU~.WC
U e
t'n
II
_
Ni
z
Goa
U .
Z
II
*
2- .
IN
J
C,
I0
II
-"Li
LU
4 ..J
JU
NU
C
_
U
_
o t
0 I
lII..~
~Juu_
. 4, 4
* -
N
3:
.~
t\ 1L~
C0II
C
eZ
CW~%J(~..
3E
U?_
9.I1L
I
CD- +,
U.
N
*;
4 i
C
c
IILU;-
o.
*
-~
#' uZ
_I
_
II
9 .9_
O 11
C
IILL-, .II
Z
C[DZ
X
_D
11z_. Z
..,
C-) U.. 0
"DU.I
O I
CLC00U
- LZ
NJ.- * C
i
C14
0
4
1NN
o 1J N
-4C2N
. L)
N+
CCU
OQO U
J
-. ..-
2,-
Z U.
1*N
NN2
ars
NI NI .'Li LUN
*
C
_
_4
I
-
-
9.*
.L
++
C
(NJ
Ij
I I_
-4
~
U
tl o~
U-Q
3
OC
tJ
C
~
+U
-A
U
.4 _J
CN
L)
_
CQ
_.
UJ
U
3
a
C
-
U.)
-
EDW
~:
cc)U
Cc C
0L
C:D
-~
CL U.
*
*
II NNA -s
-I
A
*
-J
C.
w
U-
..
CLt.
'4.
-
-
Cu
N
U
N
-
UF O0 Q
Q
_A '-"
~_-4
co
183
-4 N 0 0r
N N N
o r - I4 r
r
o- J
0N
0e
ON0N N
C4
N ff) rn
m
ro
r
r
r
r0
r0
r
f
O' 0 U
-N cv f0f
U Q
M m M t -.I 4
4 - t. r
r
r
i
4 N m 30 0
. L. L U LnLn n Up
) C
Qr
r
r
r
r
Li
C.,
(4
al:
Li
2.
U
Ux
N
(4
E
CY
U
C3
,,i
-4
N
N
*
+
-,
_J
-~~~-
-. I
W
C
O
U-
0.
0.CL
_
Z
-_J
Z
~LL
N~~
LL
LL
U.-,
.4.
_.
-4
3
U
3
U
J
LAJ
PU
U
-4
C
4
L_
3L 2L
++
U.
-x
' Z U-
O
-4zF+
-4
Z
LLJ~~~-
U" C
*.. C4..)
aJ
-.
-.
) N *N
Z EJ *ui * CD
_I
* *
Itvr'mJ
i"'I N .,a:I
J
9.
N
_
II
-_
_
L-;
J
*
9
LI)
-
*
_
) o0U" * s* *i
Nt
3]} II9. *N
N C3
M} -
0
1-jI
I C4N
4
-
-
:: tZ
*
N I I O NI
Z
0
'
9.
J
V L
UZ Z
C) _-.
D
-
£
S
*
N1 3 H No
I CD
I
75
Z
IIt:U
Q.LL
NON
N -4I I ON_ -'I -4I r ID
CLJ
t Z ft)
W
Z1 N
N
N
cc
-I
N
..J
U
C
X
2
L
44.11 :
O4 o
1
3IU(
Z
(4Lg
ZLL
00C)LN:LWL.01
U-'
U
Z
L
u-
N0.
U-' 30
W
4u
a4- CL
-CL UL
I--
X N N a:
4 -
CL Q UJ
W.--
t"u' Z'UJ
-
J
_aZ O
-
(0 O
S
U
;fL.
,110 ,...
-4
NJ7N
2::
4.
+ 3L
+ U
U
3
Z
NNL_
.
N 3:
N
(J
L
uNN_
0 WW.C ZU.
aj~~~~a
U)
Oc c~jO3
2 +LE;
C
U
3;
I,- U
_
C
s U.J
U.
CD
(40W
QY
(4C4:
U4NN
04 N W
-z
W
CX
o W-)
ff
a"
U
_.
.
_
-
U
U!1 f .1
. tn
0
j
..
U CD L.~
L L) O C
C]
-
_-
I
-
It U.
LJW
U
Q
0
-
O"""-
It
,-
-,
!1 LL
II
. LL
W
CII WO_
O
-4
CC)
LO *
U
CV
ag
184
; O
C
0-- N
V
UQ,U ,U
U
p
D (
U
Ln
O
C
' C -' N
4) '0 '0 b
r
r
r DOM
U
'
r--
O
NrO
O
O
'
(Lw w M
w W Wr CO
o
N
U
W a Ur 0
v'-~~~~~U
CY
u.
U
L.
N
VI
N
U
I.-L
Z
I--
Z
U-
U,.
.J
.J
-
4.
7
.
.
-I.
LU
-
L
.U
Li
.L) LAJ
-4
.J
IJ
A
a-
C
CJ
-4
U-
C
A.
+UL.)
+
3
_J
-,'-"
CdU)
C.;
0Z
II
- UNJ
* * --9. Ia)
)QLCL)
;L
QL
+
'IC
O
~0L
C2
rI,
'_
cc
I) C)
c
U'
'VW
U)
N
.) -P- U
¢
+
U.
O 40 1
_N .
CN
v,, rq U r~
< ,_. IN
U.; iZ
D
Z
M
,.Um
II
UNN-J C
*
* * -
U 0LU.
- 0-4
4.
NN",
CL JIL Z
ttU *N
.4s
I",,
I t ~O 11('4.CJ
O
E E
CL -4
)
I :LI _-N*- 3
I
N IC
-
IANN
NcNc
I
H 11NN
I C
NN
'-'--" _*
Z O t,o4X
I LJ
Nt
Z
O
N
*
II;S
N
-U
~L ,L -,-,_ Lo
'"Y. )
=-*I cCui
*L.)CD
U
~0 u'%-''-'LNI
II
N)OC
1,. ~a, 1It.F 11 11LL
: cs
co -r 31>: L .CO_J
(40W
Js.- V IC ,~ ( U ~~
u
sL
UJ
O..)
U U C~
iA.
"-
U
- _
Cd
9;
t'-4--L
+
N4+
II
-4U
U-
N -
U'
: -
A
LL
No4.,!',.;
C.)
Cd
VI.4
It
C_
+ U.
+NW
4 -
-
_
N
A -
. +
4+
, InOo
CD
I
L4)
O,--
N
Z
+L
+
r' J UxX >:
OQU
A..J
Z IN-J
z LLI
*
J 4C If
N
1t
- UI
...) Q
.
CL
L;J
,,-"
9
_U_.
,11
1--L)U0 (
W (9
I
_
+
m La,.-_
COU
CU
N O_
· "IfU- ID
.L.
*..
4.+
I-4e.
0
C;
U-)
--
_CC
4.- ,. L~i
.U
LL-J
4.
_.i
-
3
4U
Q-U.
U-
N
CL
i
c.+
,+(.. o
c;)
o..u
C d0uJ
r. 0 C:) J -- e 0 Li o
_ -
-) 4
C1o
0
WU
-
-4
I,C
.U-v
"¢O
o
II
II
LLN
_i) 0L IZ
.-,U)
UJ. Z.;~,
CU O
o,,oto,,
185
," -
u
O~a'
O - w a
a, CP
o
0 0 f- r-
O -_ rj t
, V 0 0 0 0 0
;C
u
.o 0 0
OQ
I0
0 rx a 0 U 0a0
a
o
m 4
o),.4
--4 ,.4
Un o F
Q
ooQo
. 0.
0U 0 x
u
00 a
0
w c o
N
n
O
f.. co
. .4.,
-4
4 N N N N N NN N '
oQQoC
0aw m w x.
pVoo0
. w 0
N
N
C)}
C
U-
0
LU
U.
LU
UU-
U.
U
U
LUJ
LL
Q
N
i.
LL
LL
~-
z
CN
4.-
CL
-J
LU
_
U
0.
-
-.
U
LU
IJ
-J
LJ
0
0
INLL
I
Z
C
O
(3
U
Li
-3
-e
4-
U
-
4.
+ 4,.
+ N
W
+ .
Q.CL
LU
N rN UJ .
rv+ +
to 4
O
C 11
N
.4L)
a
04
Pl
U
· .4.N
.J.J0!-e0
~E
ZZ
.S~e
4.4.I
CL
m LL
0U
CL)
N
.+LU.
+) NU
-)L
e 1
L)UO_
NN N
11
P O
'
Ul~~~~7
N.Z
Y N N Lr UJ U Z
- __
0,-_
_Z
Z
O~
UI
tL
AU.
- -I-
1
s Li L. z
OQ
o.-
NnNI
z
QU.I-
U 0
L)%.(-QI
NNa'O
cr
NO
C.
1_
I
I
0,
W
U-
cN
4.
21
4_
ULj
~~~
L~U.
cr
N'-_-O U
N. U.
.
-U
,0 L0
W
I--
Z
Uj
4.4.I.L
n
uj
a;--.p 1
-
<
.
U'
Q
II
Ln 0
!II
Q0 Ul
CQCO
0
(U)
C L.
U
C
f-
-
O
II '.-4N3121
NIII I !U.
..4 (i
I0PN )
. -
U0 U
I
,M 7
I
a N
Wa- INm
+
(+...J+ U
IfU(,
I I*"
L.
."_1'
LJ
GL
N
O L)UJ
n
L.
.
C_
u:
J
U-Qt
UN
CX
a
-le _
(:
C
CU
tC . . I:.N
N
_
cc
j
z
14 U. w Z.;-i
C_- J1
U
Y LLI
O Z -q
tz _-QC
t_
U)
U
O
QQ
* N
cCD
_
-.-
LP
N1
LU)
Z
* -
0..
*-, -
COW
UNL) C;UC
+
CL 4A
) V
N )-. -,=*
I
J
IL
~UU
U
Li
(N
,s0
ofX 11 LL LJI
-4
ILW
- C-
U) UJ
pp pp U-
..
U.J
-.l J
-
% N U-
UtUJ
z-42
x
-4.14
. cLCL.
Z
-4NU
-
U.E
LU.
-4
-J
-
o
·-_
UJ
-4
.. J
aC~
0J
J
-
U-
a
s-aU- U U
Ie
_
_- -
186
4 u o r U0 0
u-0' 0t~ Vu0Uu
O 7 u o~It O1S¢¢
UN
'0
-ormn u n 0 Q
1 -4'f; Mt
C x> U- tn-os x7 U
uT
~nW
~
-7 r J)
4 -Z 1 t 17I
n I
0 aa;> W o m
0
co
co
X
oo
mr 0
X
au ON-4
O ~M"}I
Np Frt
C'
w
co)
cco
W x' m
0 O
O O C
Q 0 Li .C CD
U
co
:
W O U U CU O
U L)
;= a
= W CO
C
U
C C O
O C)U 0
O
U U
O
N
0
40
0 'Q.
0(
0 o G0
ava
O O U
C.
A-
J
UL
-
CL
L.
a-
N
A
,N
LL
-
CL
N
0
_
-
.
J
4..L
0.
(~
C
I IL
. *,
,Ij
(7
I
N
CLL4u
,
CUNC'S U-
+ +
N4
4J
N
*
N
'^ U * * -
1.2
II
#1
N
6
q .'4QJ J
N440)
44*s
v__-
11II
_j
,
D
U
jC U LLLJ
- - U
z_
.0
r.-
II
.U 11.
urN
LZ r'
-
Y
it
_
94
**.5
_I
CU
-u
_.
N S
_
0
CD
E
p.
UC
*4
~r CDI
z C 9 CUO
r-.
1'
'C
~
r J I.J
IC)l~
.NOc
9-A-
C
.
N Il
It9 U.
1uj..
iUJ
-
Oto
NC
N
U')
Z C.
*. N
* * _
Lj0
1
cyO-J - LU
CCU
U CO C:)
Q
U
O
E Z Z
'0
C)
4 uLJ)
00w'
·-
'NNN; ' J
ICC tsN
O --
ZZ -
U~
j N
Y- UWj
Jct
+ +
N:E
J -..
>rN4
OO+tl9Li
IZL
Cf.
L.O)-c-- .
0'0a
I!
Z
+ + LL
:ElL
L
L
Y ew
c- S 11
U.
Li
C)
N
I
zUJ
a-
. a,
+ + U.
-J
N
tYL U
fi + V
IC
N fl)
-
2-
qN
ZL__
i
JL
C
Z Z U
U.
L.)
un
to
J
CL
U
r 9 CD
Z, r
U
CY
++U-
(~ t
o.
0.-
.S.
V
.
_J
A
N
U
+
.j
LLJ
C)
_j
CD
(NJ
0.
_.
0
co
'_ .
-J
-'A
-
*.
U
LL
.
0.
Z
zO.
YL
C) (7
Li
tj 0 0
187
. . o 0 -N
uO
"0
'0
'O
'
o
(.
srac
u.
r
p.oo
r-
0 -.N (;
m ,.
9%. or- o-
o
o-- Q
c0
r.
Wa:
o
W
o
o" 0
NCm . u a 9
-
0 0"
N
0o Cl 0
0a
Q o'.
o
-
aL
x
cr
C
U.
U-
z:4
C
UcN
N
UUL
U.)
G.
e.
z-Nu
CL
-
~r
41
9-
C.
+
LLJ
,-
v9
-J
0
+
J
-4
+.J
9.J
U.
J
U.
+
c~
_j
_L
Z: ;UE
I
t
-'.
L(%.J
ZLL
~
9-9~~C)f
uJ
L
- -
J
- .4
vE
"'"'
Z 3EZ?~~~
.N
.]CJ Z A
-
N u C:UUJC.
J
o)
390
If
Q O
)
0 U-...
r 009o
Ur~C;:C
f..~ .0
0,,
9.-
ZO
H)
i!
* a.
_ _p
N 0) 1\1x N
Z*
*::
. L
OI
0,,A
F~--
L
JL..U
z
4..
3-
-
0
-~ __
.,
.j--J3
OIw
0L O.J U
N
0C)
0
_-
un 'O
0
00
U C)
4
.
I
P-
0C)
0
-_
CL) )
0
O
- -
a:
z
NO 1.-,
L) 0
,-.d
,..O
tl
* +CO
%Q
4o zu U
"C
N
a.LJc
it.
el0 IA
r.,;
IJ
Z ZCW
IN
x
+ +4
,.,
.-1- .. IHI
L-
ODQ o~ O C-V-U
I0;--t_-
U
1-
I .:: UUWNNO
N *r *NA-a.
_
U
: a:W
ZZo:
+ +"
N
InQ
)
-II U9
N.J
I U. z
Ub N
IC~Ow
C)1LJ
I-tJ
.1J
M J,I
o
N
U
um *:C 0.1U
aj'" 0oUbJ
t.J
.J LI2 _ -7
=)
um
VI)ot U. IJ W;;}z.(,
0Q
0
xl]cIA.
U
I!
-'I
N-
I-LA.
*V
NO N
..IZILU N
N
-
O.
7
_
I C)
%--)I
0
-rlC0)
LA
-
OIl
. _ o.40. U.
Li..
* e3
-
1U
C.1 I
0Ul
r-
O CxJJ.
** *
J
-i
_4,
Li
-J-C
D
4II II
- -
_CN
_ -I
* a. 2. J
01
,-U.
0
-
_
-J
,J
C)
0
. CLU. -D
.r.;JO It (L
%r
IN ~20
,+,,0tA
+. N +, + UW
C . o.4- I C Lt,0
r~)
O~CL W n + u ,u.
_lr~
_r-*N
*
W
LL
%O % U.
~C)
a-
"Q10
4. +
.-. ,4""1--
M
+
-3
0 uJ
o;~ co
ZF.CD
+ +
-'-_-_
I
"U
a;' Oc:
O --O
. .- 0 t -..
(3.
U
CU
z
U-
L.
04
0.
-N+
0.
4I
U
*
-
-J
<
-J
0
U
-J
-
-
N
U-
w
_
-)
z
zU-
z
0.
c4
Z
,
g.
N
p4
_) W
U.'
N
C)
4
188
-. N ~
o 0 00
rta '0
Uo o
0O
=
cr
00 o oU s U
0O,0Or
(
lC
VI
- - - -4
U
'
cr
CP
a,
0%
.-
- u
a
O7
-4
U
O
s
'0 r(- CO0' 0 -4 N
N m*
o)
0
(
N fn nU M
N ON N
N N
N (N N
_U N
' O
U
O
a
C
a,
C O {'p
a Ur
OQOO'OOQQO
St
Ocr )OOQC
cp
U
n
..
Ul
M,
'
%
a,
C
Ui
''~'aa'''aa'a
a'a'a'''a~a0'aa''ao.aa''
Q
~L
0
-J
U-
X
U-
t~
(N1
U.
-
U
L..
LL
U.
-4
4
N
Z
LL
C
~-
.JL
z
c.
Z
0
-
f--k.,I
-J
-
-J
.N
4.
-j4
Uit. i 4.-..
_J
-3
CD
_J
J
:_)
I_)
-J+ZX
Ow
0S
C)
*:. 0-i
N.U
LL
9
,U-
·-
.)
L..
-
4 .-
f- + + LL
UL
0"N
,--Y YI
CL
_,
u'U
. J0_
L~rr,,.Jm'
I
-
+
. -· ,-,
N _I*- -
Z C
'
t
_
J
II -.
"
- ·
·
Z'.D I
-U
CL
C Lr)
%
.
U.
Z LL/'e +,
0L+
K .. )
(NgU.0(l
+N +
004
Ue
-l-C) Ln - -^
U
l-
0U OU
O > C W
,fi tO Ci
ICUJ
{CD
4N
Ul "N
It
11__C
C
CD
ZO0
F 0 (-4 Ii
'-* I U.
OrN
L
C
ZO~u· -4-
-.zv*- -L)11
-I
II II
Ot
II
-I^
t- ..--
.J ...
r,,
J * z
I
N Nr-4
00
O. -U )D v
L.
z~~*~~~a
'V
'_
U
iSII U-
'.3
L)
.
UcAit'
L)
_4
0
_-
_
m;
:L
N-4
ol
U -t
IO
N
N
Ol
44
Ni
(. )
Jo
I0C
ltu.X
0
0U
L O
Vu L)C-
u~
O O
-S N
-Z U_m
-.)U)-Fo
cc Z s0
C,4 - I0
CC O JZ
N
-r4-
II
)L)
C.
O O
C
O 011 11 z - C0 UL
-4
"' Fu
N.J
NZ +C ++0* CU
NN
N
NeX CL;
- LJ:D
_, -Q-. J N N+rqN
_, +
k
i
0 -,, C
-JiU-
CNNA
NLP
CJE)N
U')
L.L
Q m.~
_L 0:)
IW-4 (dl V
tN
11
. --CO(N .tN 3
C
I
N
O
0
-I ,"_
:
{I 11
0QY0
0_
Ure
Cr
189
o
o o Q
o
Q o o
o o
C
o
o
oo 0
0
C
O
,
C 7
ON
0
C
U.
N
x
-.
m,
oA.
CL
z
U
¢
~~u
U
C
sr
U
U'N
0
(.
U-'
0.
AN
-4
N
C-
z
N
CL
4-
+
u
~
-J
C.
N
l-4
0
U-
u.
U
U
A,
,- L
3
-4
~
+
4J
L.;
_i
e4
£E UL C;
+
r- 4.
.
U4..
^
-+
t
I.)"
aLL
uD
yn
Un ..)-- a;I
r
L ui
cc
N + N Cr L
Z -rl-r- - Z
U
-j1IIL-,L
C'J
U
U
-.4
y0Q I
N0C\N-.
C
)
-
I0
Q
W
o N
4
11
N
-4
^0
CU
U'.QN0.CY I-.J
NNuw z
UQ A. -_. C tD
LA
uC
C - - - Z
Lr
X0
N
o
C
"U.E
Loa C)..I
Z CCY
li I.Li
u
z
zI. I<
LA
YUIJC
-
t
"~~w~~-,
-.
O,.U j. L LA.,
L
OUL
~
C
b-
L
-4
o c o
_4
C;4
-
-o
a: a.
A-
C Z ~ (.)
n -,,
W
Li. NF- n
Qu
o
CW
Q)
-4
)lI U9I
cr -
.LA 0
II
_
_
U
04 -
X
II II .N4
J J J U-
(D
.4
o
C7 C
UC)
-4
-4
N
o~1
D
0-4oo
ZJ M
UJ U- .
-U
.
...¢,1.
X
X '04
I.
· 4.
IC:
0; Li
o
I U.
IC!
)
Oi
IOY
U IC)
c O..Lj t
00wt a
N~,..,~.
_- .IJ 5 lLA.
0 .DLU
- O- 0 A-)
O
0 - C2) O.O J ,
0r.(.7,0a,
4_
_LL
U.
C
a. a.CD
O6
D U-4--
"~) N I
I * *
ce U. U C
- - 0;
o0
-
N
z C)
4.
cc0
P'-f' U,
W
X 2U
'-"~ x .U'ZOO
N
4. U-0L
Y4.
il
Lt~
W
-4
~~
U.
~E "_ U+
tZ c! o+
.
.
.. J
CD
A-.)
U
+ U.
9.
Z
U-
L.
A.
4.
CL
L
..;,~
41L
Z
-
(-)L)*-0
! O,.C
U\1
OC...
CY,
0
-. 4
N.1
Z
t 0tJL ;liLL Z_
4
L
190
r
?Qw O0
n "r Ut
f-,
cD71
0 C,-
c,
r,
r,.'
N ,
N vr" u- z
m
Z I- w m 0
T uLL
: , ,"
'w
a a,:
0' 0m r0 a, 0o0% c0'
C,. C.UC ov
oC
0
01
cw
) ca, "
n ' r- U,
0O0 0o o0-UJ0 0 0 U
Q
m o
CU
-
o - U-4o00
M0
I C? cv LT,
r U.
u" 0 r0 U'
-0 f.
Uv
(
0
U'
C-)0, 0 Ln
U
U
0
0
Ne,
LL
-,
af~i
4.
T
-4
-
-
z-z
A-
N
-
U4.
uLL
.,.J
+.J
C
-
_.. 3:
_- 4.
+
4
z 5.
_-,,- C
Z + ,_
N
0
,cLO
NX.
UJ
II
I
II
-CJ:)
I I.,C,
U
-
_,10ILLJ I,--a
LL
.
00I
--4"
O
N-,,--,
00W
U
-4
-4
-4
0
-4
-.4
-4
I
I
I
i:L
z
0
N_
.2
Z .-
-4
CiI
,-.J
t4 1 J-
'
-. 4 -4
-4
-4
"
-. 4
-
0N';L .4 _
C:Z
001
O -
I
N_
':
O
O I
rg rg C:E, ,,
r'4
"
. ,4 . . _
w
_ .1'-L2
I
0
Z
NUJ
N N- +La.
+ .
Y
I,
..-4L.'0
.N
O
C;
_-
x 0L C
Q1.iG
(J
U,)
11)
0
N..
$0
N4KU
'-O
CN
Nr~j
ZIt ZII
Al..l'
* ,,
N
Ii
I.. Z
--. .I4
al
*NN3
*N i. *Nut CC
A5 ~
,
0
II
NN ;U, LIC. u * *
_ ). .r..
_ _ r~~~~~~r.2
2-.-4
N^
uz)
z nN :C,,i * m - Z a.
' z Ln
$
_, - LJ
n
_-_
Ln
z
IZ
CCO
CD
-3L .)
J OLL
6:
-4
-4
-4
-4
-
_
CN
, _.-_, _. U..0
G
q
-4
*. -U,.
I
0-)u
N
Q
-4
+ +, 0
N
I ^
NN
_
'4
O
VC
-
-')
-4
*
L)
O.
-
,,
C
r f,,.,
Z
U.
IIU-LU
La
0
U'- -
tt
- I L
.. JUO
_I _ _.-CC
X '4I
ZC
U-
f.L .,~
-4
0
I U.
_ -4
'
....j
L.)
-
-_
.D ,
,, N
X ..
_;
-4
;.
__ .- O
+ +.,~
0,_
_I.
_
-,,, i,,l,"7
N
II
z+
4.
0
I, _- 14
N
s N
p.4
-4
I,--A
1-4
''-
* *
N O
'-'
ALJ
-4
L3
a-
0-L0.
-JZ
Li
-4
-4
Z
_.LU
-
e,,
f
'.LL
O.,
+
4
N
O-
N
)
-4
' rz0- _,)u
0 c.)U---~~LLU.00X0U.
0
-
-
_
-
J
U U_
;
U cc
LL
LL
U
U0
,0
-4
-.4
00
o
un O
I-
-4
-4_
-4 -4
-4
-4
(% N4 -4S
_,,_Z
_,,_0
_'
_4
4-4
--4"-4
_.4
_-..4.4
191
0C; N
o _, - - -
r
n
-
-
4
-
0 U
0 r-)
- N JN
-
-
-
Nc
o 0NaO
r
N
j
-4
NE
N\
-
N
o
N\
-.U
n
M
-4 -
M
M<
r
f
Mt
- - -
0
Mt
M~
<r
- -
oN
.t -at -T
-
0
A
2.
cc
C
U-
C
CL
U
J
U-
-
I
U-
-4
-I
LA.
c4
0.
0
1L)
LrP
-
LA.
0
Xn
U
VI
V)
0
A
.J
N
LU
N
m
4.
+L
_J
-
0
J
Z
+)
LU
L.
UI
J
LL,
LO
JU
of
LL'
4.
W
U
0
0
3
U
CC
I U-c
_+ _
CC
-
C) NU
Z L.)
0
N
CC N
+ .+ U.
0.
-'.
"'. N
t N
;^1
,,
*'~ ;:
I
I C
_
_
H
S4O
,O
a-
co
V)
CL (N
a.
C
(.J
-- II) -O -,U
-4
-
4
-4
-4
-
N..
U.
L-
U
o0
,, t, :E::[ f_
II It
0
L -*1
NJ P.
O
n
Z
..JCDx.
,]1 LL
U.
U
X I
UJ
N
N
N* 0* LU_
3. N N c
z;D_I
O
N(D
c
t
_
N0>
M) tn
_
4 -4
-4
-4 _
.-
-4
,
(
-
0 CW
--
U¶Qo
,--
-yZ
I
.
< D 2~
0-11 0I Uw-Z -WZ'~(f.U
Cowzzo
v
II
£W * . N
I,
V
-.4-4 -4
k- F-
-4
t-.- *.
I LJ
_t _ C OW
3: LL
C LIOWO
ZL
00C- - U
v0a
a-U
f-
,'.,
0 -- 0
2 0 U%nn")"-;2 -
3I
- .
C -,
-_
-4
L)
CD U CL-
JZON-^
-.
II, t
N
N.Jc, .S_e
N N cz
If
U,0:
4
N
I'I'
NN9-A
I
C+-N L
CN
Zc 2. -4.1.4. ,-.-.
LLA
N 0 r-
U
LA~ ,
,
Co4 +
U.
r; I+ +-U
r )
N 2. * * -4
II II '- -' ,
.)
u
cc +
N -J
Z r-- u
I- 4 1-JCD
U 5' "-J ,,-_-N1
U 0;2)*_ *_AO
C
_
N N
_ _ _
( fUirN
L :fe
+I Ne.J
+I L_)
,-
.,
_
L
).
Q
- Z U.
ZZU.
Z
LL
+ LL.
+ Z
.-
:a ]X
r~4
UNc~4
X rJWuJ
) Y
C L:
C0¢
0O
zI
m N.-O- :
-C.o4
0i--U.
CL
(N
..j11_C_,,_
11 LL
II
~II0Q
..
L) (; ) O
c, 1*
_
J
X
uLL C
Z
LWI4U,L
0co
NU
P-
-
9-4
+
-40
Z0L
G
-4
192
t
t
'4
o
0
U
1
aJ
p
V
U
U ' Qz',U
U un
U
V C
O Q
-4
-4_
-4
-4
-4------
c U _ Ns M ¢
O " - r-P r '-
r
Q 0 0 D0 0 0 ) 0 C 0 0 u Q 0 0 0
-4-
-4-
4 ) z I w r o - CN
r
UE u r. w ( o
U'
No r-
r
r
- _
_
w
Gr
L
- X
0 0 c00 0 0 0 0 a Q 0 0 0
-
-4
-4
-4_
-4
4
-
Oc
C:
~L
Ua.
-
U-
*
cL
-4
UJ
OY
CL
LLJ
LU
5
.N
N
Z
Z
Z
z
N
0.
L
-J
-4
_J
JL-.
00
4.
-
:F
- Zza
~+ +
4;
CL
kUII
,,.. .. ,,,
3~ u,,,
£ *
-'I
110
J N a:L
Ns * *
It
-DI
CL ED
* uU-
-'-'-4
_I
_
_I
I10
,- r
-4
-.4-4
00
+ + C
O UJ
fl-
-4
-4 >,~~~~~~~-
II
U.X
S
- U
I-'--0-w
'5 N
I4-
-4
-~.0
N* 0
* II
sU-4
li
_ U_C N
-e
It
NN N; .
LL
I
-4
-..
c~
-
CCI
C,
t0 t a l I
r,..;j
f r..J
LU
u
L,
>
-U~ :E+D
Q 4O 11
S.1s-.
u.s
Lf
( L
_
LU
- 0
U
-
F- < a.
* NN
* NN
* I-
0 0NN
N
: ZZ
. _ _0
. _A
-,
Co)
--4
Wn
.-J4
--- L
-4KCL
a,
N
N
a:C
i"'N
! el2Z r4N
*0rC
LU
,LDU
-.
-4
_ L.
U
li -
H 0cZ
o _
0-4X
Z z LL
N * *
IL4
IIULU
w
C,
-I
L.)N
U..
6LU
U-
N
HU~C2~
*2 *z -s,
ta
ŽX
-4
Nl
)
a
00V :0
-4.
- LU
-a.+
II
3.L
LC
N
-40NN
.a --
--_L -L4
t,,
i
4 I11lL
--
C
a:
+ +
40
U
_c P, uJ
-
N r.. -4
0 NJ _J_0.
-) IN
4O_
N~ ¢'%
..n , ·-.
_,_. O
. J,,.V
t"',4:
:E
W
t i n U
U
-
_
LU CL
N+
+U+
..-i .j
-4
-4w
a
:L
U 4. L
O.
Lr
.
--
-
_,, *
_.. . _..
*~
N
4.
.
j...
. --
4-N
-,
-
tCL
-G
_J
U
-
Q C U
C"4
-4
3
--
~JCCN
~_,
-
L.
+J
--
N
-4
193
2
N "I
Ln
¢
u
G M U c
Ua c
s
M au au M: cc co c0
O
U
CU
O
O
,4-4-4__
__ "
O
_
Q
U
CW % U'
N
a% 0%
O
_4___4A
O
4
O
Q
___q
4-
M a,
F
N
Ct
a,
)
,
C)
O
-
O O
Q
4
4 _-
LO U - 00 a
) O O co °
-
___
_
_-4--4
_ 4-4
-
0 _ N M r W) O
--- 4. -.
4_-
4
_
4_
_
-___- 4-
-
_ _4
4
_
4-_-
10
CL
-
*
a.
N
z
0.
z
CL
N
U.
CL
~Z +
X
cL
N
0.
_
J~
+ + N
ul~~~u
+
-
..
J
V-
USJ
-U
;4 +
N
Li
ID
CN~Ii _*
i'
- : A.
ir ]IL:E
~,
:E ]LU
--
Q%r
N
11 N^0If
Z
C
C::
N
UZ
I
Zf~
I~ .
.
~ ,,k
Ck..
J-I,
i ca-
xm NC
+ C)
z0 44
NU-W
-9E
U.E
O
rN
U.
,U_ Z0(-
C)
+ +
ZN
Z co
ItO II
UOL
UU
LLZ
o .J"t-_]l__
J _uII' Z
(0U
IILJOLu..
N
_4
DUJi
-
-
-_
-
. '"a:~-,a:
'*
0
U
ZL CV
N 2 * * _
II
Z
,N
0 ~"
*_
, Ul'
I
.
-.'
UA
Z
U'N U-i
_N N
o U.J
O C
nU
N
N
0
-
. 4 d (J
W
vI
2Z
UNN
I
N: E :EI U'--I
:E
+
I' C~+;. IIU
Z:)
~' 1,'"" 1'-
II tl1-.-- .
9$
:3,
tv'
Z (N
~ CX.
Z+ aE
N N '. N U.
*Z,-UJ')
S
Z
x
0',-I_Iu
C" + + U
CD U
O L)
Ln O N
-
_~~~~~~~~~~~i
3:
N 2L
.L
44
\a:
^0(
cie
_O- O. .U~
4 UJ
E:J SU_
V20:
O
_(_4
I
-4
Ja + + U
LJQL+
A N CJ U Q
N
N
N . r' :l
t
0N 0L
uDJ
U
U O: ,'CJ
_J .I
O
I--I
J.
Z ,~: Ii.
JiL
5
-4
V
zU
1t I
-.J
N
,,i.
-4_
LL
C
Q
I
Y-
vi
L.
i-
-4
++
N
N
+
~Z
a.
-- I
U
'V
w
',LL
_
~~~It
f
0U
40 0. U M
6-.uiK L) ( U . ,
UO u-
_N
194
r.- x
cr
-_
-
-
_
-4
-4
uC..) N
- 4
4
N
-4
N
m.
N
.
N
4 -- -4
,
u-
~-4
-
,
C
N N
___-4
--
r - "
N
N
4-
-_4
--
4
4
-4
-
'
-
--
-4
4
4
-4
---4 -4
-4
-"
'"
m
-4
-4-4
ri
-4
-
-_4
-4-
~4-
-
. -. 4
c
o? L
N
'
_
4
-4
444-4
4
4
4
--
-
-4
-~
-4-4
-4
u'
-4
--
Lf
.
-4
~
-4
:
0
U-
*
U-
CL
.C
Z
-C
c.
r.
L
N
[D
0
Qn
4U-
-_
-
'0
-
C
-
' NL.)
·e ' NU r~4'e
f
-
-
z
LL.
L-
.,2
U-
.
.L
O
+
-J
LU
-4
1
-J
-
U
-
J
z
"Li
_
x
U.J
_J
w
IL
-
L
C4N
-4
0
-44
_
,
O- hCD
N
CS0"0
U
0
0
,LN
L.J
.
4. -9.110- N-
i.
-4
II
3
UV
Z * * -Z
*
.
ui
$,--4
CN
U-
N N4
1
-4-4-4
w.
N
I*t
*
*
_
_
NJ N N
i
I
-)
-
CNNNL
_
.
C -
Y.
t1
UN
U_
-00
j
oV
11UU
C-J J1 -
r. Z'0.U"_N--,'"'
:E C90
'-'
L)
- C
11
D L!.u
Z Z.,
tia
Li0
~';J
ZtU#
3 Ni
N~ Y
N
iN
em.-
os
C,.
-'--r
L
_j O_ I .. U-N
--U U
S . W--4 .J
.e .:)I
L Z
JzN1
-G _ _ U- IU
E
UUJ0OZ
U
-4--_-_P.) 0 U 9.L
U-U..)U
CO -UE U
NL
2.Z)
<
4
-r
,
C,
ut, <~~~ Z
.d
+ + L.) 2.
IL)71 I O
LL
.,0
C- C- U -
* i ..L)
N N sr N s
UU.
zz
_-4 -
,
UE C) 0.. U- US C.
.E
L
._
_C
CrN Ns .4 Q.
-i
LI~
....
1C
?..
* + _
-4N
_.
0'0Z'OU, i
:
ZLU
2C
*_-~ +_
-4
-j
A
-21
-4-
:
IAI
Uw
-4
-
N
I .7* *'--61.
1 0-
'N
t
, _
J..5
ii
. ?,~JII
L- U ,.-..
N--NNN
---
U-_)VKt
-4
-4
-
..gI 4ULI
C3
U(I
_ _-C.)'-'_ 1_1.
.
.),
_ ,
N
4
-.
rN
P-
O
NN
4
,,
195
Un u, UM
tn
IU,uLn
_,.4 ,_t
_
.oU
so
4
, a°srrrFF~Fr
._4
W W =0sCZ)
,.
W
-
-
_4
_,
C.
U.
u.
u-
-
LU
,0
r-
2:
N
LU
+
LU
L
-
z
L
ihL-J
.U
iL
J
U
-JF
-J
-4
--J
C:UJ
L
U
I.-i
0
X
-C
u
-W
+
o1
*- .. i.4
-O'
A.
__
_
,i 0+
Q~.)
Zca:
UJ
Z
_ -+
LL
w
,I L .U Oq I,
L U
Ch 61,
U U
N
X
o
--J
0
11
2 ::
cs
z.
4 D.-ct!
N
C)4
,U,.4C
'"- - ,IQ'.C ,4
-0- -.. LL
- I "N
c. N -J
Ui U
C. C.UJ - 'I
-J
L
Li
4
-42:LA
*NN
L;2
NN
*i. w
NJ N V
N4wU
Nil itU-.Ng
UL)wQz
O Q L
-_<.)
0
Z U.;U )
L
;*
(%N
L * *W
O-
0L)
N U'
N C r.
C
N
0
N
.
L-
N-"_
;
^m
*
- .
-, -*
J-< It .)*
NNN
O 0_*iLW UWoNXiIZ
-;c a.
a
-0
NN
-
O
-..
N
f
l
0
N
N
NN
CO
-
-4
--
-
O Z Z ~.1
N·
NNN
"
-E
U.
+ + L
W
UJ
N+ +
NE qJ
N1
+
Y x
)
W
Yn4 U
N
+
11
.
+
0
It
N 4. +
N
r ;J'!'
0-"' 2;)
) rl
m V N .J
a1,
-1
2. lN1.cc
aCU'O~
cl
N U'
NQ
r~
*' c *
. LA
- N . 1F 2:
NJ* *
N N N
5x
F1>* U _
Y"N -O
I1 NI Lj.S LUJ
X I I IL
a: S A1
N C
N
rV I
_in
-4
u
O ,.LL~ .
i
Q_- i
ED o LU <: I OU' W
ON
Ot
ii OU W '-a,
XL UC-%
iLU
1-
t_
0.z
-J
-
0
--Q
_
U.
Z 4m,,, U.
0~ 0-,4
l-,,
eUQO
-
CL
x
'4
C1
(-_ _O C
tf
h---
rZ
.- cur
LL
OIj,*W
L ~-.Cj O
2
U
W
44
c)
2cLAUL
N
* *NA
O0)
xX
+.. )
C4
_~~
UJ
CC
2
U,,
-
CO
A,
-~
-.
-_
J.
CI,--
-4aO
CL~~~~~~~0
-
N
Z
+
+4
-~~~~~~-
U
--_, .
C
U
0
Z
UN
11
x
"L
fl
0LJ
Q
m
_ _.
C M-4)
;) o O; C
U'l 0
o cr C
N
N
-4 -4 -
f-
<a
IN
"
'I
i
3: L.
0
-
of
U.
,_)
0
...
i
196
T17 '0 r'- m 0' 0 - N M IT
0' 0 - N
r
Uc')0, - N mr%t0
a O
0' 0 - N
N NN N N
-4 -4 0 0 - - -- 4o 000
C o 0) 0
cw cra 0% 0' m VI ('
N N N N 0NN N N N N N N N N (N N N Nr N N N tN N N -4 - -4
- - - - - -4 -4
-4
-4-4
.4-4
-4
-4
4 -4
-4 -4
-4
-
--
-4
4
-4
-44_4
-4-4
-4
-4
N
U
c
J-
w
C
Q.
LL
-.
I
rr -4
J
F
Cx
~A J
-4
L.
CN
7-
U-
(N4
OF
0.
I
U.
UJ
0
0
--9.
LI
uJ
U3j
O
-4
I.
Q.
N
Z
J
~Z
~~
_
UC-4
,,i.
-J
+F
4.
4.
-J
X
-. 4 r
,,, ;IL "-
N N 0U N 2 + + 0
II
Y Y 11 0 c * 0
N _ _. _
+ + rj
U'
a.
kn
's
9'
o
--
.j
wN CN -J
V) LL
a U U' U
E U*X
2It - lI_*
j
N
(NN0 *n
- O-*
Um F. N a * *
(N -4(N3.
I
* * -A
:Nn n
j_*JlC
..
N
u
0.
tN
Z
O
UzoL
i.Z ,..i c
I
MI
-.
U',_
_
9 O
U LU
-oW
Q
9L6
_0 04
-4
_ A 0LU
F..;_'"
-,
-
-
i
O4- O.~
-4
-F
J
zZj
UJ
-
U
4K
a; _Z
>.iN
UJ
U
l.J
_,.. a,
(_
Q0
C -4
~+ Z
~.
+ ULZZU.
0 4.
0UJ
L - 4 i.-LL
N
0.
.,
_
L4
O
Oz
_P
CN
rCu
*Y 0
1
C...
O79.-
Z .
-
+
ZLU
U
31L r'~
_~f
U.
3_ t9-
z
w
L
+ +(
-4
...
II CD
II LL
U.J .L
-4
_ uJ t
_c..O
LU
.. N, 4, ,
5
U
oF.
- -
r'r._
,_ a-o ~ ,)_-LL
I lL O
a:N
Z N· -tn O
:J
7
LL
LL
£
;_ ia rL N (N J
t
O-4
x
U
Na +N N
VU.J * Z crU, r.3
+ L
Z n
w.a. * *
Z_N L *() O~
* -U
* tt 9I ^.
I, t 9~
_
_ -I.
CLN .._
Zi. '- x. I I O)
t NN UJ
U)
0
-4
-4
V
-4N
3. .q'U-~u
ZW
I1IL Z e n 11Ai-
-
z -Z ,
N N UNC
Y
t I A.W
L.)
+
Lr
N
Q.,
_.
_ +
C
z-
Iz
m- LAJ
+ O
0
N
3
U
-J
V
Z O
- U, ,
X ~ _ +~lCl+ .-rv
:
3E ~ a.n
N
- ..3
3.]
"- _-
-4
0C C
-3
U4
Li
.-
U-
C
A
'-a "
U
W
0
V *.>
N
,
Z K-Z
-- wW-J
.L.r -,11U1
0 cX'z9
W.)-4 .-N
r L; 0LtI
LI,L
z
Z 2.
C--m
0
O
-4
-4
4
-O
197
UCN r
U
r
-N
-L-N
- r m a:a
in
omo;run sv%-C
U fc a
,U
N N N N N en 1 M
rn
r "I m ?m - T
7-. %t 'r 4 4r
-. 4 n U' LN U,' Lln
f
koU U.U n Q
N N 0N N
N N N NYN N N N N N N N N N N N N N N N N N N N N N N N N N
-
-
4-
-
-_
-
-
_
--
-
-
-
-
-
-4
-
-
-
-
-
-
-
-
-4
-
-
CL
C
*
U.
O
1-
*
CL
$2
N
z
-
UU-
o
o.
tj
w
IL
N
C
CL
N
-
.
z
-
z
Z
U.
4.
N
-.
N
Z
-4
-4
.J
U-.-
zI-
4.
N
-
0.
--
+
_1
-J
r.J
z+
..64
A
CL
-4
J
)L
--o
'LL
-4
.J
,J
-I
C)
C)
cxz
cr
-
C
C
y
rj c
i-4
a-
.N2- 2- U.
C)
L
-4'4
I,.+ + -4 _7
w
-Z
-
U
_
-~CC
U
")N.- _ N N
t'-
IN + I - tJ
W
*6 -
0
I
I l .
C,.- A_
N
u-uuz
-4
-)n I IC)U
_J
N 3 I
N.1
."
cDQ)U
-4
tl
,- ,-- N
- 4.
'
if
11
Z L
L U + UJ
I CD
*
_
I0
11
O
I C.
0 -4
_ -
,L
-
L) U
U
-4
I-
4
-
U-U) Um
Z ZU.
_c
L3
-
C
O
+ . U.
,.
_
1114
+ + QC::
ULJ
U
Z + LU
C)
C -- C)-4
+
)U-C
oCL
~-UO II "
If.
N * * -N
* * 3: ON-
US LU
N
-aJ.,)
It
U N y;
NNA-
-4
Z
LU
zx
,Ll
C)
m
a-
I-
.. I-C
U.
s C
,.n--,
C
U-
Cx C C)
U L)
r~
.
,,.~
II _~" o" ] 11--°" o"
,-Z
UL U O Z U L
NKti
U) U-)
II
-Q
I,IO-U
- -
-4
-
L
-J
) N 4
*N N
* CCN) N N
11
Li
OC
U C
4.4:
_
m
U 1L
OC)
.J V
I$tU-_j J U-
-j
4
~1 0
_.4 -_
LL C)
I-Li
+ + U- +
_W-_ 0 0L I+ + Q m +
LP ZZ11
LI'
- U- - U U L) z '
rr< *
*
O
L:O
LU
C)
I.
N
-
-
V)
~U.
l'
4.
w
LU
-.CCUW
3U_ 1U C)_C)_j
n
_ L) uj
CCI I U)CrL.11-
LL f,
L.U
U
O N
,.
0
-4
_
II
II
(4U-LU
LU
-
r-.
:~r
_t. t
if
a
i'~t
- -C+ UL
O--
U
O Lu
-D O O C11
-rC)
>
-O
- xY
ULI
£,0
f-
198
_
r
.Xu
b C (7 ) )--, N
¢r
<~'*
- N -4~~
N
r Lun
m
'~ Q
L) N 0 'o f m r -
' m'
,O ,O ,O
Q
n a'L?
aaUa,-_ N
r W c>
- N
arcr,' a,
Wc0
aUff) - u0 aW
c
m ' m" m
' r W-" a.
Cu ° o
(N N N N
N N N N N N (N N N N N N N N N_ _ N
_ _ _ _ _ -_ __
_
_
-
-
_
-
N
_
co
0c
0c (
0
0'
N _ N_-- N NN N-_ _N N
N
_4
N
N N
__ _4
-
x
U-
,
U-
*
z
*X
C
2.)
4.
I
Z;
r~
I
X
IX.=U
0XI
/)
x
IX
x
.-
4.
a.
_J
J
-J
C
,
0
-
L
xO
0
I3):;
X
X--
>uou
I
-.0
ff) V
a:
-
" "
N
en"3(..
Li
O 11.I.
.11 .-
rn II AU
:E^II U
;,U ,iL
J U W
-c _
f. C',W3
.tt L.,
' ^.
*N3:
4
v .*4.
Qrr'
UJI.~
I
W,- r.4
Z
C Ur
I.
I IU- _,,, .
3U
' zJ(JUOWU
cro
_-
_4
co
0
M
1-
mW
N
fn+U+ + -u.
I
-4
I
I
4
.
-"_l
,1_ :_
_ ,
.A.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-n
I~
i
J1.
0.
(N
O"4
(' k. 3E -
a.
-t
II
,-
,
'4
N
j;
Jr L
a-
*
; ~"
l
I
zX
"II
U
LL
W
V
CC t )"3'"
-IZ
C)n
r(7
(. ;:
O: L; L~ Z
G
;2
I
II
1 '11 U-
a.re IU .
.
0 X
Z
-O
I N
rn
u
- _~ ('
_J (.- 0
IVLJQ-
.
I::
Cl) 0 IiV
-U
i
M I-IUrn fl
* r
t.. 3L
IU _I
Li
_L
-)II
·
II
r
_
0
I-
10
N0I
C
U
.-4 '4. LU
(N + *,
3
-
m
iL,;L 'Z U-
ml ",-
C
C
f
O
3
C
m
i- -l
m
11
O X -l
10I- 0
N
_I
; L
(lV
U_zU Om
O: LU
U {
It
199
a,
o ' - N
c'
oo0 e
v N N e
,-
e
-4 -
LO
C
.40
0,"1L
0,.
l
e
n
(
en
-- -4-4-4-4-4
Cy, - ,1
N o"
,. .t
= m
-"N
,1 .
ll
ll
- N
000NNN-~-~NNNNN
eNnen
en
en e e e
m m1 e m ) m n e
("
nmn
en m rn M m en
en nr
-4 4-4 -
-4- -4
-4-4
n
-- 4
- - -
4 -4
-4
X
x
U-
.J
-
-.
L
LU
z0
0
eN
*
*
I
oLi
A
x
l
-
-
-
fn
-
;'C
xLi
1
-x
X
.L
X
X
NU
x
v.
-
x
.J
Lx
N
all
O
A
C
cr
en
NU -
-4
Z4
+
,-4
Li
_I
u.
-
_4 C
n
L)
10-
Li
4. LL
-
O
C
C
L
0It
l Z U.
n
C .1. t1.
N
N
-o
LL .
n ii H U. _4
9. -4 -A3 U. 9- --
Di#
00 00O£
LiCK
i i
U
aOD
0 cr
"I
tv.
enen-
-4 N
'
0'
0'
cr a
-4
-4
-4 -4
4
I.1L0
-0 "
,L a.
-4
2
0(0.
L, I14.
II
O4;J LiU
1lL
II 2:
cx.
Li
O
A.Jt
+ 4+ LiX-.
9N :
t- t.U
-JIB
n
n 11
* tl
_ +I+u_
N vK
N .
Q·
,-i
L..)
U
N __ _
< l.en*- -* 23
2:
B-- 0Li x C esn - C'J
l
-sen
-Z.., - O
0
O
tx
-J
*,-4 It
0.
+ .ZL -
N I:
m C-4
e0
*
-l"*
ofI
+X
14 i Li.
of
r.
N . *_
,ill
- 1 tt _, _- *
e e ~ 3c
.Y
"I C
2 _* _.. tu
._
X 1 cr'
--)
O
+ +
Li
0 -4
$,U.m
2.0U
MI
,X O-
-J
1.)
>
-
-Li
+
aL
U0
0CD
-4
+ u
Ix
CD
-
aL
+ U.
x
A
-4
+m+
zO
-
L
Li
0
en
(.1
Z 0
Z L .r II
Z..-Z
(n. i U. -
-. 4- · O
4-i'. - _: i1
u
AL0
Li
-4-4X ALtD-_. C2
f,. NC
,-,-~.
At.X
_
Y
j
9
L
l
.za
_
r
2- CDU
LLtz
U
O0
11 U O u
-_ .4
Y.
2:
L)
LL
IC_
eUJ
en
W . -4
Y. O~ CL
W o U; a
j
c 0 iz O
O
.IY:
- U
u%
U,
-4
N
cr4
_r o_:
-4
_4
-
(70O
oo-
--0~I
.
O
200
me
m (
n
Nu r
n m"
w a u - N m
m m 7
ST
'.
u Orv m m
4 ¢ 4 -4
t
N
uN!f
U z - Wm
n U U t .Ci N
N
n
r
----4-- ------- 4--_-4_-4---444--4- --_---4
m
m
Im M n
r
4
m m
<
V)
n
f
o.
VN
4, e.
n m M EM M M
'4 '
' '. 'S ' ' ' '.
rr nM m M 1 rn -1 e M
0
L
z0
3:
X
-J
00
I
(0
>-
cL
4.
0
0
U
.J
A
,,'.-4
14
-
I
-4~~~1
- a.~~~..
.
X
.N
,
4. ;4a. U"
..U
U, -- ,-4C
.?
I'~ rn
('~ UJ,W
L
(.C:
x
N- C1
'_
O.r* t;0 yI
ICG
W4 ¢
e C Wj
u.F + zFc-
N ;k >
.*
I ::' .
O'-- kO
-,-,I
4,1.
LU
- .
r_
e
4
4
4
r,4
214 a_
N
_
J
-J
U
..
C
m~jX-.
+L
rN
-U
- -J
03
:I
U. I-
0
4
~-4
o VU .-
.4.) U.)
.
-
.-
N
-4
--
,m4
4
,m. r4
U
C-c
XU L
I
LJ CD
WCK
4
(r
4
-4
-
-r4
e..
-4¢
-
4
-
--" -I
4
44
-?
It-¢
-4
I L.J
ZJ
f¢ ~--..- -, _ z
,U'
:_ U. OL. U.JZ
z0 &C. CZ)V
U-
-
..
_ _
-
,_
L.
I
t1N
.. J
WC
.0
6.4(
L),
U.
'-U
+
C.- 4- II ,1- - I .;J -.
IIt
Z ,,--4-,L
.a-4-4
Ir t A_
4- C:- It -It u. _ U,
-..J4
4
L'.4 --
+ CM I-
U-
-
0
0
w:
.
C
.4
i"94~
If .1
0411
O
-
0
0
-
i'
iIICIWi
>-J
- iLU
.
Co
~
ZZU
m
U
3L
.1U.,
.Z~
U.
0
I.-
x
Us
c
tt)
-J
0
U
iLk.
,
o~
1-. 4Lji (LkW
0L
LLI0 ~.'~Z
' I-)
! -44.L Z'-lC)
;_)~.
_. .
+ * _
~F~
*
I 11..Jl
.-, - O . ~,~0
..~ D.:
U 0
Q Q
'.0
-*'.)
N1
~~J
X
e
4
04.J
NU
I
_-d ~ ~
-J
0I.-)
0*
C'
X
(0
x
4.
*-
x
C3 ¢:3
C
)
0.. ".
(7'o
_.. rN
_,',1_
201
C c - r M
r 'r Ln
zo r
r-
_ 4
r-
r-0X OC
r
I-
I-
-4 -
- N m , u- , r- a) VI 0
cu x
-4 4
-
w
-4 -
co W
c
x
CD
a O
0 - N
0t
0
a, U,
-4 4 --4 -4
- -_ -
-
0 0 M r -IQ-
O a% X m
O (%
1 00 o (Z) 0 0
- 4-- -- - -
Q.
X
X
,-4
J.=
<[
a
P
.-
.-
32.
0-4
3
x
U.
U.
L+
-L
U
2:
C
U-4
:
L)
-.
-4
:E:
N
.-
a:
>..
L)
'C
.4.
U-
L
I-
.9.
X
U
J
_J
LU
C
.
'C
Ir
'C
.L.
J
J
-.
-_
U
Z
aX
V.
2E
*
X
X
,k
)
J
W
L_
0
U4U
L
Z
+
C.4
l
U_
. ,,IL-""
U~~
U
Q)
*U
-
li a7Li4
~-_
11
:
0_
N
U
-
;
LL
N4
-4
4
9
.-
_+ W
i - CD
_+ Li
-
Q
- 4L)
J
r :
· O.
Z
e- 2.
_*
I_
mYrli )e
f1lU-
Z.
J :U .
COw Z*ct
OW
ilf_
-(L
.4
It
Li
+
- tI
W
+
_
-4 -
U
+
-
_
'-0
LL
+ +
NC
rq .4+
_.+ (1)
- U
I%_-+ ,-..
N
+ rj
t"l-
.
. - .-a : ,-4
Y
C LL
Ij
O X
s
Y
V
*= 0
CC.
1
_
- -4
_-- -
-4
I ,
9"NNA4') NNA
I 0 Ce -
_
: X
N-
-
0.2_3_
it
2:
* *
-
90L
_
I
I
_ _
I
C
U m
U
t
*N
a.
. OL
he
w:
i
m: U
L;
-
'.
IC t I
O
C U U uJ
H
JL
U.
Ci Y
0
CI
-4
-4
_
17~~~~~~~~'1
11
L)
L.
N
_~
NN
W
C-O1 -U.
I L Li
.
2- x
Z Z U. Z Z
Q
-4 -4
N
I
Z Z L
fi-
,,_ _ _
I
IN
I
+
-I.4,.U.J
+ + "
-4 -
F
U-
,-4
ZZ
.+..
+ + ULL
-4 C"
.
3.11C O W
-
N
N CN
-.4
Li~~~~~~i
0"q'"."
o,4
Z
11
--
FU.C)-J 1!
cc
Lz L
Qi.3.t It
1
i fi o li
eC--_Li
-4
U
-
X
0
-
WZ
U,
Z tzu.
LL
-
,-4-411.
0~cI-.
.- .
a] 2.
U.
ZZU.
*,.4
JLi
3
-4
C)
-
Vj cr
-
202
Lr4.r
O
-
U
4
U
4~ W .
~
w1Q 1-
a a
_)
4
".
1- -_"4 4-_ -4 -
- -.- 4
-
".4.4
0 - o
, L('UQI.~ a a" a0 - N M r % 'C
0 0 oN
_
_
".4-
_.
.
~- - -
N N
-
N
NN
N
N
-4 4 4 -.-I4--4 _.4
f o
j
U.
__
0_
:
_
_~~
0UJ.
LU
<
;cJ=J
x
x
.4
4I
LI
X~~
uU~
U)
*
~~
_
x U.
U.
o*LJ Z.
A.Li.
+O
,,
ry
-
++U
L.)
U. rCl-1 -,4.,j
a.
z U.
U- 04+
LU
,I- i. .U
+ +i ?,j
If
-
"XNNJL
L) N I:
U
U~~~~
~0C
:uOnr
0
Zo
+
JJ
6. c~~~~~~r L.
.4~:~
Z4
-4
_i
0
*N
U
x
U
x
X4
+-.
.
a.
x
x
C
4
I
_
_.
--
z
0
0
4- - - - 4-
.4..4
-x
1
< e 4 s I'1
.- 4
CD
CK
0L
r
c I-4 ,.4N
- .
-i
L-
I
.
CD:'YIC.
t:yLU -,4
-U
-. Iu-d4Z
U.C
- U
1..
UZ
:L
_
Y.
,
9l LL)
-
L-)
U+
z
I
X
U) --U
NNX
X
1U)
I I
N
...-.4
CD 2".4'
+
~ Uc
U.I
4
1.
*U".
U.
.4
zz
C
~
-
.4
:u
~~~~~~~
U0
n
O_ _
4* 4.U9.
. I I O)
I-IJ
U.
.4N~u)- Ui.
- i0 *- * U
L
C:)
uL
U
C
t
tt NN
U.
^- -2..
.4
LU
'- .
~r'
II
11
4 11
11 UL
L)-U)
0
N
3 4- J
N4
QCZLL
0 . ifI
Z 4
UJ
-a
X U.
0CD
OJ
I
I
iLXo
XaLJOJ
~-'
M' L)
o
1ox;".4a
0J
u
O)
J
XL
U
M N
-
0 _ _
-
4
4
S
-
4
C.
E1 YI
:
O
203
-
.4
-r
N
4 't 4 4 '4
-
-
4
a OP 0 j e 4 u, 0
W O 0 - N Mr
k
Io
rO 0 -3* W
4 4 LALn tn U L L Un S UL'0 'Q ' No Q ' '0 'Q '
r-
O r
-4 _
_ _ _
--
-
-4---
N rM W LA 0
r
- b
r
--- -4--
-
_
X
I.
a.
UUU-
C
0
L
Z
C
4:
U
...J
U-
Z
.L
J
X
0
V)
X
.1.
X
*
>wj
cx
U.
X
NU.
(JJ
U
0
:0
0
U-
A.
U.
+ ZO
+
+
LA*Z
0
_
U.
_4
LL
N1 N LL
LL
CQ Nll U.
4.
-J*
_
4
U N
-
I
,a X
A.
_
-
4
·3
v
O
I
I r
''-
t! cr
,,4 t- 31C
I, -
J
'Lr
L)
1
I U.
J ;;
LL
C - U
'0
_4
uu Z
)Y
U
LUfL
Z-j
a.r UJ
Z
I-
_
X
0U
0IXIt
0
-
_J
+:+
¢uO
4
Q.
L C)1.
cr
U
U-
.4W
-
-4
"
cU.
11
m cc
Lt U
*bU -U.c
m
_1
SI
LL
'.
N
a4
- NU
9
-0
W
1Z-4
,$.- -<
It
t t"'-,-L-LZ
JI3~3r
',
J -I
_ NLL
i
.L-O
L
I i C-
I.LI-- -Z
_
C.
g V.
00C: c
-
-,-
OJ ,
--,
c
C.
a
N
r r4
r4
CL
MCS0'o
r~
yt aXo. (;A
t n
t
-
X
C.
-4'-0
I I 11
O -.4
Y
. 4-.
0.
0 UII a.
Na^a 1
Na a z
_ _ - -4O - -3EZ Z _-0 ,Z Z--_
tt
II IIU.'4
It II II U..
_ J . I ,- 4.13:
.0UOLJ
:X -
O
X
O) ~*l
O~C
(.-
C0~O0
C.
- y
r_
4
cO
-
:4
I Ol-s
UA CL
-
,4 X1
aIl O O wU
rh
14
_X
- -4
_
X
0U W
E 3[
' tI t I. 3E c
* * X _ E * *3
m ~,, C
C
- -4
X2
a
Li.
O
LL
oLJL:rJ
C -3 -
N
>, 2L -
E'L)
4~.
*
(N
U
aL
--
j
U
* a. : 0 N , ;.L U N au-4.N- I U-00IS
* *
a. * *
N 2 * -
_,. _d C
O Z
CL.
z L_ --4 II;t
' o.Ca-_~~t4
.-j
U.
UU - CD
U
"I~ OL
I-4*
II
uOu
U
J
A
.
S
rn
LL
N
0
a.n
f
.
Ne N
,,lr
X~~~~~,I
.U
:D
204
0 Ic I J' If- ,:) _ . ,
' .j r rr
-J 1,
_'__'_V
___ o
r x np
- - - - .- - 4 - - 4 - -
- N
A r'4 aa)
cr v _ N,4M am
mX _MC 0a, T 7'
=X gr W
m
r- .-, r ,~ X, rW
r-
a
.. -. -
.-
-4.4-4- -- --- 4 - - -4 -
h
r Z P- GU vr G ,
("
r
if
-
-
-
i--, r
L r
-4
-4
-
__ _
U-
0j
-L
X
(tJ
-
r LJ
a
a:x
a.
w
4A~Z
.-A
_-
4
2o
+
4.-
-J
C-
4
A
A
-.
fr)
0
-
,C
W
C
W
.,
+.
*,c
--,
a.
4
---I. -I Wj
0esiL
- _ -J a _
_-I.~.C_
... Cr
N ix i0
C
V -
-
x _ *
r,,
C- -
-J
C
4t~
(J
.at_
C
_s
L"
-
I, 1LL
cc
ZU.
a:L^It.' H
fi
4Y 1..
U - 4 II il
I.
. ZEr.c a.
z- , *
_
g - -* C A
-_ C
-Z i*
-_
NO
2-
r''1Y ., .
oL. 4. L
*~' :f. ~ ;L
-'
I-
~-
N
-0
*
J'- Z- -*
--.a.C_
t, 4.
N X
5~ SC
9-.
r-,
NN
-
s O
U- X Z it'
, _~ *_
a
'J
X: Ni
-
14
,
NY..
0'
A
3U-
4
_
-_
-
N
0
.r .r'
aa.aI.-_
I11 x
it'*-
4_
X
P N
-,
>,
_.
a. gZ-
-
!1
a
t~II
C.Zal~
l-
*_L
,r-
_
C~C e) a: C - - U-z
0
U
0
-
-
Ii ii
2.N 0,
3_'
i
-
.4
az
"-
C. - -
_a
-.
I
L.-'
- r
_
N, N
*OC CJ -N~
-
NsN f)4'
arN
-
-
J
C'aL
.,g
?.~
i
z
-
-
a-
r
-
3
4-
C
C
r.
205
m --r
_-
uno
- w Ocs
-4 N
.
m
n .
--
N _
_
N
_- -4
) n n
U
n Ln
-4--4-444
L L( U. Un U
-4-4-d--4
-
r- W a o _ N
N N N ?InM
n
U
LO
-4-4
-4
L
W
o
-N
1' 1l 't'
- UvLUL n ) U
n
-4-444--_-44
.
4
4
e
.
.
P %QI
1 .
'
U1 L UX t
44-
-4
.
U U UN
-4_4
-4-
UJ
I
F-
O
Cx
L"
U
(.3
La
U-4
UU-
C;}
.-
LC
_
I-
w
a.
XJ
er
a-
+ A.
-
-
z
-
+ C
,I
U.
I CJ
·-
U-
2
Zr. -,.=-4 I,=,
La=
I
;' *,*..
2:
-~4
.
cc -4I
U'
Nz
-W
- UL
ow
lJ
=" U
Q ) ULL
OCaI-N.
LaL_
U.
Yi J Yi
_a,
O
(\j
N CN
0
U)
-)
-4
_-
-_
-4
_4
L
2:~~W
1a U ff Q x L)LJ
U,) -
-.4
11-
Z"-'.
UC3 Z
II IIU..
¢ ur~ G_9.-
_
-42:
ZZ 5. C::,.'_-. *
' 0
UO II
VCKa C)
--
Uf4
,.4
I,-
ILLL N
L
.I
_.=
_ t.J
_ -
z-4CLO
z
0- L
-4
Z
a
-
UI
t! tL, Idr
_4
_ _U_
Jr UJ
W
a
Va 0--
0
U
U _"~
t.J
'~
*. II I!
aI. L.
.H
-4 , Y
.4
. Z.JU- UtN
11:
_
zzS-..
' J 4 .C
I+
_ CL
+It 4 -~
4.
a- -IW a- a-0
- -4IE -,eL
lIE
e' I1 -4
~[ -4
]
0.X --
C_ r.
I
' x_
- -4_h --U. 4-4
Lo9Z-=4 11-4=-4i
-= L.
c
L.
-c
,I_I
U.. m_
NZ
9.
Et
LZ_
t X I U..I k;r IA
X
W -a r
ZI ~m
I
-.-trII
Z1 3LLI-U
~.cZ
N
LL
.
L.1
- L""~.) J L) C C
U 11
,
0 7
ULM,)j
U.~I,.C- - ,
II
LzwL a
we
I* * A
;_
LJ
LL
L
,-~
WC;U'Q
-
+ uL.LN
I,CLL,
+ _(
. 4 .
x c. z z cu
cu
I
Z
N
.
-4
II -CL
C UIJ
r4
_ LLP-L_
L). CLJ
2:
uWQ
cV U.N Um
LU.
U.
UL
L.)
U-
U-
L&.
J
-A
-4
-
_
owr
,4.
U
CK
-
IZ
C
U.
UC
(J
CL
A
0-
-LJ
3W
w
I
$V)
-
L.
.,L Z
1.
_- t.L
X
-
4.
L
4.
:'-Z
X
-C
A
.I
ULIw
~.
,J..
U
_J -J .
'
J
_-- L.)
U~
,,,7'
'3.
O N
U,
U-% Lo
uU)
UN
_
-_
-4
_"
206
ul Ul VI~ LA ulA VI Jl us LA VI L
L
V
,-
-
-4
,-4--
jlA V
-="
=
UN M
I-. r- I--
U
(7 c-A
r
N M -2 V1 % f-M0CU C)
4O
Q 'U Q %,UV
Un 'U %U
-r UMO
A L
V un U--,
-D N
cr
V4.U.
LA LA
i-
o r-
0
L
LA
LA LAN
U LAU
"
"
"
L
-4--
-4
,
4
T
N
0%
av c
0
r- w w
r
rp
-
L LA
U~LA LA LA LA LA
"
"
=- ,-4,- -
LA
w
LA
L
j
-
U.
x
Li
UJ
o
uJw
N
_
zo
J O
Z CL
-
k_
--
_
~~
'*T.
-4
z
=. ~.
O
L}"
-
-J
U)
-Jw-I
CUJ;
.=J
Z
_)
U
U
-J.J
C.
.1.
C.
.
A
#..
Z
u.
-J
x
X
'
.1
LA--
X
4
4.
'.J
:,,
.
CZ,
U)
J
.
C
U
-J
-
3
N
CN
0.
-3:
I.
40Z
.U-*0 LW)L
U-C.
.·U-.
U
'...-
C
O
-J
'4
A
%.
3:
C
P.O
4D
-
i
-4
X
C x Z *
L.)
Z
N
I
LA
U
.L-4
Ua*U
.eX - *..-I .1,
- Z
*L
* o.
,--t
UJP,,
O -~1C~l,i
O
r Z
I u~ o,,
'.4
I-Li-u
-
C.
,.4
. 0
)
X
I
of
-tN*
WA -4
-[. It
'
U
1
'~.
.
.u
B=
-owZ
U)LO
U0,)0
r -LJ
11
LA.
IIt~
tx
;E
_.1
:i;
- _tZ ,-
Z11Z1 -U
_J XB U.
O Li "J
~- U0
0
LA
O.
II
L0
L)
- 4
-
-
Il
~-O
-+ Zt
I', Z + ^., O _E~r/
+-t +ZY-- .--1, N+
IIU3_ 4 CLJU
U)
C)
NM
UU-LJ
U
UI
*'--4-4LA1
0 ON
CL
NU)
4.-3
-U
~.
_ cc
LL
:H
CL
0 N t
O
a;t -_ Z.
4
-- ~i
Z
if
-J
Z
0 fIt 1
zif - O
J
X
U
U.
UJ z
- _
L
O
L.,
< c:
O
-4
-
JrLL
(XU
" )
CD "
UI N
m.
N
U U'
Y
_ _ U C
- -O
,
+ + _
O
-) -
-4N
;
*)U"
*
cru a..
*:
:
O''C
'0
"
Ye 11
M 1 U" 0
N
-O * 4 - X
I N4
)N*
0.
LL
.
*.
*3EX
N
N
IU
CX
C
IL
A
LA
B
IItt
-'
L
I
Go
C uiW~
11=2 Z-.
I3.;Os sn o * *N
U, .-%A
j;CI,= w:
_ I.- ZJ,11
-4
(Xc
u_
uj 1-4 I
Y. Qw,JI _
* :&. 3I.
zw
~J
-
X
." U)
_.J N
-,,I'
4.3L
N
,4.
I
L.
U-.
II
3
3. .OJ
-4
N
.- J.g 3I. ,,. -
_. SI."
X
_- 0
_
X
.4.4X
-4
O
11
+
c-J
CJ
.+
3.
C.
v"-°
C- -Q
V
CX
_.
-Q
N2 X
M
N
NN
X)
X.:4.t-sC
-
N *
I
W
3E
N
N Z
U)
, t
I 11it
L Uf) I 11
NW 11
ZZ-11 Lk UN
-4 _J XI LL. LA X%-J i U. LA -J X U.
0
C
U r
- -
UJ
- -
I~lOW
U0
O
-
b
Cow
-
o
C
cr
u~
-4
0
:"
207
oooNa -q
Ln o r-,
'0 -I "Cr
LC. r, f" ,- o0
m aONom o o
ooo o0oO 00 m
un n Un n Ln *n
- i-. 4 -I-I ,-i -i
n
In
,
on
o
o
0o 00o
Ln
n
--.
,
o o
ID I'
u) Un un I' ID K.
-I-i
,-i ,-i -t,I ,-- I-- -q-I
u)
-I -i
n
,,o r-
o O,'.
o'
~
-t r-
-4
-q -4 -4 -4 -1 -q r-
O
0
O
O
O
- -i r-
7
OO~~~~~~~~~~~~~~~~c
)
L f~~~~~~~~~~~~~~
C
cr~~~~~~~~a~~~
O
O
cr o
b-
c
,
--I--I .-I -I - - - -i
s1
D 1 %D O %O% 'IO
I' I' I' I' %D%0I'D
0~~~~~~~~~t
r
"
o
<
:1
O~~~~~~~~~~~~~~r~~~~r~~A4c
0
_,
X
sC
LA.I
z0
t-CL
LLI
a)
-x
x
n-1
-J
£Z ~~ ~ ~~
OJ OJ
--.J
-0
Z~[Z
0
O
U
0
I
3
I- -M
^J
Z)M
U
~
0Z-U
_~ ...1~,
.L
~ f~
z
~
(lE
0
U _
U-t
Z-
43W
+3
m~~~~~~~~11,-J
-O
-.A
i,.
0as
a
3
Y
b-%U X
I
"
(
-4
D
CY.
aO J
LA.
~
U
1-4IX
HO~
Z¢
I
I~
--
XX
Liu
-
O~
UU
CU
OJ
0
0
...
u
U
<
;1
,--*
t
J
O"L
J
J
I
_~
,
,3
lU
-:
,-1
.
O
-M
,
i
..
_1
,
...-4
L
.I L;-4
-.C.-,,-4
*
~,
_
.,- ,.-
t.LO
h~,
Q
-'-
OU
Z
Z 0-
+ O _
¢Y0
+_
I
L
7 7w
u
Z Z LLI(
0C L
YI
,
:9JF~;&
_J
L q-~ ,:
.,~
4
)
4 -
_+
-
& G;
OU + C
I
CL
-,
-4
C-
C
(Xi
-L
,,- 1~ '
Z
U
+
J
t
.-
U
C.
_+
CL LQ
4
J
- s
G
Y OI
aI
-4
3 LA.-- J X LtYJ -3J
LA
LL
0 () t-
'
u Cy C-
n~~~~~~~~~~~~-
C)
C)
Lr
2' 2
-4
u
uz
Ut
C(
Li
Ci C
O
h C,
I
C I
-.
F-
0,
IS
C uN
-
C
Y-
r
Z
CC
-
Z 2V
cs
*-
4
.-
a L" a4.Un-- 4LL
v
'-
-
CL Li
(y O I
Z
-
H Ca
o U
*a
F 1U
1
I
C
-4
CU
X
f
U
U X 4
X
X
(
U
Y Lk
U
u I
LL
y
C
U u-t " F-.- x
S
(2(4(
YI
I- I
'<
CU u
a-.
IY
X
U
av C D GI
U F
J I.
-sC O LY
2 IULL
4I
CL
Ci J C LLY
or C.)
Y C.
-
4Y
X
I
-- Y
.O
i
L {D
3
Um
F-
0 Ufl
O-.
O
tO ~-~X
,-%,WZ u u
m
; n '- Ml'-U ,;, LO
N" ) u-',
y* LO
Nu 4
_t
-WT* t F L0 :3
y* L0
1.t Yn
I Li X Y
( X 3:i XN1 *TS dC C. ~ I
C r
I"C Y,- Y t * C-¢ -t
IUICU
LL
~[
~
U
L f
-Y
-
C
W
6\; Y
(
\ W
I -(I -i"CLa0- 4
2N. Z
4- O.
- 0-JZ x;-C-G-44-Z
4-S
Z
ta it&
I H N LL <C
1 LL
m
N1 - 3 9
t>
I" ',~
-
+
.H ,r~r~
'~~
Z Li (XiZ Z LJ J Z Z Li..
U
+O +
+ 0
E
.-10
~, J~ 4.,~
( UL
LMtt V U
A")
U 0",- LM
I'L
_ 0a Y
Li 01
Y *3I 1L ( 3U
1 -3i * C
rt I
C
C
yO
0-4a U
Z
~-
*~*
-WLi
,-
_~
00
X0
I
Y-
an LA
tJ
F--
0
x
-.
X
3-0
[
0a
0
HO
~
I
-
0
CL
-MIs
~ Ou
- ~
F
y
.* . '+. .+ . 4- U3
WW' L U
4
0O
~
2
I-
1
+ LW
_-'~
~
Ctj
ou
I-- I-
z
x
q
'-4
X
208
1--iC-4 m
n
cs
14 M
C0 Oo
m
t
\ b
N
cs
L D b am o O
i Cm
oo o o0
; U
n
m m m m m m It It -T -I -I -I -It -1 It -I n un n n Ln un U1
c C C C" " Cq Cq Cq m
\0 \D-O I'D
'O
0 %D\. \0 \ O %D\ \0 \1 D \0 O 'I \0 I' \1 'I D \. \1 %O%D s. \ %O\1 % \D0
1-
Lr
-ir-i
-i
-
-
-
11
-
)
-
L
r4r1ri
f
4rAI.
-
-
i
-
+
L1r
-ir
-
q
14
'-
-
+-4
0
+
LL
LL
:i
i.
0
U
U
I
I
-4
-1-
-
0O
U
+-
I
nr
-3
Li.
-'
'-4
z
a:
I
-4
I
ar
-
'-4
-3
2
(Y
z-'Z
LL
4
vj
u
L.L
U
-4
i.
-36-4--
"
-
Ht- -
U
'-- II
,
-- 4
N
cc
-4
r
uLi.u. .1~ +LL
,
-
C
1-4
C
H
'--LL
w. L
LA
iti2If
CA'-, j
LL7
z
z..1
z
0
U
--a
L
C
L
*
C,1t
-
-il
'#
11
'-
2
-
C
'
~<
ILL L
~-4-4C
LL.
O
+ JU
-} - 0 0
1
1,
-
-
J
· LiUt.-
L
'-4
L
l: Li u- C
[ [
O.u
C0 a: YI Y
I-
U
,.4
il _ _,,_-Q-
a^ cr
Yr 4 LU
3(IS II
i
-4
U HLiuUS
-
U' _ C
4 w
a:-
0
. O- L L
C)
U
U
)
- U
-
LL
Lk
11 if
I -L; '-4 Ci Lu ILL U--iC C
'3
~
O
J
C: 0u
c
LLf W tL
0.
i,- II,,. Ct
11 I.
'-4
LJ
-
.
-(Y U. U.
L. Li.
5-
U
LL. L
^
a:N
.Y
)IC;U OCI-T-- - + a
II
-J C) C - -4O _ _
C)
4O J
ULVU _- ,-4 C_ C"fC3 U U L) ',-
'0
aro 0
h
U
Ri-'}
UL L
tt:
Z 2Z t~
La Z
lA
C
UI - -
'-4
I +
.- cr
4 LrA-4 LA
4
X l.
L.
+
U U
I- U
14
"- S- 1-1
-% %r
L
Ck ff
L
Li L
a
4CJ
O
-4--
~U
U U.Uu
_ IIEC,
0
0
LLZ
3 a 0
-4
-J
O
('
I-.
'-
U U
<IZ
0
'- -~',U
'Z' )
zt
4
Z'J
a:2
O C, l
OrtLt.L
-LL
L- r- 4 x
",_
I- C.
(.U
00
-4
L
0,
XI
-J
I
'-i'-4
W---.;z-
i -II
+
-4
<LLC Li
FL
'- C.
l'
C
-
* ft
+
N--
+a
ar.1
-4-4.4
'-4<+ -, _i
z
0
0
X+ +-I
N
u
0
U
0
-. 1
-. '-4+
+
U-.)
4' ,j"
U
0V
N
i-+
n
o
am
'-4
cr-
-
U.
W.
C:
<
-
b
I
-,'3--0L14+
- --+4-
LL
LLI
Li
I
IL
-
2
0
l,
-3
+-i*
w--
-
a
+
-
-
+
'-4
a.
- )
-1
-10U
0 Ci C.)
0I
209
-
o
Wo m
u)
Ul)
n
D I'D % D
,-~- q i
-- cj m -o i
x.o r, w
o
CN m n
,o r- w
o
-I N m -, Un %. r- oo
o
i C,
%D%. qD I' %D'I O 'IO 0 I' r r " t- rr
- - oo oo
oo oO oo oo oo oo mo m o
%.
£ %D0D . \D I' \D 1s \. 'I \D D %DI' \D %DI %D\C> sD~o \@ .
D
\\
'I
'I
10
£
\1D
£
kD
1
i-4 O0HicO.I- i - - -iri- 4 - -i- 4r OiIr- i-
~-
LL
U
I
s
-W
v
UL
IL
w
U
N
a_
S
C.
+
+~
0
0
cu-
x
LL
LL
r0+
0
a:
-4
'-
'-
r
a
z
u
z
*
0
X
U-
0
+
-4
-3
-
'-4
z
0
£
CU
cu
a
U
0
q
I--
G
U
0
3
U
&
0-
U
I
LA..
- <,
U L
UC1L
J *L
0
Cr
-
LL~~~L
-)
'
_
0C 4
It:
y CL
Yr y
-t
La.~~~~~L
l u
C))
4t) A-
CW) )
(V(W.
Z I
) -
O
0
__Z
%
-
L
UUU
II S
-3-)-)
NL
LI,
LA)
j
:.
_
IL La. La
l, U u
--
1
'-
2
U
LC)
_ L1,5 -
IC, C;
-4 ---
:
(1)
G \L) Ck
U
^-
z
0
4
-t-IF
U Z
C')
-
II
U -
C) -4
0: Li n
4
Cu
0
'
I.,
.
-
O
>.--. Q2
sU
>
_
N
l'-
IC-) >iiiUU)C N
s
UG'
CuFU ,'Cu1 8Ct
O)
N
- 2
-,
1LL F -
}
I
--
0C 0: L _s4X
L Z
C) C)
)
L) ) C' Ut]L
0
)
(Y7
lo.0
m
It*
100 00
I -- 4
-
HaE U
'N
-U;
(1.,
(7
'j
L)
L)U
3 > C:7
Cu
00
4
J
-
1
Y
&)LO
LmN
(G)
&&.
0U
L-
LL. C
U
-
CE N
0
CL'-4
Ic
r 01
I
'-':a-,
U'
)-
i
y cc X
z:
(u --I
"'
> C
6;
(U
(N
·
C')
r)
3o'
oi'
oc
oc
t
- . 1I
w
W
m- L'
>
"i
C'
(y)
U
-
-
L.
2£L.I
LOn
o0
.- 4
8
U2:
L"
_
_ ,,4'|
V
r
][ : UC
i: C8-
b-'
1
o
cr
I-.
U
11 U
rua m
Z-4
a-
U
U
C)
-
4
N
)
'-4
Z
ql--
,x L
00
* I-
1:
-
-
0x
LQ
=
`> (_
CS-
) .4
0c(, no
-4_-
-
210
m -It
00
00 000
0 cr%
ma
m
0 C,,
0aa%
m
On 0'~O~
a
0
r-It
,--"
0
r4
r-i"i
r'
r~
-4r-q
,"
,'
r-q"i
-Ir-I
--eq
",e4q q
q q- qeq4
"
II
a:
W
-x
cu
A
Cu
C']
Cu
+
+
0
3
v
)C
.4
x
Mu
'-4
A
Z
Cu
7
0
0U
EL.
7~N
cu
U
(I
Z~-4
R.
LOC,
Cu
Z
O
U
c-3
0I.-
!
*
-4
Uj
-0
cT)
O0L
.4
IL
IL
LU
C)
IL
Lid-
0U
0
U
u
+
X L!
-4
II
-
II
-:
(V 4 ( ()
---
C",4 '
.
C) (
cr) "V(
0,.a ,-i
0' rU
i-
*
'-1
4- (\.
I
a-~.:-
',' R
')-
.U
1U--
Il>F- Cu
LO
(I
U
L-_h
,-
cx
I C'- , i :F
r-
4
LVJ
1IK
XI
.AI
40
-,
'.l
Ii II
N
. 5-
C) -
LU-:
.4
It > 11 r
IfL
L
ok o
b_-
0
5.
kf,
(.
1 --
LJIi0 I
L'.Lf
a,)
LC
OC oc
00
oX
Cu
7"
>
7 '- !-.4
-u C
- ? iX.; , 0
I--
U
01.
> l-
(U wL0
i
-
-4 i)
0. cL3 F--
LO
F-
I
3"
)(
-
:L
4c
.-- wafa
Y')-ICIL
x OC
u 'I0
I--
Li{% f',vW > -'
ifj
4
#
x
tJL.
-3
araC
au
L' C( C' (
o.
>Z
IIuo,yli
XZ
-Z
- '>-4
'-4
-J C) LL~- LL
M-
-'2~Nfr
0,
0 F-
I
L O
I
IV
N
-3
C
0.,
C
4-
a LLI
0
C)
LL
-j
I-
Ln
(%J
21
O
LAn
fl
'--
'-4
a
af
Ni
7-
:\V _.
211
o
ON
C n
1-
"
m
n
un
r
o 01
1- cq m
N U- I' ro
m m m m m -. -I -1 'r -r
-z -t t -, 't
- rl r-
't
'
o
r- I_ r- r r
rcmq14I-- q - Il4- 1- r_ rl-qcm1-
rI-A
- -
- rl
1- r-o r
14 c4 m It
e un
n
i
n
I'D
00
r-
n
L
0
n
Ln
-i
q
m
It
£, £. D
n
r--i-cmoi14
rl 1- r"-f- l r"_ 1,_ !"-i. r_ r-i r_~11
r-, Il_ r_ 1_I1.P
CU
o
0
uz
cU RU
._i-
-)
)
I
-- M4
.
*UvIC
C
' *)
- -
*
*
I-O IO
3
.
I,-- I,.
CU
Cc!
oc
(u
'-4
a~u
r~
)
,
--
C
- LAI
(U
A
C
00
02
... _,~
-4
N
'I==,
i-
(U
.20~
tl
¢,41
cu
U
DOli L(n-"'
cuCuCuOd (k 00 C,J 0
.t.'.')~J
- J
I--I
I
*
4fU
*
"_
<
O
-4-#
4_
I-i
£
cmi
4)
--_
-)
-Zj
-U)
H_
o~
LO II , (I-4" ..
-r4 L)_-")
GC
O
I
,.cr'''L
P_
.-#*-U
,
7"b3t
rrC,
Cu
G
L ~
=- z40
.,-- 0
a'
"'
C
C--
*G__
O, l
C,
C
* * 4tCD e=
It
· : ¢M
*. _ I*''.> Z l-zI
-4 -fl,
:'~,"Z
D
_.l
X
W. -M
o3 -7 U-
'I ,Xex
a
w-8*
Z 11-C
-
4 .4i
,40O
(IJ (U
- z I -')
(U
* I
I---
i u
Ou_
u
# C;O
-4-4 t--
I-I
X
1Q
- -L,
S' m
C
C,- Y_
(
I S
"
'-
X
U .-
11
cm
-
_4
N'
.
1 Z
=Cur
EL oc (-2¢IF-- 1--u UuQ_ 0
:DLI) "C_
nC0C
U) ) (1.1-Iq<
-A5
)
s4u44
.-.--.
-oo'2
SLA0
---0
on
'O D
- 44Z4Z
--4 , 4 (IC
C .-"7
U
, 'I I ,LI')
-.- ~-,.
2-4
*
F-'-I
X4)zA
-'0
Y FNtx57
xa W 0 wj Zk-4 Xi
~Zrrx
ca
1
DXfl-JC
LLO0
LLJC
d
a a- · Iq¥LLIZ
-4 C
v-i ~
~ )c~ -- ~i a ao
,;,1_6
6h r,
NO
C'U
I..If *I
4 40
* cm
) U
ouZx
(U.C
Z
.-
I-
I
r\l~ O
'LLL
I--
CL LI)
'-4
oUod
0 If,
~' tE. Oj CH
00'a-4
.,- c
I (.-J 11 lC<
(XOOJYU
ON"X
O
It'- UH
_~
. I
n.
CtC- U _L_
U GTi '4 4
1 Zr
L rr
Ln I4t 4
m.
.",
: .~'
-4-'
< -z
..-,~2r-'C
4I0n
I- .O.
"~
-) .
110- h I
,-I U)
Cm 71 Cr-...
2I'--4
z. u '",II <14t
OL
I_ L
a
[:
CU cu
ItCIL
_
-U)
OdZ
I -U.
"
0')
-4
0 ,
r
r".
I-
.#
C~
,,,
~
20 ¢
-.
OL
A
~
zra.
cmcm
'-
.-4
--4LL
U
U
O
,.C
212
iu %
CX:
~o
H cO m-t
00
4
o -
c
~D
r N
-
0 m0
0nO
n D
,-4
'-4
a
a.
*
a.
x
U
.
I
ou LA.LO
+I-0
[1I--011)
-
0
L
' N a.C 4 N-X
ii ;n
4 zn
is a
x C
s
a
a-4
,,
<~-k
': x cu
1', (_. u 4,toc,
L-_ A--
O -a -a L,.
Ul
0 L.
U U -
wI.-4
;
R.
.
.
S {J '
<
L.
Li
ri CU
(.
-4
LI L
q
LO Z
I.D
oo
0"
00oh
Oh
,-I--
,
-
t,")
~. ,,
'llJ
4X
X
J
D -a
.N. L -,tL IE
C'JLL
:'
(-'
-
)"- C:1 X
-
I
a- U
LI
Cu C UU
Ia
L-1
6o
S U
-L a . 7
LZ - LL
{; U
N a
C0 4-J
.4
LL
" U
3[
J_
0^; -. X
-
CD
Lai CL
,,- X Z
') (. CUL
,,.-'
-
a
I
', LI CU >,4 _ 24 U
< aCY E- 4~ . a 4
c a . a
IS
.LL
II II
Cu
_' Ts ,,
-
'..
0O
0 4,
(')
(",: I,'
U.- ,J "
0,.
',,-
-
I[
Co
I 1 1 C
s-
II
*', T'
r,,
oo
l 0"
t,
Y-E
II
c-
Cr)
LI, '- CL 0-
SIC
(k, ¢'- gst 11in~, JU "--U'
I -L
Ov5
a
SI
9Ln _.
,--
_ D (NLLC
L, :X
*
.,t
4
I..
_J
* -C
LL -
'-
,-'-I---4
a5 a) at I
-,r
I
C
0D
-
a--aUa
-
CL-
*_-
-
Lo
L,.
cu-
-I-
<4
lXLf'
+<
+ CC
I
La C^,
I
X-J
I
I.iI -s
X
-J
-
.,- (VI
X14
: C~
'-a U
a
*X
* +
,.i
'--4_ LL
a- u*L_X
-X< (
N
_A-
&.
-nI
)G
IL Ia i:
4
:LL
-I
-r
LO
fX4n la
C)
4
.
-
-
+
0..
'
·
4
t
x: cv, a.^
L) . U I,,:1- m r frn
_ XX
.XX I-XO _ X I a_. M
-N -0
c
N -. (. a~ i,
-
+_
-
1-
a-%.
41-
X
-
I
-
(U
CU
n0 I
,
A
i-
inE N
U, (OU L) -aL
,
W,
NU IN
Cu l
UDI-.
-
a
M:
C
I< -L,JA
La.,
I- "- a
CLA- U
GU ,,- l)
S If:
fr) M
OQ 'A
213
c m
oo ooo o0 H C
o oo
ooo o ooCo
o oo
oouoooo
oo oo
oo
o oo o
o
n
I-4
1-
i
1.
1-4
"I
ra__
aN_
J
LJ
X
x
'.0
LA
N
I
a-
a-
a-
Ol
OA
Cu
O
-
J
0
N
:LX
* II
Iz
OUL
LLi
D.
LO
7ru O tL
(f
I_
x
I
-- LL
Cu
<
X
IX'I
-
_
wa
. 00
1~ if:
D<: Ir
u LLI4
Li L - Li
3 L.
Cu G 1s (ki
It
Ui
U
sIt C4
&
LO
X
(1
ckQ
(
O
'.L0-T,
C:U)
7
6 I0
:tOCOC
#s
a.
CL
*_ U
4
I
a.
'-4-
L(- L1
atu
I. IJ
X
irZ
X.
_
&L
'.(,'0
L
EC_
-_
O
j
UL
_
u * <
JA
LLu _D
· "x
0,
a
'-
=
:
0-4
)C11
"I
I
4I :
l r-, , 0,,
t
00
.)La.?_
L''
GLAU
F O: (tlrCF
O30~
(
(T.
--
S
F .S
- X X
L
7O -
F
2.
3 ;
C
x- tV
7-t I O
0I C'. ; ,
(:r
a:
Gs
O.4 CI
OfuO
7L(
"iin
tO
(n
r
,t
--
z
I -
-4
-
X X iLI01
a
_ _ x
- U Cr LUCJ
_
-
uA
v
,--i
C c. (_
ram
c CD
tj
-
l; o-3 0I
[.E I
{a. _
- CuI
.J j
0
LI) Cu La.
-
L Z L Z
o cu 0 Li 0 CU O
- C U
IiQCU UU ,'-~
L
21
_
4
L
_D _
i. al
_
0 O0t
CO'
.JA. JL
4 r_
I,,
4 -4
0UO
-
-4·
Q-.
fU U#
S
-'. < 4u LAOa6; r.
bZ
:.- ,Li Z I-C£
i-NCu
Y -JU
i
0 La.
) fr t- L <
F
-' IN
L-
O
o
L
-J
4
L
U
I.
II
tU
00
LA
-
R
0
C04ID
CufU
is
14
LL
_-
I-4
O
U)
s
i 3
Li
-
L
jr7
1-4
"-
r'
ri
'D
,
-b
'I- C 0)
I ~3'L
.- ; - I
-
,'_
LL
LL
.-
i
r-4
0 c ) Ic
I-
V-4
6- I
T 2.,~
,
1-4
'
'
t
I
'-
--
'D
--
4
;'I
214
r-
W M o
r-i 04 M -Z L \0 r- W ON o
r- C M -,t n D r -
M
r-I "
M -t
n
D r- W M 0 r-I "
(U
uI---C
0-
C -
0-
N.
'4
· 0
4-
*
-)
U
_
or)
m.7.
U
.
U)+
0U) -
4'~Z
_-
-_
L.
UL
'--
'C
+
~~~4
'-4
U
30-
4
-4=O
'-4":t
in-'-4
J
_k
Q ._
NI
.*
a-
*
4 O
_C
_
*0O
a:,u.
4-
U -
A .4
A
.4
(y_
(
~+
_
'-4
3
* Lr,
f0E
3
-4
4
_
am_ A
-N
C
* *
0
0-)
Ri
+
-U
a:-
-
+4
U
-3
-4*
_
X
t_
4+
z.-
4
Z4
O.4
i
t_
xRiI-- T- o
1-
-
wwa-a:
m-a:*
4 X
*
-4 C-
a
-4
3b
tt_
*v4
LU -+
14t J5
uCUjI**
-4
II1 i
3u 3I
0
,4
-4
-'-,-^
r.
-3
,
-
H(N11
>->.44-
zz
xUu u 0
u
* ".:* U_4f
3 45
3 .U
o-.
4
Z
u
-4
IC) fC
xI f
Ri4
i-
_
i-- II4
iflX
'-_
<r
"'-43
ci ,,,)L.
t
il C f.
~,
U)'-
It- It fl- Z-:E: D
'u4-I--)L
--- *r bJ:
Ri
* N
('-'I PI-N
C' - X
H
N
4 I-,
nE-)
- >
*->4
z Z._ -
II - H
II (A-;
Nr
_
4
'.
"
f
-
N -fa- D iDg*
3 (7) 4 Z rfn
'..4
'-4U
ZaL
:0 <: ·i -- N
'CJX
-4
Cu
< IN '-
p-
C
0-I
oC
30
N
i-
/it
. i.- .I- z
(XI
(.o ) -4I Ir w )Y .-.)
F--
j -
UN
fr, -
I
U
# XX
_ .<
Z- 0'-4iZ0 -24
.i
,
i
-
I~ .:
1l 4
II
II tl x
11 li
,
O4
Ri U:1
-4
3NU@
_
L~.J
C- '-4
''-,i-
* * I- I - -U -
cm*F
a
·,~
,,'.
*t4N
D-
* *
ff
Cu
,JN
3
40
_0*S
,-_4
a. N
ofn. Cu
.N
-3
X
,_4
I---
U,
t
U)U)
*
'-4
C-)
-4
cL
*
u-uj
U) J +oNI)7C(0
UU
4-
&-4
nL-
'.4
'-4
u+ Ui
-j
-j
-a
* -
3.3
Ri
·t n*
OJ
-N
_
_ t
NLo
('3M
'-4o r}
:iat
*
C'"
:
--I
< Ci ::
V- -,
l U
-A 0
<
- (' i-
-
,
NU 4 Z
P
-e2
I-
<
-_ u L Z
ZU
u Li 3 0 0 U LJ
>U ,
r--
s
,
N
N in
Ns N
N
.4
'4
w-4
'4
0.1
4
RI
U
4
N
L(;
Ri'
-
30
N
'4
'.
%C
NNf
'.
RI R
s
RfIx
215
M t L<n
\o
0
0r- o 0 rq " M -It un1.
"C-
0 o0
M
0- 0- 0, oo. 0-.oo. oo~0 0 0 w w w 0 0 0 w 0 0 coY
Ln%o
MM
" C 0t U
n
00
0 0 0 0 aa0 0 0
O00
aoa
0
A
cu
:3
(U
·4
14
cu
+
N
W
Z-
3L
zUu
LL
U
nJ
a
It
4
ru
IL
Cu
'
CU
4
.
*
*
0
3-
M
*
n,_.
.4
C1.U
Z.U
_
Cu M
0*T
-.4
M
C'
rY
IL
.4
Z u
Z('-
00
0 4
-0
CU
5-
-CU- I
*
40
D
LJ
U)}
a
-
4Cu
* -.CU
,-~
UJ(%i
0
r,
I"a)
C) 2300
o
e
&
ZU
.4
Z
4N
)OC
Q
C%
cu
CU)
4.
*
Cu
Ix)
a.
*CU+
u.4
) u.
.4
--
Z
_
-* * CU
4*XX
'-I C) CU
-44(
.4
v IC
F
Ww-Iu
£2>
Zw4(U
oO0U
(! Ui 4.
._
In MIr
-4
rU
CU
.~q
M~tu
u Mn- ,r4,~m I-u:z
~Z
_l
Z.~
" 0.M
-* CU
- l+ U
- G LU
Cl -,of 4 r>L
.~__Li Z
·
~
~, .,
L (JCU
,,,J
CU C c
MI X X
£0
3 33t
.4
M ru
Ci
Z
<
0. U
> Ci
I S N 11%
if
-
Z U CL
<
.>
",0 4.1
00
O>
>- £
r'OC
XImC2Orla
_t
o4
N
-4
4E
!*
* * M+
4 L
-- M
4- a. >
44
LcrL* *i
-.
-U
S<;t
LL.
Li
C
I
M
I
00 Cw)
Z
l Z (% -U0.--UV000
M M
n>
-- :
.
0
O 0 LA A 0 Y4Mc4
* - £ C)n >
O-)> *
0
4
ZZ
UT
*.4.4
* .4 (14
_* *>-)-
-3-
D
.,4(.4
Cu
+ +
U
.
ZO--) ...
s
W
>
CuU II UL, UN
C Cu
9p
-
UU
I-
+U 4
I.Ai or
-'-.
_L
I -)
Ctr
',=[
>
*
cu
LSi
.3 * -) * fs * - 4
-
.- CU
0O
Au + M' 4
N
-3
WI UM
C
tU)
L(')C
ZLC-,~ 4
Z
4 MU-
C)
P
_I-
-0 *cu M
.40
CV
-3
z
_-.
.4.
M
I-CU
4! .X
_
11-
DO
-4.
1L
Du
1a
Ck
.4
CU
G
4
(14
0
216
cr\ o
c
m -I
L .o
- w
rl
m 0
N Cn
-
Ln
I'D
r- w m
N m -I
rl
Ln
%.D
r- w m
C
O r
-1
I
n _L
n- CL
N
N
-4
N 0
N
N
-3
,_t
y0
I-
-a
I:*>
Z-
cu-t
Cu*
--
'-4
C
x0t om
*
3
'4
Li.
LL
LA.
N
4
A
(U O +
- .- -
Z I: (5, ~-.
u'-3t-"*
-3
4
o0Cu
0
w
bJ
'-4
0
LL IJ
Z) N -- c z (n h I T
u F
Y
-
0
0r # .kI
a
*9
w-4L
a-l
O5O ,
CU
CL
,~JCU
-4
U
ON
a u
-3
i-- x
I 2:
x t,-
N
G
*
.4
O0
4
ax
1·
3
4
-4 It
* . I- *
U S, - Z '-4
'-.r 3
'-4
-3-
I-
3*>C.-_j
*-4
''
a J
-'
0-
-4>
fr-
a. - <
'-44 Z
ZZ
Z
Z
<
_ D-} .. -4 XX
O
+3-_.
.4c.< 4-mo
-4X
NXU
11
I(
+ (N- X
'-4 3' _-L.)
tY
a Cu
00 -*
U
,.- -4
--) -Oc4
C Cu CuF
7-4-m3
-3
m
+
-4
O0 _*- z
C *
3-4J
0*l F-Lu*3+
+
X
u
X
4 > * >a(
_--u C
u '-4 1 C ub-C u - aU i- I-Lw
L u ? Lw Z
C) 0
1 0 3 0 I- 4
LO
G'" O
V 'C
C '")> >
u I U
PZ
0
CU CI..-m*0
3^
l ***m
o
-3
.
a -- m)-& x1
U wzL-J
07
< ZO 4
* >-4
0
* *
_ tU '
t-_.
a -4
-.44d
.cu<[
a
at--i
.4
t -4
+
)-
II II If"U>
0u ,u E-a CQCu. UI I U
N-
IJ
I4-
rl axo
*N--.3
0,--3
I4t
<3t
0-
4'.
-3,
-4
'-4
CD
M
.* 3
O
N
CD-
a
4
nt
Cu
t4
I
-3
'-4 a-
tU
a-
o
LL
c
-4
*7-.
-
* U
-
3
--
X-
a
'-4
-4
CU
*
*
CD
C,)
-00d3
;-)
-
M
0*
a-*
I.
i', N4
-4
a
7
_.l
0-
cr
EC,
I-a - .tcf
C
u
CD U
mfl
3
.L
LL-4
L.
_§
O~,C , ..
a_
U
:r
OC
ru
(U
Cu
-3
*
*
):*
Z Z '?,4 x m
44
-
Xx oU4 - 4
'-4
_. u_ 0.
P, -
)t<CJO )
044
ZZv- -4
<
.4)
o'h4
-
<:
3
+
. ,
x
LI? F- I N 0
00
a%
ro~
-4
-U
Cu
N
217
un
I'
r- w om o r
c
m -< un %Dr- w am o
cn
r-
": Ln D r, w
a cm o
-
cq
X
Ln
m
". r
o o
a, czt-, a - min
an
nt-I-4"-iro%o
o,
%p
or-a r-i --iO"A-rr-q
a C%rra %-i%cO( oO0
Y% mr-Ir-q
rI a- n- %-C0 r4
r-no%rq
14
-o rI- Nori6a -4r-IrO anL
m -Ar-Ir%
O %o
%
a oN%o
C
m ri
a %14
CY - r-r-%%A
0m
A
n
C
.e
>
_
Lu
*
.4
u*
I-.
1.'4
U
U
-L
.4
:
M
IA-
'.4
w
U
.4
3
CUj
CU
a
MC >
t
4Z
Z
4
4X
U
-
U,i
~-
v
Sa
I.
.4
cu
LU
I:
1
5r# +
-
I-
'-4'.
.0 cu
U
X
CU
_+
-O
%(
W
-
X
O Jr>
-a
I- U - -*4
(U
tn
CUI:
X
+ m(UI
-4
I0UO
C')
-4 # X14
,,-~ +I
0
Ifr) v +
-% U '-4
-k
IU & .U-%4U C.
~V > Si X
>>
U -. >11.4
0 :1 (U X
Cq-4
* S
'.4
COX
N t U S NI - L
I-- I
I 0-.s
Si- S
+2 ISi + IwU w-.
CXb I HO __
O
Cu
CN
MU
U)
.
U
UC q.4
U4(uw>
>S
oy> TG
p cn
<
0Y Z4
C :ve u
14
V
'
-U
i
Y CU CU4-Y
.
(
Y,
'-4 ACU0
CU CU X 0
'-4 CU (U J CU C')cu N C'
U ' 5 4 X
4a
UU
U
IC
OU
II§
44
l, mu
IC
(O
) ( kI) CU U '- (U 4
> 0
> 0 > > 0> >
.4 oui 4 40Li' .40 Ui44
44-
(U
-4
~Zl
'-4
.4
Cu
U
[I.
zz
T. in
m
1
Cu
'.4
CUi Cu
4-
(U
Cu
0
. 1-
I
* -y Z < .)
z zs
U
w
(I
_
'-4
Q
OM- CL{<
-Z t-
0 Cu
U
'--4
P')
I
0UJ
II-
0
--(U - Z4
Ju ('4(' .- JI >
*-0C(U
Y _L_.
~~~~CDO
,,_1 Ov U __
a (U
>
-4
I
U
_J
+
-4
'.-0
X
-UN
('4Wu
-A >
.44<
~-0
-4
0J
0trU
0
- cu
4
LL
Cu
Li
U.
U
.4
-
Li
4
m I-41
Z
Li
U
.4
X
U.
P.-
LL
U
U
Li
a
cu*
U
U-
0
->
-'
I-
-
0
-
-
Cu
(U
Cuj
Cuj
tUn:
Cuj
cu-
I-
0
-0
4
0
U
0-
z
> 0
U
F- 1-0 04
>
<
G
Cu
IU
cu
cu
ru
4
u
-j
218
0 C
O0 O0
00
O
q Co
" M On
0 0 O
00
MO0
~
-I
O ~O 'O 'O~
~ ~O
a ~ ~O O' 0O
O ~00
O0
0
0O0
0
OH
O Hr C4 M f
Ln
0 0 000000
'A
'~
I
a-
-- 4.
-CU
I
>
--
I
> ,_
P-,
cu
-4
)
c(
4
'-4
(u
LL
IL
I-
0
'--*
-4
-~
-
-
1-
4-(I LI
a 3Li
M.
,-4
· '- . 4
,*
Clj
.+ C-"
- -G
- C
- (5
> t5'-4000
O
cU
u
N;'-4
U J'-I
'-44
LiI * *
Ui CU 4 a
U
2
I-
i-4U
4-N(U
,~t- u u -oI,"O
.Y N
CJi
CU to4 N Er Z-'-...
Z
C4
4.
(3 '-4
,,-
*1
>
-'-
+
4(51
>
0Z>
C
oU
4.
4.
(U
CM
NU
U)
4-
4 _X>
x
Z.LLI
*
N
ou
-2
F,j,
: '--4
M x
-('3-
Z 0
xaU)LIPxa L- L
J
~
f-4
X
3-4
X XX4
X
0_Z_01
X U X)(U F44_-4
xx
X-4X-F-
C. - OX
(U4t
0U'-
LO)
NU
'-4
1-4
1:
I-
Z -
-b
0
-
C.-.-
(U
w
+
x -
'~rCU X
2. .4 X22
,,~a
X
X
'.-
.4
1r
x
x-,O
0
(U
0UlX.
. 'n- 0,-4
LO it
x> x
"-4
x
LO %O
_
aN
x
ruM'
z I- Z
z0 z w
,,-41
£CUlIJJ N -
O I('3
_
a
x AU
M - M M
4*
4 0U-I : I -
9-
(u
(S3
N
> 0>
-40 LI 4
I- 0~cc
>_
'-4
.44-
I l' II--FI
ILL
:> .-.
-JtV :E3 > (U
It
S (
s- 11s*
C>0
-- cu
UU x
xx
CQ
_
<
4
-- 4
U
0
0
M
uLu
(U
u .4-CU
o
U
(U
u
_Z Z
-)
.4
_a
_
U
-
'
O
0
L(U CU U
_
5--
-
>1
_J
J
IuN < -4-
4
*0,
c Ur,>
#
z (U*- 4 Ca
-4
+
Cv)
1,
-.3
fu
(u
>
'-4
0> II U
Li,0 LU
1+- Is
50
---
-3
+
CU
.
aQfL
I-
I-
<
I--
U
_
t.4
R
-.4.
_-
-
'-41 ^ IMIM
L) 0.71 -
-
'JI-0,J
P09J
- C
N CU
C
Lk
iRI
Nn
-l
cu
00
U
O.
Co
c
o4
Nb
:0
219
Iq
Cq o'
ci m
q
ur
n'
ol o c c'h m J
C' C
a oN
Nc m ID
C nI'
Ch
o\ O
I N
eI
H
H
u
aL
:
a
A~~
3
_
I
-'
A
x
*
0
,-CU
.
0-
'0LU
a(
xz
'-
M
4
~ ~ ~L.
Cu
xu
X
UL)
-
x
N
'-4
LL
L
N4a
#
a
Ll
-4
LL
I
Y-.
E
--I
'-4
0
X ~
X
A
n0
N
Cli U
e~~Z
ew
x.4 x4
T-U(D
*XX4
* r
Z Z -4 003
ZZ
-)(X
_UXW
-_~
xx
-- -U X
-3
O
uUU
W) *-I<
z00
X X
Z
Z -z
Z
"
X
--3 00 -
X
u UC;n
N.J'J-4
Z
CM- X
-4
X ,-^,' _
i, U
u.-4 -4
.- 4O-On
-X
4 -...
a
<
z
x
- I- *~IZ
"-"X
Oc O
X U)C
0104c
'-.-
C uJ
Yu- Z Z I-
C.
_
.
o w J.,
Cu
0`
,
Cu
Cu
m
a)
cu
-3
a)
Cu
L
-r
CU
)
Z
Z
-4
.-')
I
4
4 <
Ii Y
Cr X CU
IXIt x x
Ml x4f
iL~.
&
Z P-Q
La_ a.
I- 9-
'-4
C0
~
&I
C')
u
m
.Ru
B_
xx
-
<
C0.-4000
LU
LLI
) L (Tu Nm)
c li
u
_.
X X
WLAJ
r -I--
-
-U ul.
"
CU
a~OO2
Lt
a
Y_ I-
ua3
- -
Z
'
I- I- f-
33 3
2_
I
L
C 0 0 - o I- - Ya
Z *-' M
-'
0 :_ 0 - U U X 0 -4 r C '-4
U _~cLi X i- I- 3 L LL
o -U
~M
. _
-
xx
Z_
U O
3 (D
3
-
.-
Gl4 Lit
X
_n
N
I4,. ,
1 I'-u N
%N
I- ^
'-4
4 4 O
]E X~
Z 1.Z '0
OJ N
.. FCJ
i N X
JXi
IiX-
_i
I <II uC
ZI
:3
-3CL
0S
I OC
uu
0
ux yX C
4>
Cu
r
OO'- 0-
Y-O
-4
4 Z0Z
zz
+01
O
N
x-X
")M OU
z
-
J-I
I--1--4
z
J
4
'-3'-3X
+ +
C G
i
>
4
xa
-4 Cu
UU-4 -4
-
zU--z
z
+
SI
-4
- CU
OU
0CaCu
m'
219.1
n 000000
nS CS
C
0
0
C
00000
000 C
C
C
Cs CS C
C>
0
C C
00000000000000000000
0C
S 0 Ca S CS c) 0 CS 0
0
zZ.4
:S.
L
A
o]
4
_
1A
~
U
z
oJ
Li
C
G
L
LU
-
L
Lii
(I)
oc~
X
X
C, 4cu
0
xen
·
-
OQ DC~C
v
LL
30
I!-
#
n
4
tu
--
N
4
<
I
N
en en
X
_
*
t-
I
'
-
*
,
LI
. .
'-4
-4
30
A
L'o
l.
U.
w
~J0+) -
,
I-J
0+1I-
I
X
.4
a_4
Z Z
I U
-1--I-
I- Cu 1,4
O O
Z
WOO,
WOUJ
,
M,
-4
-,0
04
u
a
o
0'_
.-
Cu-1
Lt '
0 0C)N ~--M
4 U
M
Z ..
= 4
...
-N X
l:Z
Z Z_ Z ^ Z
- E
w
a)
S
Z
C
X
J.
, N.,
C I-,-X; .' fI-,
', .
-.0-
·-,-m ,,.-
k
~
n -
4
L. - -I i-
Z D
S. -_
.
Cu
_ _,
0
_
_
t-00
S &
M
eIsM
(
u
ii
L
O
x
'-
0O 0. 0
r.
Me 0
IL
Li C 3 Li .-
e
~~~~C
cuI--
')D
0(
VI
w-4
4
&
O7%
Cr
-
M
F uW4 C' C i et
: C (k C)
s
Cu
(k\k,
l- I-C -
_-tu
-4
F
C lii ct Wi 4
IL.3
en
(j
XX
--
x LL
Li 4 ;- Li i
- b- < ¢r) £: - L,- "x- - .~._.:
t.:1
IU
1-4
Lit .I)
*_ O) a:1C
'I, Ct C
0
Ja~-^X<L
L<
-
-:l-
&i
:
F~~~~~~
I-
&
F
I'-Z
.4
I- Li. C
-
-4
__FI
CU
_ + :.4 ena)
IN '.n
Z
< -X Z
O Z
-:1-
:I O
t:Li
I.
I N .
41 MJ
Cu
x*
_
-j
F-4
U
U
M
- LJ
*
w
IJG-4 . 4
I- &
* en
I
W
(0
-
'-4
IL -0
0G +
1Ld
C)
OIL
)
C
+
L
-,
-
0
I-
al 1 + + I.Z
r: Z- ,J -4
-
LLI
O
'-4Z
L.
_
zJ
U
.--jji
-W LJ
x
J ~3u
LL..
zZ
U,
Lb
0
#--- -_ '
-0..,- C. .-I~,
_L
-Z.4-
:0
.I
'-4
Cu
Cu
N
CQ
cU Cu
Cu
eN
219.2
N
00 0000
a
o
0
0
C)
a
LL
_J
-J
.-4
C)
I,- >
4
LAJ
LJ
-jJ i
0
J
.-
'-0
~r U
1.
W
x U
Z
I-I)
X wb
N Z
- ,-, Z
auJ Z
C W
D I
O C t--
4
U]
*\j
(V
mk
r
Y)
(Y)~
APPENDIX
G
ONE-DIMENSIONAL SOLUTIONS FOR
TEMPERATURE AND CURRENT
The Temperature Solution
In Chapter 2 the equation which governs the
temperature distribution of the cable insulation was given
as
1
T +
r r
2T
r22
?
2
k
(2.1)
Considering now only the radial dependence of the
temperature, Equation 2.1 reduces to
1 dT +
rr
2T
dr2
k
(G.1)
and this may be consolidated to
rdr r TdrJ
1 d
k
(G.2)
Expressing the volumetric heating term q in terms of the
total dielectric loss per unit length Wd, the governing
equation G.2 becomes finally
220
221
d
r- dT
dr
dr
Wd
=
(G.3)
2rrk in
)
where r1 and r2 denote the inner and outer radii of the
The boundary condition at the
insulation, respectively.
conductor
is
dkr
qc = -2Trlk
dr
1
(G.4)
= W
c
1
where
W
is the conductor
loss per unit length.
As for the
boundary condition at the surface, consider for the present
that the surface is at some arbitrary uniform temperature:
T(r 2 ) = T
2
(G.5)
.
A dimensionless formulation may be obtained by introducing
the variables
r
T - Toi
6
(G.6)
W/k .
r2
where W is some arbitrary loss per unit length.
governing equation is then
The
222
d
dE (
where
Wd
d)
dE
27rW
(G.7)
n
1 = r/r 2 , and the boundary conditions are
W
dO
c
(G.8)
I
2-W~ 1
and
T
e (1)
0
=
-T
.
w/k
-
o
(G.9)
W/k'
The general solution to Equation G.7 is
Wd
6()
4rW in
(ln
)
2
+ Cln
(G.10)
+ C2
Substituting this solution into the boundary conditions G.9
and G.10, the arbitrary
constants
1
Cl= -2-W (Wd+W)
C 1 and C 2 are found to be
C2 =
(G.11)
00
Putting this result back into the governing equation gives
Wd
0(U~~
e1 = 47W
)
i
)2
n
(ln
(Wd+W
)
d
- ln
21rW
+
o
.
(G.12)
223
Equation
G.12 is then the temperature
distribution
for the
case in which the cable surface is at some arbitrary
temperature T .
0
However, this arbitrary temperature may now
be eliminated by applying an energy balance at the cable
surface:
Wc +Wd +Ws = 2r
where W
2 h(T
(G.13)
-Toil)
is the sheath loss per unit length.
The
nondimensional form of this is
W
+W
+Wd
=
W/k
2r
(G.14)
2 hO0
from which
(Wc+Wd+Ws) / k
(G.15)
hr
2 W
6o=
The one-dimensional temperature distribution for the cable
is therefore
Wd
4W=n
4W
n
2
1
(ln
)
(Wd+Wc)
-
2W
Wn
+
(Wc+Wd+Ws)
k
2TW
hr2
(G. 16)
224
In terms of dimensional temperatures,
T(~) - Toil
Wd
4rk ln
(ln E)
2
-
(Wc+Wd+Ws)
(Wd+Wc)
27rkin
+
1
2wr2 h
(G.17)
The Current Solution
In order to obtain the one-dimensional current
solution, the stationary and variable components of the
temperature distribution are again separated:
0(E) =
D(i) +
C(E) ,
(G.18)
where
Wd
OD(E)
4 W in
2
1
(in
)
W
n
Wd
Wd
in
l2W
+
k
(G.19)
and
(w +w )
eC(E) = - 2wW in E +
2rW
(G.20)
The current I is introduced into OC(E) following the
reasoning of Appendix C.
By analogy with Equation C.39,
2
ec(E) = ec ( )
I0
(G.21)
225
Co(E) is the distribution obtained by using the
where
arbitrary current I0 in Equation G.20.
0()
The total solution
is then
2
e(E)
+ OC0 ()
= OD(E)
(G.22)
2
I0
Again it is desired to have O(E) take on some maximum value
emax .
However, in the one-dimensional case, since there are
no heat sinks within the cable, 0max must occur at the
max
location
= 1 (at the conductor). Substituting this
information into Equation G.22 gives
2
emax =
D(g 1 )
+
C(
1)
.
(G.23)
0
Upon rearrangement, Equation G.22 yields the current
/I
\
=
0 max
e-D(1)
(G.24)
eco (
kIo
It is then only necessary to insert the appropriate values
for
D(C1 ) and
C (C1)
2
from Equations G.19 and G.20:
emax
Wd
-W
2
(k
hr2
In
-_
27TW
hr2
in E1
1)
2
I
20W
2)
+
(G.25)
so 2
kwWhr2
226
where Wco and Wso are the conductor and sheath losses,
respectively, produced by the arbitrary current Io .
This
expression may be simplified by using the relations
W
where Y
C
= I2Ry
and Y
S
W
=2 IR(Y
Y)
(G.26)
denote the AC/DC ratios at the conductor
and at the sheath, respectively.
Making these substitutions,
Equation G.25 becomes
(
md
max
2TrW
k
ln
hr2
2
)w
276
(--2)n(G.2
7)
2
2)
in terms
c hof
kFinally,
temperatures,
2~rW
r dimensional
12
W
F-
(~h
which may be further simplified to
2W
-
Wd h
nr
l _
(max
/Ysk
2=
R
RFr2-
n l)
2 )(G.28)
i=
2
(G.29)
Yi
Finally, in terms of dimensional temperatures,
2,Fk(T -T )_
max
oil
i2=
/Ysk
Wd~ ~ i
d
-2
1
2
~~~~~~~~~~~~(G.
Rhr2 Y
nE
29)
APPENDIX
H
CONSERVATIVE APPROXIMATE SOLUTIONS FOR
MAXIMUM TEMPERATURE AND CURRENT
General
Conservative approximations for maximum
temperature and current in the two-dimensional conduction
problem may be achieved from a suitable modification of the
one-dimensional solutions presented in Appendix G
The
conservative assumption to be employed is that cable-cable
and cable-conduit interactions effectively insulate
appropriate portions of the cable surface, thereby reducing
the perimeter available for heat transfer.
Furthermore, it
is assumed that the entire cable sector subtended by an
insulated arc on the perimeter is also effectively insulated.
Any losses which occur in the insulated sector are
then referred to the remaining undisturbed portion of the
cable.
Consider, for example, that a 60°-arc of the cable
perimeter is taken to be insulated.
for heat transfer
is then reduced
and all losses in the ()-cable
The perimeter available
to (5) its original
size,
must be scaled up by ()
in
order to have the same heat flow or temperature as in the
original problem.
The maximum current or temperature is
then computed from the one-dimensional solution, using (6)
of the original one-dimensional losses.
227
228
The Temperature Solution
In Appendix G the one-dimensional temperature
distribution was given as
2
Wd
4Trkin
oil
1 (n
(W +Wd+W )
2in
2r2
r+h
(Wd+W)
- 2k
)
(G.17)
The temperature drop from the conductor to the oil is then
T()
Toil
1~~~~
-
- Toil =
T
-
in 1
4Tk
W+Wd+W
)
(W
(Wd+2Wc) +
2 hr 2'
2~ihr
(H.1)
where T
0
denotes the conductor temperature.
Equation H1
that the temperature drop (To -T oil ) varies
Now let the cable perimeter
linearly with the cable losses.
available
It is noted in
for heat transfer
take on the value
P' = fP, where
P is the total perimeter, and f is some fraction.
The
losses in the undisturbed portion of the cable are then
scaled up according
drop
to q' = ()q.
(T -T oil) varies
o oil
up by (/f),
linearly
Since the temperature
with loss, it too is scaled
and the conservative expression is given by
=
T0-Toil)* =
(To-Toi
(o
i)
(T0 -Toil
where the temperature drop on the right side is that
(H.2)
229
produced by the one-dimensional solution.
Equation H.2 may
then be used to conservatively estimate either the maximum
allowable oil temperature or the maximum cable temperature
in the two-dimensional conduction problem.
The Current Solution
From Appendix G the one-dimensional current
solution
is
2Tk(T
2
i2=
-T
2 Trk(Tmax
)
W
_ in
k
1)
hr
oil
l~sk
=
Rth -
\
nY
n
Y
hr 2
c1
2
(G.29)
2
.
n 1
Again consider that the effective perimeter takes on the
value P' = fP, and that the losses are scaled up according
to q' = ()q.
Since
current-produced
losses vary as 12, the
latter quantity must itself be linearly scaled, along with
the dielectric loss,
Inserting this into Equation G.29
gives the result
2
I*
f
27Tk
(T
max -T
oil )
*
f
/Y k
R
1______1_111________----------.--_111
-d
(hr
k in
\~~~~h2
_
t(H.3)
JI
s-
c1
Y
in
----------
_
230
which may be rearranged to give
-T
)
27kf(T
max oil
2
I.~~Y
(
W k
d)h-r
2
_
2
i
n
41
(H.4)
/Ysk
hr2 -Y in i
Equation H4
is then the conservative approximation for
current.
The Effective Perimeter
The size of the inter-cable conduction path is a
reasonable guide in selecting the amount by which to reduce
the cable perimeter for cable-cable and cable-conduit
interactions.
This convention was followed in generating
the conservative comparisons tabulated in Chapter 6.
For
those 16 problems the inter-cable conduction path was
chosen so as to subtend an angle of 30° on either
surface.
cable
The following cable perimeters were therefore used
for the various configurations:
open - 330° effective;
cradled - 300° effective; equilateral - 270° effective; and
equilateral-pipe - 240 ° effective.
effective
perimeter
of 360 °
It is noted that for an
(f = 1), the one-dimensional
solutions are recovered in Equations H.2 and H.4.