NASA Tec hnica I Paper 2685

January 1987

NASA

Space Shuttle Main Engine High Pressure Fuel Pump Aft Platform Seal Cavity Flow Analysis

S. A. Lowry and L- w- Keemn

/NASA 'Technical Paper 12685

' 1987 I

I

National Aeronautics and Space Administration

Scientific and Technical Information Branch

Space Shuttle Main Engine High Pressure Fuel Pump Aft Platform Seal Cavity Flow Analysis

S . A. Lowry George C. Marshall Space Flight Center Marshall Space Flight Center, Alabama

L. W. Keeton CHAM (NA) Inc. Huntsville, Alabama

I ACKNOWLEDGMENTS

The authors would like to thank Jeff Ingram, EP23, NASA MSFC for his work in compiling the property curves in Appendix B. Special thanks also go to Gene Teal of Lockheed Inc., Huntsville, Alabama, for his assis- tance in providing the boundary conditions used in the model. The authors would also like to thank Loren Gross, EP23, NASA MSFC; Dwayne McCay, EP26, NASA MSFC; and Ashok Singhal and colleagues at CHAM (N.A.), Huntsville, Alabama, for their technical advice and support.

TABLE OF CONTENTS

I.

11.

111.

IV .

V.

VI.

VI1 .

Page

INTRODUCTION .................................................................................................. 1

PROBLEM DESCRIPTION ...................................................................................... 5

A. Inlets ............................................................................................................. 5 B. Exits .............................................................................................................. 7

NUMERICALMODELSETUP ................................................................................. 7

A. Assumptions/Model Details.. ................................................................................. 8

TWO-DIMENSIONALTESTRUNS ............................................................................ 10

A. Two-dimensional Test Runs: Boundary Conditions., ...................................................... 10 B . Two-dimensional Test Runs: Results and Observations ................................................... 12

19

THREE-DIMENSIONALTESTRUNS ......................................................................... 19

C. Convergence Characteristics and Computer Time.. ........................................................

A. Three-dimensional Test Runs: Boundary Conditions ...................................................... 20 B. Three-dimensional Test Runs: Results and Observations .................................................. 2 1

SUMMARY OFTHECURRENTTESTRUN RESULTS ANDOBSERVATIONS ...................... 49

CONCLUSIONS .................................................................................................... 50

REFERENCES ............................................................................................................. 51

APPENDIX A. PHOENICS SATELLITE ANDGROUND ADAPTATIONS .................................... 53

APPENDIX B. PROPERTYCURVE FITS .......................................................................... I17

APPENDIXC. CONVERGENCE CHARACTERISTICS ......................................................... 125

iii

LIST OF ILLUSTRATIONS

Figure

1 .

2 .

3 .

4 .

5 .

6 .

7 .

8 .

9 .

I O .

1 1 .

12 .

13 .

14 .

15 .

16 .

17 .

18 .

19 .

20 .

21 .

22 .

23 .

24 .

Title

The Space Shuttle Main Engine (SSME) .................................................................

SSME High Pressure Fuel Turbopump (HPFTP) ...........................................................

HPFTP turbine flow paths .....................................................................................

Aft-platform seal cavity ........................................................................................

HPFTP second stage turbine disk with blades ..............................................................

Computational grid .............................................................................................

Two-dimensional basecase results ..........................................................................

Two-dimensional basecase results. expanded view .......................................................

Two-dimensional reduced coolant flow results ............................................................

Two-dimensional reduced coolant flow. expanded view .................................................

Two-dimensional 0.2 I b d s leak flow results ..............................................................

Two-dimensional 0.2 I b d s leak flow. expanded view ...................................................

Three-dimensional basecase results: vectors ...............................................................

Three-dimensional basecase results: vectors (close-up) ..................................................

Three-dimensional basecase results: vectors (end view) ..................................................

Three-dimensional basecase results: temperature .........................................................

Three-dimensional basecase results: temperature (close-up) .............................................

Three-dimensional basecase results: temperature (end view) ............................................

Three-dimensional basecase results: mass concentration .................................................

Three-dimensional basecase results: static pressure .......................................................

Three-dimensional basecase results: total pressure ........................................................

Three-dimensional eccentric (0.003 in . ) rotor: vectors ...................................................

Three-dimensional eccentric (0.003 in . ) rotor: vectors (close-up) .......................................

Three-dimensional eccentric (0.003 in.) rotor: vectors (end view) ......................................

iv

Page

2

31

4 , . I 4

6 1

8 I

I

13 I I

14 I

15

16

17 ' 18 I 22

23

24

25

26

27

28

29

30

31

32

33

~ LIST OF ILLUSTRATIONS (Concluded)

Figure Title Page I

25 . Three-dimensional eccentric (0.003 in.) rotor: temperature .............................................. 34

1 26 . Three-dimensional eccentric (0.003 in.) rotor: temperature (close-up) ................................. 35

27 .

28 .

Three-dimensional eccentric (0.003 in.) rotor: temperature (end view) ................................

Three-dimensional eccentric (0.003 in.) rotor: Mass concentration .....................................

36

37

29 .

30 .

Three-dimensional eccentric (0.003 in.) rotor: static pressure ...........................................

Three-dimensional eccentric (0.003 in . ) rotor: total pressure .................................................. 39

38

~ 3 1 . Three-dimensional eccentric (0.008 1 in.) aft-platform seal: vectors .................................... 40

I

I 32 . Three-dimensional eccentric (0.008 1 in.) aft-platform seal: vectors (close-up) ....................... 41

I 33 . Three-dimensional eccentric (0.008 1 in.) aft-platform seal: vectors (end view) ....................... 42

I

34 . Three-dimensional eccentric (0.008 1 in.) aft-platform seal: temperature .............................. 43

35 . Three-dimensional eccentric (0.008 1 in.) aft-platform seal: temperature (c1ose.up) .................. 44 I ~ 36 . Three-dimensional eccentric (0.008 1 in.) aft-platform seal: temperature (end view) ................. 45 i 1 37 . Three-dimensional eccentric (0.008 1 in.) aft-platform seal: mass concentration ...................... 46

38 . Three-dimensional eccentric (0.008 1 in.) aft-platform seal: static pressure ............................ 47 I

I 39 . Three-dimensional eccentric (0.008 1 in.) aft-platform seal: total pressure ............................. 48

V

NASA TECHNICAL PAPER

SPACE SHUTTLE MAIN ENGINE HIGH PRESSURE FUEL PUMP AFT PLATFORM SEAL CAVITY FLOW ANALYSIS

I

1. INTRODUCTION

I In working to improve the performance of the Space Shuttle Main Engine (SSME), the engineer is confron- ted with the difficult task of analyzing a complex engine system running under extreme operating conditions. The temperatures in the Shuttle's engines range from 37"R up to 5000"R, pressure vary from 20 to 8000 psi, and the engines' pumps rotate at speeds up to 37,000 rpm. Direct measurement of the engine environment is often impracti- cal. Indeed, the particular area under consideration may be virtually inaccessible to instrumentation. Fortunately,

1 the capability of modeling heat and mass transfer using computers has advanced to the point where computational fluid dynamics (CFD) can provide an alternate method of analyzing the engine. When used to model the various components and processes in the engine, numerical analysis can provide the engineer with valuable insight by allowing him or her to examine a wide range of operating conditions. The effect of a change in geometry, of a change in flowrate, or of a change in any parameter can be examined. Even a simple numerical model can demon- strate the sensitivity of the engine system to such changes, and a sophisticated numerical model, especially when used in conjunction with measured data, is a highly effective analytical tool. '

I In the current application, a general-purpose CFD code named PHOENICS, developed by CHAM Inc., is

used to model the temperatures, pressures, and velocities in the SSME's High Pressure Fuel Turbopump (HPRP) ! aft-platform seal cavity for a variety of boundary conditions and geometries. This cavity is located downstream of I the fuel pump's second turbine disk, between the disk and the aft platform seal (Figs. 1 to 4). It is an annular cavity 1 where 1400"R combustion products and 150"R coolant hydrogen mix in a complex flow pattern and then are vented 1 into the pump's turbine exhaust. An understanding of the flow field in this cavity is critical since there are at least , two known problems in the High Pressure Fuel Pump which may be linked to the environment in this region.

Specifically, these problems are ( 1 ) cracking of the second stage turbine blade shanks, and (2) hot gas leakage into the stack behind the aft platform seal (Fig. 4). The first problem, blade cracking, can severely limit the time a pump can operate before it must be rebuilt. The second problem, that of hot gas leakage, is potentially more severe since, in the extreme, it may cause the pump to shut down prematurely if the temperatures or pressures in the coolant liner behind the aft-platform seal exceed certain redlines.

Accordingly, the primary purpose of the present analysis is to investigate the two problem areas mentioned above. In doing so, the study addresses the following questions:

I 1 ) How severe is the temperature gradient in the region where the turbine blades are cracking?

1 I

2) What would be the temperature of any fluid which leaked from the cavity into the coolant liner?

The analysis addresses these questions, not only for the pump operating under normal conditions, but also for a range of off-design conditions since even a slight departure from the norm might have a radical effect on the flow pattern and temperatures in the aft-platform seal cavity. As such, the broad objective of this study is to develop a model flexible enough that it can examine the effect that boundary parameters such as clearances, pressures, and flowrates have on the flow pattern and temperatures in the cavity. Such a model must be general enough that it can support future analytical and experimental investigations of the HPFTP aft-platform seal cavity.

e E ." S I

OR1GlNAL PAGE fS OF POOR QUALITY

Figure 3. HPFTP turbine flow paths.

BLADE SHANK

STACK BOLTS

(LOCATION OF HYPOTHETICAL LEAK)

The boundaries of a lculat ion domain

/ LABYRINTH SEAL

Figure 4. Aft-platform seal cavity

I I . PROBLEM DESCRIPTION I The region being modeled is the aft-platform seal cavity downstream of the second-stage turbine in the

‘Space Shuttle’s HPFTP. A close-up of the aft-platform seal cavity is provided in Figure 4. General views of the ’ shuttle engine, the fuel pump, and the fuel pump turbine section are given in Figures 1 to 3 . The dashed lines in the 1 close-up view, Figure 4, represent the limits of the problem as specified in the model. The fluid properties and either i the pressures or the flowrates at the boundaries must be input into the program. Unfortunately, the only available

1 measurements of these parameters (i.e., temperature and pressure) are far removed from the inlets and exits of the aft-platform seal cavity. As such, the boundary conditions chosen as inputs rely heavily on an existing one-

! dimensional analysis of the HPFTP and must be used with caution [ 1 1 .

Inspection of Figure 2, the HPFTP, will show that the aft-platform seal cavity is an axisymmetric annular cavity defined by stationary walls on one side and a rotating disk on the other. Flow enters the cavity through two inlets, one at the inner radius of the cavity and one near the outer radius of the disk. The flow leaves the region via the gap between the outer radius of the aft-platform seal and the blade lips. At high rpm (up to 37,000) the flow is a turbulent mixture of hydrogen and water at temperatures ranging from approximately 140”R to possibly as high as the turbine exhaust at 1700”R. Flowrates are on the order of 1 lbdsec and pressures are in the range of 4000 psi.

The inlets and exits of the aft-platform seal cavity are described qualitatively below. The specific numbers used in this study, e.g., flowrates, pressures, etc., and the assumptions used in defining these numbers, can be found in the section on numerical model set-up.

A. Inlets

1. Coolant Inlets

At the inner radius of the cavity, approximately 0.3 lbdsec of liquid hydrogen flows into the aft-platform seal cavity through a labyrinth seal. The source of this hydrogen is the coolant circuit which is fed by the discharge of the HPFTP (Fig. 3 ) . In the two-dimensional model, this flowrate is calculated implicitly based on the pressure drop through the labyrinth seal. In the three-dimensional model, in the interest of computational economy, the coolant flowrate through the labyrinth seal is not calculated internally, but is simply set to the value predicted by the two-dimensional model operating with the same average clearances and flowrate through the blade shanks.

2. Hot Gas Inlet at the Blade Shanks

One wall of the aft-platform seal cavity is formed by the rotating disk upon which are mounted the second stage turbine blades. At the periphery of this disk, a mixture of coolant hydrogen and combustion products enters the cavity through the gap between the shank of one turbine blade and the next (Fig. 5 ) . Since there are 58 blades in the second stage disk, there are, accordingly, 58 holes available for this hot gas mixture to flow through into the aft platform seal cavity from the high pressure side of the turbine disk. The flow pattern of the fluid entering through these holes is complex since the shanks of the blades are curved and the disk itself is rotating at up to 37,000 rpm.

In modeling this inlet, the 58 separate streams entering through the disk have been “smeared” in the circum- ferential direction into a single, continuous axisymmetric source. The flowrate and fluid properties at this inlet are prescribed based on predicted values, and the angular velocity of the fluid entering the cavity through these passages is assumed to have the same angular velocity as the disk.

5

6

% .- a e, c -

a u

a c 8 %

M

i B. Exits

1. Exit Gap Between the Outer Diameter of the Aft-Platform Seal and the Blades

In this study, the single most important parameter which affects the flow pattern in the aft-platform seal cavity is the gap between the outer diameter of the aft platform seal and the lip of the turbine blades. This gap is very

I small, on the order of one hundredth of an inch, and it supports a high pressure drop of over 500 psi between the ’ aft-platform seal cavity and the turbine exhaust. Any slight variation in this gap clearance will have a strong effect 1 on the total flow and overall flow pattern in the cavity. In general, the actual flow exiting through this gap at a given location will respond to changes in the overall turbine discharge pressure, the circumferential variation in turbine discharge pressure, and any changes in the width of the gap. The latter could be due to a number of different causes, including: sideloads, dynamics , machining tolerances, eccentricity, or thermal expansion.

In the model, the exit pressure outside the gap is fixed at the best estimate for the turbine discharge pressure. The pressure drop across this exit is then related to the flowrate based on a loss coefficient times the local dynamic

; head.

2. Secondary Exit Hole I I

In one of the test runs discussed in this report, the aft-platform seal is modeled with a second exit in order to simulate a postulated leak. The leak was assumed to be around the bolts which secure the aft-platform seal to the lift-off seal stack (Fig. 4). The hole size, loss coefficient, and exit pressure of this second exit were chosen such that

i the resulting calculated flowrate would be approximately 0.2 lbm/sec. The leak rate of 0.2 lbm/sec was chosen because it is the maximum flowrate which could be leaking past the bolts. This last conclusion is based on experi- mental measurements of the pressure drops in the coolant liner cavity which is downstream of the postulated leak.

I Ill. NUMERICAL MODEL SET-UP 1 I

CHAM Inc.’s general purpose computational fluid dynamics code, PHOENICS [2], has been employed for ’ all the numerical studies described herein. To use PHOENICS, special purpose “satellite” and “ground station” sub-programs must be formulated whereby the built-in features can either be turned on or off or modified, as neces- sary. One set of the sub-programs adapted specifically for the HPFTP aft-platform seal cavity three-dimensional studies is listed, in full, in Appendix A. Full listings of the other adapted sub-programs used in this study are given in a separate CHAM report [3]. All of these sets of sub-programs are extensively annotated (via built-in

~ “COMMENT” statements) so as to make then self-explanatory when read in conjunction with the PHOENICS ’ User’s Manual [4]. Consequently, no detailed line-by-line description is given here; however, the most relevant features are described below.

I

The two-dimensional calculations described herein have been performed by using the two-dimensional ylz, polar coordinate option of the code. Figure 6 shows the selected two-dimensional grid distribution. There are 1 120 control cells, with 40 and 28 cells in the radial (IY) and (IZ) directions, respectively. Due to the (initially) assumed cyclic symmetry of the problem, only one control cell is required in the circumferential (IX) direction. However, to enable correct account to be taken of the wall shear stresses acting on the fluid entering between the blade shanks, the circumferential extent of the calculation domain is taken to be equal to the space between 2 consecutive blades (i.e., an angle of 1/58 X 2 T deg, where 58 = total number of blades).

In the three-dimensional calculation, the full three-dimensional x/y/z coordinate capabilities of PHOENICS were employed. The identical y/z grid distribution of the 2-dimensional calculations was retained with, in addition, 8 cells in the circumferential (IX) direction, such that a total of 8 X 40 X 28 = 8960 control cells is used.

7

, Figure 6. Computational grid. 1

As depicted in Figure 3, the “cold’ liquid hydrogen coolant enters axially, through the labyrinth seal, at the I

inner radius of the cavity. This “cold” hydrogen then joins with the mixture of “hot” hydrogen and water that flows into the cavity from between the blade shanks located at the outer radius of the rotating disk. The combined streams 1

1 of fluid then exit beneath the blade lips, as also shown in Figure 3. I

A. Assumptions/Model Details

The major assumptions and salient features of the physical models and the boundary conditions employed are described below.

1) All boundary surfaces (both stationary and rotating) have been assumed to be adiabatic.

2) The hydrogen and water mixture are treated as a single homogeneous fluid with mixture properties (density and laminar viscosity) and temperature deduced from the calculated mixture enthalpy and specified hydrogen and water property curve fit data as described in Appendix B.

3) The turbulence effects are presented by way of the two-equation (k-E) model of turbulence. In this model, two parameters, viz: the turbulence kinetic energy, k, and its dissipation rate, E, are computed from differential transport equations. Thus, it has the capability of representing both the local and history effects. The effective viscosity is expressed as:

rn1GINAI. PkGZ .13 OF POOR @JALT‘Y

I bhere pg is the laminar viscosity, C p is an empirical constant and p is the local mixture density. In addition, four bther empirical constants are assigned the values as recommended in original publications [4]. I

4) All boundary surfaces of irregular shape are accommodated in the present calculations by use of “cell porosities.” In this approach, each control cell is characterized by a set of fractions, in the range from 0 to 1. These bractions determine the proportion of the cell volume which is available for flow from the cell to its neighbor in a $hen direction. This practice is much more rigorous and accurate than the practice of using rectangular steps. 1 I 5) The wall shear stress is calculated by using the conventional wall functions which are based on the assumption of the logarithmic law of the wall. For partially blocked control cells, the wall stress is calculated for the Projected surfaces parallel to the velocity components.

It should be noted that the PHOENICS (1981 version) built-in process for determining wall shear stress is restricted to a finite number of special regions, to be set via the satellite subroutine. For the complex aft-platform seal geometry, many such special regions would be necessary, in excess of the built-in maximum, and a special 8PHOENICS user subroutine program was written for the current problem to overcome this restriction. This user sub-program (GWALL) performs the identical job as the built-in PHOENICS “WALL” subroutine but is used via ‘the PHOENICS ground station. A listing of GWALL is included in Appendix A.

I 6) In PHOENICS, an iterative finite-difference solution procedure is employed to solve the governing I differential equations together with the above mentioned relations. The method is based on a fully implicit, con- I servative formulation. As a result there is no restriction on the selection of the grid and the magnitude of the time l steps.

I

I

following: The variables calculated and/or solved for (and printed) in the seal cavity flow calculation include the

a. The fluid velocities in the 3 coordinate directions

b. The mixture enthalpy and deduced temperature

c. The (mass) concentration of water vapor

I

i t I

I

l d. The turbulent kinetic energy and its dissipation rate

e. The static and total pressures

f. The mixture density and separate densities of both the hydrogen and water

I g. The effective viscosity.

7) Boundary conditions are: i l

i except for the two-dimensional solutions, in which case the flowrate through the labyrinth seal was computed based a. Prescribed mass flowrate, velocities, enthalpy, mixture ratio, and turbulence parameters at all inlets

on a prescribed inlet pressure.

b. Prescribed exit pressure at all outlets, with the pressure drop related to the flowrate based on a speci- fied loss coefficient times the dynamic head.

9

c. The incoming fluid enclosed between the blade shanks is assumed to rotate at the same speed as the 1 I adjacent disk surface.

run was made with a reduced amount of coolant entering through the labyrinth seal in order to determine the sensi- tivity of the solution to the ratio of hot gas flowing in at the blades relative to the hydrogen entering at the labyrinth

8) The (phase change) freezing of the water is not accounted for; any water at temperatures below freezing is I I given the properties (density, etc.) of liquid water at freezing.

I

I 9) The effects of viscous heating have been ignored.

IV. TWO-DIMENSIONAL TEST RUNS

A. Two-Dimensional Test Runs: Boundary Conditions I

1 . Basecase 2-D I

The basecase two-dimensional run uses boundary conditions and operating clearances taken from a one- 1

dimensional flow analysis provided by Lockheed, Inc. [ l ] . These boundary conditions are tabulated in Table 1. It should be noted that for this particular run, the boundary condition specified at the labyrinth seal is that of a pre- 1 scribed pressure boundary from which the flowrate is then deduced based on the following relationship [5]: 1

1

MASSFLOW = FC * AREA* SQRT((RHO"PO(1-(PN/PO)**2))/ (NUMBER OF TEETH + ALOG(PO/PN)))

WHERE PO= UPSTREAM PRESSURE; PN=DOWNSTREAM PRESSURE; FC=FLOW COEFF.

(Note that for the basecase test run, the above equation when coupled with the PHOENICS two-dimensional model predicts a slightly lower flowrate through the labyrinth seal (0.26 lbdsec versus 0.36 lbdsec) as compared to the Lockheed one-dimensional model predictions.)

2. Reduced Coolant (Labyrinth) Flow

In the second run, the basecase two-dimensional model was modified by reducing the clearance at the outer diameter of the aft-platform seal while leaving all the other boundary conditions, including the hot gas flowrate, the S ~ X . 'Wiefi tk gap size is reduced, the pressure in the cavity goes up and the coolant through the iabyrinth seal decreases. The purpose here was to determine the effect that a reduction in coolant flow would have on the tempera- ture field in the cavity.

10

3. Leak Through the Stack Bolts

The final two-dimensional run of the current study simulated a 0.2 Ibdsec leak through the stack bolts. The boundary conditions for this run were the same as the basecase but with a "hole" at the location shown in Figure 4. The loss coefficient at this hole and the hole size were chosen such that they dictated a leak rate of approximately 0.2 lbdsec.

TABLE 1. TWO-DIMENSIONAL BOUNDARY CONDITIONS

Variable

Rotational speed of the disk (RPM)

Gap size at the labyrinth seal (in.)

Total flow area (360") between the blade shanks (in.2)

Clearance between the aft-platform seal and blades (in.)

Loss coefficient at the exit near the blade shanks

Enthalpy of the H2 upstream of the labyrinth seal (Btu/lbm) (Resultant calculated temperature - degrees Rankine)

Enthalpy of H2 and H 2 0 entering through the blades (BtuAbm) (Resultant calculated temperature - degrees Rankine)

Density of the H2 upstream of the labyrinth seal (Ibm/ft3)

Density of H2 and H 2 0 entering through the blades (lbdft')

Mass flowrate of H2 and H20 entering past the blades (lbds)

Mass fraction of H 2 0 entering through the blades

Pressure at the turbine discharge (psi)

Pressure at the labyrinth seal inlet (psi)

Loss coefficient for the second (leak) exit

Total flow area at the second (leak) exit (in.2)

Basecase

37,000

0.1069

3.877

0.0108

1.5

278.3 ( 145"R)

3558 ( I466"R)

3.574

0.93 I

3.649

0.474

3582

4254

1.5

0

Reduced Coolant Leak

37,000

0.1069

3.877

0.0102

I .5

278.3 ( 145"R)

3558 ( 1466"R)

3.574

0.93 1

3.649

0.474

3582

4254

1.5

0

37,000

0.1069

3.877

0.0108

1.5

278.3 (1 45"R)

3558 ( 1466"R)

3.574

0.931

3.649

0.474

3582

4254

1.5

0.00019

B. Two-Dimensional Test Runs: Results and Observations

1 . Basecase

According to the model, when the HPFI'P is operating at full power, centrifugal force dominates the flow field and pressure field in the aft-platform seal cavity. This is not surprising when one considers that at 37,000 rpm and a radius of 4 . 5 in., the hot gas exits the blade shanks with a centrifugal force equal to approximately 175,000 g's. Figures 7 and 8 show that there is virtually no penetration of the hot gas down into the aft-platform seal cavity and, as a result, the temperature in the cavity remains cold, at approximately 375"R (-85°F). The flow pattern in the main cavity consists of two large co-rotating vortices which maintain the cavity at a relatively uniform temperature. The liquid hydrogen which enters through the labyrinth seal at the inner radius of the cavity flows radially outward along the face of the disk and then abruptly merges with the hot fluid stream exiting from between the shanks. While it appears from the drawing that the cold flow then recirculates, in fact all of the coolant which enters through the labyrinth seal must, by continuity, mix and then exit with the hot stream, resulting in the sharp temperature gradient 1 at the blade shanks especially evident in the close-up view provided in Figure 8. The actual local gradients that the 1 blade shanks would see would be more severe than predicted here since, in the model, the hot gas flow is treated as an axisymmetric source which would tend to smooth out the temperature gradients at the trailing edge of the blade ' shanks. In reality, there are 58 blades shanks between which the hot gas flows into the cavity. The cold fluid which 1 is slung off the disk, up behind the trailing edge of the blade shanks, will be sheltered from the hot flow entering from between the blade shanks. As a result, the local mixing of hot and cold fluid will be delayed, and the local I temperature gradient will be even more severe than shown here.

2. Reduced Coolant (Labyrinth) Flow

During the course of the study the question was raised as to what would happen if the proportion of coolant to hot gas flow were different that that predicted by the one-dimensional model used to define the boundary conditions [ I ] . In order to answer this question, the boundary conditions in the model are manipulated in a somewhat contrived ~

manner in order to change the proportion of hot gas flow to coolant flow, viz: the coolant flowrate is reduced by slightly reducing the clearance between the aft-platform seal and the blade lips. The result is that a reduction of only six ten-thousandths of an inch (6 percent) of this clearance reduces the coolant flow by over half. This change, however, has little effect on the flow field in the cavity. As with the basecase, the flow in the cavity remains dominated by the centrifugal force. As shown in Figures 9 and 10, the temperature in the cavity has risen by only approximately 150 deg, up to 525"R, which is a moderate increase when compared with the hot gas inlet tempera- ture of 1466"R. The conclusion is that the flow field and temperature field of the aft-platform seal cavity is relatively insensitive to the amount of coolant entering at the labyrinth seal relative to the amount of hot gas mixture entering through the blade shanks. However, the pressure and coolant flowrate are extremely sensitive to the exit clearance at the outer diameter of the aft-platform seal for a fixed hot gas inlet flow.

3. Leak Through the Stack Bolts

For the third two-dimensional test run, a "hole" is simulated underneath the bolts which secure the lift-off stack. The rationale behind such a study is that a flow leaking past these bolts into the coolant liner might be one explanation for the erratic temperatures and pressures sometimes recorded in the coolant liner. The exit area and loss coefficient at this hole are adjusted so that the calculated leakage rate is 0.2 Ibdsec. The flowrate of 0.2 Ibm/sec C O ~ C S from the best Isiiiiiaie of the upper limit o i what the leak rate could be, based on the known temperature and pressure measurements in the liner [6].

Figures 1 1 and 12 show that a leak of 0.2 Ibm/sec through the stack bolts does not dramatically change the flow field or temperature field as compared with the no-leak, baseline case. The temperature of the main cavity and the fluid leaking out past the bolts remains relatively unchanged at around 375"R (-85°F).

12

BAS E CAS E

DSK 2D BC

2 - D SOLUTION

i

t

O.D. GAP = .0109"

I . . . . . i . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

E X I T PRESSURE = 3558 PSI

FLOWRATE THROUGH THE BLADE SHANKS = 3 . 6 5 . lbm/s

RESULTANT L A B Y R I N T H FLOWRATE = . 2 6 lbm/s

--

TEMPERATURE ST AT IC P R ESSUR E

Figure 7. Two-dimensional basecase results.

13

v) 1 E n

n

a d e

" % . C

9 9 0

I I

W , 3

c e a W a r W c

a

\ \

\

\

14

DSK 2 D B O.D. GAP = .0102"

2 - 0 SOLUTION (.OlOS" - .0006" )

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . y............ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....

Figure 9. Two-dimensional

INLET

reduced coolant flow results.

15

DSK 2 0 B

2 - D SOLUTION

I

E X I T PRESSURE = 3558 P S I

FLOWRATE THROUGH THE BLADE SHANKS =

RESULTANT L A B Y R I N T H FLOURATE = ! .o loa i i - .C006")

.12

VECTORS

IF!F?==

\

Figure 10. Two-dimensional reduced coolant flow, expanded view.

3.65 l h / s

1 bm/s

16

SECOND E X I T

DSK 2D 2H

2 - D SOLUTION

3 . 3 . GAP = .Ol08"

E X I T PRESSURE = 3558 P S I

FLOWRATE THROUGH THE BLADE SHANKS = 3.65 l b m / s

RESULTANT LABYRINTH FLOWRATE =

SECOND E X I T HOLE FLOWRATE.= .20 l b m / s

. 3 4 1 bn/s

STATIC PRESSURE

I 4180 P S I -7 I

I N L E T

Figure 1 1 . Two-dimensional 0.2 Ibm/s leak flow results.

17

1

I,

. - n o . . 0

i L n w c

ul \

E n - 3 N

II

W c 4 z 2 L L

W i c - + x W

0 z 0 U W v)

-

18

C. Convergence Characteristics and Computer Time

dimensional effects either known or suspected to exist in the pump. One of the most important of these asymmetries left unaccounted for by the two-dimensional analysis is the circumferential variation in pressure which has been measured downstream of the exit of the fuel turbine. This exit pressure serves as one of the boundary pressures which regulates the flow in the aft-platform seal cavity. In addition to this known pressure variation, there may be I

I Numerical solutions of flows involving rotating boundaries are notoriously slow to converge for a variety of ' reasons (not to be discussed here) and so it was deemed essential that careful checks be made to ensure that the 1 PHOENICS solutions being obtained were meaningful. Thus, before the two-dimensional production runs ' described above were fully completed, a series of test calculations were performed to ensure that the solutions were t converged to an acceptable degree. To this end, various runs were made for the basecase setup with different initial

guesshtarting solutions that were quite extensive. The results of these investigations are presented and discussed in Appendix C. As shown in the latter, the PHOENICS solutions are clearly converging to an identical solution in each case, as should (and must) be expected.

As depicted in Appendix C, the two-dimensional basecase was run for a total of 500 sweeps, at which time all calculated monitor flow variables had settled to an acceptable degree (Figs. C-1 to C-8). All the other 2- dimensional runs reported here were restarted from this basecase solution (i.e., the initial fields for the starting of

1 the iterative calculation procedure were taken to be the basecase solution, rather than some simple initial guess) and then run on until, again, the solution monitor values were suitably settled. This usually required another 150 to 200

1 sweeps, at most. Computer times for these restart runs were approximately 35 CPU minutes on CHAM'S Perkin I Elmer 325 1 mini-computer.

t

I

~ time on the Perkin-Elmer 3251 machine.

All the 3-dimensional calculations described in the next section were also restarted from the 2-dimensional basecase solution which was symmetrically duplicated in the circumferential IX-direction. Again, converged solutions then took approximately 150 to 200 more sweeps and required approximately 5 CPU hours of computer

I

I V. THREE-DIMENSIONAL TEST RUNS

A. Three-Dimensional Test Runs: Boundary Conditions

1 . Basecase (Geometrically Axisymmetric with Asymmetric Exit Pressures)

As its boundary conditions, the basecase three-dimensional run uses the same operating clearances, flowrates, pressures, mixture ratios, and enthalpies, etc., as used by the two-dimensional basecase analysis. The

19

only exception is that the exit pressure of the turbine is no longer uniform but varies circumferentially based on data taken during Rocketdyne's engine test 902-279 ["I. These boundary conditions are the best estimate of the operating conditions in the fuel pump at full power (109 percent). The specific numbers used for this run, and for the subse- quent three-dimensional runs, are listed in Table 2.

2. Eccentric Rotor (Rotor Shift of 0.003 in.)

The Eccentric Rotor (0.003 in.) case was set up to simulate the effect that a rotor shift of 0.003 in. would have on the flow field in the cavity. The shift of 0.003 in. was chosen because it is an upper limit on the distance the rotor can shift before the shaft starts rubbing against the labyrinth seal. Such a rotor shift in a given direction would open up the exit clearance between the aft-platform seal and the blade lips, while at the same time it would close down the clearance at the labyrinth seal. This effect is simulated in the model by, on the one hand, directly adjusting the clearances at the outer diameter of the aft-platform seal and, on the other, by adjusting the flow rate at the , -

labyrinth seal. All the other inputs remain the same as for the three-dimensional basecase.

TABLE 2. THREE-DIMENSIONAL BOUNDARY CONDITIONS

Variable

Rotational speed of the disk (RPM)

Flowrate at the labyrinth seal (Ibmisec) 1:00 2:30 4:00 5:30 7:00 8:30

10:00 11:30

Total Mass Flowrate

Total flow area (360") between the blade shanks ( in .2 )

Clearance between the aft-platform seal and blades (in.) 1:00 2:30 4:00 5:30 7:00 8:30

I0:00 1 I :30

Total Area

Loss coefficient at the exit near the blade shanks

Enthalpy of the H, entering at the labyrinth seal (Btuilbm) (Resultant calculated temperature - degrees Rankine)

Enthalpy of H, and H,O entering through the blades (Btuilbm) (Resultant calculated temperature - degrees Rankine)

Density of the H? entering at the labyrinth seal (Ibmift3)

Density of H, and H 2 0 entering through the blades (Ibmift')

Mass flowrate of H, and H 2 0 entering past the blades (Ibmis)

Mass fraction of H 2 0 entering through the blades

Pressure at the turbine discharge (psi) 1:00 2:30 4:00 5.36 7:OO 8:30

1o:oo I1:30

Average Exit Pressure

Basecase

37,000

0.0323 0.0323 0.0323 0.0323 0.0323 0.0323 0.0323 0.0323 0.258

3.877

0.0108 0.0108 0.0108 0.0108 0.0108 0.0108 0.0108 0.0108 0.307

I .5 278.3

( I45"R)

3380 ( I466"R)

3.574

0.93 I

3.649

0.474

345 I 354 I 3697 3622 3606 3592 3476 348 I 3558

Rotor Eccentricity = 0.003 in.

37,000

0.0095 0.0323 0.055 I 0.0646 0.0551 0.0323 0.0095 0.oooO 0.258

3.877

0.0129 0.0108 0.0087 0.0078 0.0087 0.0108 0.0 I29 0.0138 0.307

I .5 278.3

( 145"R)

3380 ( 1466"R)

3.574

0.93 I

3.649

0.474

345 1 354 I 3697 3622 3606 3592 3476 348 1 3558

Aft-Platform Eccentricity = 0.0081 in. 1

I

37,000

0.0323 0.0323 0.0323 0.0323 0.0323 0.0323 0.0323 0.0323 0.258

. 3.877

0.0165 0.0108 0.0051 0.0027 0.005 I 0.0108 0.0165 0.1800 0.307

I .5

278.3 ( I45"R)

2280 ( 1466"R)

3.574

0.93 I

3.649

0.474

345 1 354 I 3697 3622 3606 3592 3476 348 1 3558

20

1 3. Eccentric Aft-Platform Seal (Aft-Platform Seal Shift of 0.0081 in.)

The third three-dimensional test run models the flow field for a highly eccentric (75 percent) aft-platform ' seal. In this run the clearance at the gap between the aft-platform seal and the blade lips is adjusted so that it models I what the gap would be if the aft-platform seal had moved laterally 0.0081 in. in the I1:30 direction. Note that the

rotor itself has not moved but is still concentric with the labyrinth seal so that the gap between the labyrinth seal and ! the rotor axle remains at a uniform 0.003 in. (In general, the clocking positions used in this report correspond to the l convention adopted by Rocketdyne in Reference 7, however, in this particular test run the decision to move the

aft-platform seal in the 11:30 direction is arbitrary, and is based on convenience rather than any physical justifica- tion.) The choice of the magnitude of the eccentricity is also somewhat arbitrary but the reasoning behind the shift of 0.0081 in. was the desire to choose a large aft-platform eccentricity in order to observe extreme effects. An aft- platform shift of 0.0081 in. is 75 percent of the total aft-platform seal clearance.

B. Three-Dimensional Test Runs: Results and Observations

1 . Basecase

A comparison of the three-dimensional basecase results (Figs. 13 to 21) with the two-dimensional basecase ~ results (Figs. 7 and 8) shows that the addition of an asymmetric pressure distribution at the exit of the turbine has had

little effect on the flow pattern in the aft-platform seal cavity. While some evidence of the influence of the external ' pressure distribution can be seen at the outer diameter of the disk near the blade shanks (e.g., Fig. 16), this effect is ' small; toward the center of the cavity the results are nearly identical to the two-dimensional solution. At the

flowrates and small clearances of the aft-platform seal cavity running at full power, a circumferential pressure ' difference of 220 psi as modeled here represents only a fraction of the over 600 psi pressure drop between the ' aft-platform seal cavity and the turbine exhaust. As a result, the 220 psi circumferential variation on the outside of 1 the cavity has little effect on the flow pattern inside. In addition, even with the circumferential variation in turbine

I exhaust pressure, the centrifugal force in the aft-platform seal cavity still dominates the flow such that the influence that is felt due to the pressure variation is confined to the periphery of the cavity. For an example of this effect, I examine the lines of constant temperature given in the close-up view in Figure 17.

' 2. Eccentric Rotor (Rotor Shift of 0.003 in.)

Perhaps the most notable feature of the aft-platform seal cavity flow field (Figs. 22 to 30) with a 0.003 in. eccentric rotor is the small change as compared to the three-dimensional basecase with its centered rotor. Even with an eccentric rotor, the temperatures in the cavity have risen just 75"R, indicating only a slight increase in the heat transferred down into the cavity. Again, the only significant effect is felt at the outer diameter of the turbine disk where, at the 5:30 clock position, the hot gas actually flows down into the cavity causing a local hot spot. This hot spot will be felt by the blade shanks once per revolution, with a corresponding cooling in between. In general, therefore, a rotor shift of 0.003 in. results in a slight warming of the average cavity temperature, and a cyclical variation of temperature at the outer diameter of the disk of approximately 600"R.

'

3. Eccentric Aft-Platform Seal (Aft-Platform Seal Shift of 0.0081 in.)

Of the six different two-dimensional and three-dimensional test runs investigated during this study, the most dramatic results (Figs. 3 1 to 39) come from running the model with an aft-platform seal that has been shifted to one side by 3/4 of the exit clearance (i.e., by 0.0081 in.). With the exit gap substantially closed down on one side, the hot gas which wddd normally exit through that gap must, instead, exit at a different location. The centrifugal force

21

EASE CASE

DSK 32 BC A S Y M T R I C A L

3-3 SOLUTION PRESSURE

(f-:.:: * 4 # 4 . - & I . . - 4 . . . . . :;:::.:;@) ... ...... -. . : .- : : : : I : .- I :.- , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * d A; \ . " " . ......

. . . . . . -.

~ ~. ............ r,- I - . . . . . . . . ..-e ,, . ............. . . . . . . . . . . . . 4

VECTORS

SIDE V I E W

SYMMETRICAL

GAP ( = .0108")

E X I T PRESSURE = 3565 P S I

. . . . . . . . . . . . . . . . . . . . . . . . .-a . . . . . . . .- . . . . . . . . . . . . . . . . . . . - . . . . . . . --I . . . . . . . . . . . . . . ,-, d . . . . . . . . . . . .**If(

\ . " . . . . . . . , , J - - . . . . . . . . . . , . I

L,:; . : : : : : : :;I.. . , .. . . . . . . . . . . . . . . . . . . . . . . . . 4 y,

\ . . ' . I . . . . . . .

E X I T PRESSURE = 7 3645 P S I

. . . . . . -.-a - . . - . -..-

. . . . . . . . . L, : ; : : : : : : : : * ;;* -% . \ . . . . . * I . . , , , ( . . . .........

22

Figure 13. Three-dimensional basecase results: vectors.

I I ' - . - -

R c c

I 1 \

1 " I Q

23

BASECASE

3SK 32 Bc z-3 SOLUTION

ASDQETRICAL

EXIT PRESSURE

VECTORS

END VIEW

(FROM THE T U R B I N E END)

I

SYMMETRICAL I

GAP ( = .ClOS'l)

: 30

Figure 15. Three-dimensional basecase results: vectors (end view).

24

BAS E CASE

3SK 32 A S Y M T R I CAL

3-3 SOL3TIoll P RE S S URE

TEMPERATURE SYMMETRICAL

SKDt V X U GAP ( = .OlOS")

E X I T PRESSURE = -

Figure 16. Three-dimensional basecase results: temperature.

25

E $1 c

:: " RI

I1

W

3 m v)' w v )

v)

a

a n n

II

W

3 v) m ' w m

m I - * - u ) X r n W

a

a n n

0 m

OD ..

H \

26

8:

BASECASE

5SK 32 Bc ASYMMETRICAL

3-3 SOL'JTXON PRESSURE

TMPER ATURE

END VIEW

S Y MMETRI C A L

GAP ( = .0108")

n

2 t 3 0

Figure 18. Three-dimensional basecase results: temperature (end view).

27

BAS E CAS E

DSK 32 EC AS Y M T R I CAL

3-D SOLUTION PRESSURE

"2O

W S CONCENTRATION SYMMETRICAL S I D E V I E W UP ( = .0108")

474 E X I T PRESSURE = _.

5:30

BAS E CASE

DSK 32 6c

3-0 SOLUTION

ASYMMETRICAL

EXIT PRESSURE

EXIT PRESSURE k-

4220 PSI- )

EXIT PRESSURE =

3615 PSI )q=-

STATIC PRESSURE (PSI)

SIDE V I E W

SYMMETRICAL GAP ( = .0108")

EXIT PRESSURE = 3565 PSI

2:30

4220 PSI <-

EXIT PRESSURE

3645 psyj- - 4230

-4210 PSI

Figure 20. Three-dimensional basecase results: static pressure.

29

8AS E CASE

DSK 32 8C AS YHMET R I CAI.

3-3 SOLUTfOU EXIT PRESSURE

1

EXIT PRESSURE

4230 PSI

EXIT PRESSURE =

TOTAL PRESSURE (PSI) S Y m E T R I CAL

SIDE VIEW GAP (= .Olea")

E X I T PRESSURE =

E X I T PRESSURE =

Figure 2 1 . Three-dimensional basecase results: total pressure.

1

i j

t I

i

t

i I

I I i

t 1 ~

~

i I

;

M I 32 EC ~SYWET~ICAL GAP

3 - D SOLUTION ECCENTRICITT t .003"

0.0. SAP t

. . , . , ,,*w ........... r . I ' . . . . . . . . . . . . I: - 9

. . I ..* . . & 4 . . . . . . - & & . . . . . . . .# . . . . . . . . . . . . . . . . . . .

1 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . \ ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

O . D . CAP = .9108"

. . . . . a 1

$'\ 4 9 .

. . / - a . . . . . . . . . .

. . - A ........ . . . . . . . . . . . . . . . . ........ .......... 3) . . . . . . . . . . . . . 4 . . . . . . . . . . .

VECTORS SYWETRICAL E X I T

SIDE VIEW P R E S S U X = 3558 PSI:

O.D. SAP t

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I * . , , # , . " . . . ...... / . . . . . . . . . . . . . . - - . . . . . . . . . . . .

\ . a . . . . . . . . . . . \ * . * * - - - - * . - . . . . . . . . - . , . I (:;:a4J . . . . . . . . .

. I e , ,.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 \ ' . . . . I . . . . . ,. . . . . . . . . . . . - . . . . . . . . . . . \ " " " " " " . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 22. Three-dimensional eccentric (0.003 in.) rotor: vectors.

31

1

i I

d '

I I e - \' I" . .

1 3 I

\

\\

32

3:

DSK 32 EC ASYHHETRICAL GAP

3-0 SOLUTION ECCENTRICITY = .003"

~ ~ ~ ; ~ ; ~ . ' . F. 1-3 VECTORS OF POOR Q&iLKW

END VIEW SYWETRICAL E X I T

(FROM THE TURBINE END) PRESSURE = 3558 P S I

l l r 3 0

: 3(

Figure 24. Three-dimensional eccentric (0.003 in.) rotor: vectors (end view).

33

03K 32 EC

3-s S O L ~ T I O N ECCENTRICITY = .003"

1 U 6 6 O R - O.D. GAP =

11130

SYMMETRICAL E X I T

P?ESSURE = 2 5 5 8 ?SI

OF

I a % 3 I

I c

0 -7

ry ..

e ? 0

POOR QUALITY

3 *?

m ..

I

a *. " 8 9 9 . C

0

35

0:

: 30

: 30

Figure 27. Three-dimensional eccentric (0.003 in .) rotor: temperature (end view).

36

DSK 32 EC A S Y t W 5 R I C A L GAP

3-3 SOLUTfOn E C C E N T R I C I T Y = .003“

474 O.D. GAP = _. .Ol08”

8:30

OmGlNAL PAGE IS OF QUALITY

H2° SYMKETRICAL E X I T

PRESSURE = 3558 PSI

W S CONCENTRATION

SIDE V I E W

G7G O.D. GAP s _. 2:30

Figure 28. Three-dimensional eccentric (0.003 in.) rotor: Mass concentration.

37

DSK 32 EC U m R I C A L GAP

3-0 SOLUtfw ECCENTRICITY = .003*

4220 PSI-

4210 PSI

O.D. GAP = rc

8: 30 .0108"

STATIC PRESSURE (PSI) SYMMETRICAL E X I T

SIDE VIEW PRESSURE = 3558 P S I

0.3. GAP = .3108"

- 4220 PSI

0.3. GAP = .5078"

r

38

OMGlNAh PAGE IS OF POOR QUALITY

1

DSK 32 EC m R I C A L GAP

=CENTRICITY = .003" 3 4 SoLurcow

TOTAL PRESSURE ( PSI) SYMMETRICAL E X I T

SIDE VIEU PRESSURE = 3 5 8 PSI STATIC

4440 P S I - 0.0. GAP a .0108'

2:30

O.D. GAP =

Figure 30. Three-dimensional eccentric (0.003 in.) rotor: total pressure.

39

DSK 32 ASH WPWHETRICAL GAP

3-D SOLUTION ECCENTRICITY = .0081"

O.D. GAP =

;#. . . .-. ,,. . . . . . . . . . . . . . . . . . . . . . . . . I , I , , , . . 0 , ..4 . . . . . . . . . . . . .m\ . . , * . . I * * ,..-.a \ : x : . . . . . . . . . 4;[ ....I /.

, . . . . . . . . . . . ..I\ \ * . - - - * * * - - * ' . . . . . . . . - ~ . .I

. . , . - . e . . . . . . . . . . . . . . . . . / \ \ \ '

\ - - .

O.D. CAP =

SYMRETRICAL E X I T VECTORS

SIDE VIEW P R E S S U R E = 3558 PSI

O.D. GAP =

/ff( << : ...... ::::: 2:30

I . 1 . I , ..*..a . . . . . ,..A1 ...... .i

~ ..~ ' . : . - . . . . . * J

. . . . . . . . . . . . C'

; . . . . . . . . . . . . . . . . I . . . . . . . . . . . \ . . . . ........

. \ .

O.D. GAP =

. . . . . . . . , . , , . . . . . . . . . . . . . . . . . . . . . . 9 9 e . , . . I , . . I . I , , . . , . I . 1; 3 , . . * I . , , , ,

I . . . . . . . . . . . . . ,./q (*.*#14;l , . , " , " * . . . . . . . . . . ..,. .,e. 1 ' .

. . . . . . . . . . . . . , . J \ . . * , , I . , , , . . * I , . . . . . . . . . . , . I d . . . . . . . . . . , . I . . . . . . ',\\ f

. . . . . . . . . . . . . . . . . . . 1 ../It

Figure 3 1 . Three-dimensional eccentric (0.008 1 in.) aft-platform seal: vectors.

Y > w

v) 9

E

Q 0

c

9 I,

k

V Y t; a

w u u w

z El G A 0 VI P m

ORIGINAL PAGE .IS OF POOR QUALITY

I I $ ?

I1

a 4 c o m m . C

9 9 0 Y

a * E 0 -

N . O

9 9 0

0 m In

. C

9 9 0

n n ? 2 0 V - W

v1

U u 0 >

n c .- - 00

8 e 8

0 L E

0 0

.- U

c1 m 2 1 M L ._

41

8:

VECTORS

3SK 32 ASH ASYMMETRICAL GAP END VIEW SYMETRICAL E X I T

3-3 SOLUTION ECCINTRICITP = .0081" (FROn THE TURBINE END) PRESSURE = 3558 PSI

11:30

2: 30

5830

3

in.) aft-platform seal: vectors (end view). Figure 33. Three-dimensional eccentric (0.008

42

Ol?iGINAL PAGE E OF POOR QUALITY

3SK 32 A S H USIEPIETRICAL GAP TMPERATURE SYMMETRICAL E X I T

3 4 SOLUTION ECCENTRICITY = .0081" SIDE V Z N P Z E S S U R E = 2 5 5 8 P S I

O.D. GAP = .Ol08"

8830

A l466'R O.D. GAP = -

Figure 34. Three-dimensional eccentric (0.008 1 in .) aft-platform seal: temperature.

43

- a D x v ) w v ) m

0 I-

C b) 0 0 b)

.e

*

i GR:G!EAL PAGE FS OF POOR QUALITY

TMPERATURE

32 ASH M I M E T R I C A L GAP END VIEW SYMETRICAL E X I T

3-0 S O L U T I O N ECCENTRICITY = .ooai" (FROM THE TURBINE END) PRESSURE = 3558 P S I

_ . .

: 30

Figure 36. Three-dimensional eccentric (0.008 1 in.) aft-platform seal: temperature (end view),

45

DSK 32 ASH ASYMMETRICAL GAP

3-D SOLUTION ECCENTRICITY = .0081"

" 2 O

MASS CONCENTRATIDN SYWETRICAL EXIT

SIDE VI% PRESSURE = 3558 PSI

O.D. GAP = .0189"

: 30

474 O.D. GAP : _. .o 108"

8830

070 & O.D. GAP =

Figure 37. Three-dimensional eccentric (0.008 1 in .) aft-platform seal: mass concentration.

46

I

GWlGFNkL PAGE 6s OF POOR QUALITY

DSK 32 A S H UbParRICAL G&P STATIC PRESSURE (PSI)

3-0 SOLUTION ECCENTRICITY = .008l" SIDE VIEW

S Y M M E X I C A L EXIT

P R E S S U R E = 3558 PSI

O.D. GAP = - - .0189"

/-4270 PSI ,-I

11:30

.-

O.D. GAP =

- 4275 PSI

8:30

-4270 PSI

O.D. GAP =

Figure 38. Three-dimensional eccentric (0.008 1 in.) aft-platform seal: static pressure.

47

DSK 32 ASH

3-0 SOLUTION

M-RICAL GAP

ICCENTRICITY = .0081"

'P- 4500 P S I - O.D. GAP :

.0189"

.0108"

T3TAL PRESSORS (PSI ) SYMMETRICAL E X I T

SIDE VIEW PRESSURE * 3 5 5 8 PSI STATT,'

4500 P S I - O.D. (IIP =

2: 30

<-4320 PSI -9 I z'

O.D. CAP =

b 4320 PSI

Figure 39. Three-dimensional eccentric (0.008 1 in.) aft-platform seal: total pressure.

48

on this hot gas still works to confine it to the outer radius of the cavity. However, the pressure differences in the radial direction are, in this case, becoming large enough that they are forcing more and more hot gas down into the

1 cavity. This is clearly evident in the velocity diagrams (Figs. 31 and 32) where, at the 5:30 clock position, there is a 1 strong inward flow of hot gas down into the cavity. The temperature profiles also indicate a dramatic increase in hot

gas in the cavity. The temperatures at the center of the cavity are now up to 675"R (21 5") which is 300" warmer than for the basecase. As with the other three-dimensional runs, the most pronounced effects can still be seen at the outer radius of the cavity. Here at the outer radius of the disk near the blade shanks, there is a large circumferential I

1 variation in both temperature and pressure with the temperature cycling 600"R and the pressure varying by 20 psi.

1 One further observation on the results from this test run has to do with the static pressure. The observation is that for a 0.0081 in. eccentric aft-platform seal, the pressure in the cavity has gone up by 80 psi relative to the ' three-dimensional basecase. The implications of this pressure rise are not simple to determine. The difficulty lies in the fact that such a pressure in the cavity would reduce the flow rate through the labyrinth seal, which, for the three-dimensional model is (numerically) fixed based on the flowrates calculated earlier in the two-dimensional I basecase. Since the total exit areas and average turbine discharge pressure for all the three-dimensional runs are the

1 same as for the two-dimensional basecase, this is a reasonable assumption. In this final three-dimensional case, however, the assumption leads to a contradiction. For the eccentric aft-platform seal case, using the flowrates from the two-dimensional basecase, the pressure at the exit of the labyrinth seal is calculated as being higher than the pressure at the labyrinth inlet. In other words, if in the three-dimensional model this boundary had been specified as

I a fixed pressure instead of a fixed flowrate, the model would have predicted reverse flow through the labyrinth seal. ' That there could actually be reverse flow through the labyrinth seal is considered extremely unlikely. What would ~ more likely occur is that an eccentricity in the aft-platform seal would raise the pressure in the aft-platform seal ~ cavity, reducing both the hot gas flow past the blade shanks and the labyrinth seal flow.

i

I I VI. SUMMARY OF THE CURRENT TEST RUN RESULTS AND OBSERVATIONS

The axisymmetric computer model of the aft-platform seal cavity indicates that at 37,000 rpm the flow in the 1 aft-platform seal cavity is dominated by the centrifugal force caused by the rotating turbine disk. The disk drives a ~ recirculating flow in the central region of the cavity, creating a core of nearly uniform temperature. In general, the ' temperature field throughout the aft-platform seal cavity is dictated primarily by convection (as opposed to conduc- j tion) as indicated by the fact that little heat from the hot gas at the periphery of the cavity is conducted down into the

isothermal core. As a result, the core stays relatively cold, even when the coolant flowrate is reduced by over 50 ' percent.

i

The most severe temperature gradient in the aft-platform seal cavity occurs at the outer diameter of the 1 , turbine disk, near the blade shanks. At this location, the hot gas entering from between the blade shanks mixes with ~ the coolant flow that is being slung off the face of the disk. The temperature difference between the two streams is l over 1000"R.

I The three-dimensional computer model of the aft-platform seal cavity shows that, for normal clearances and i '

1

operating conditions, the flow field in the cavity is relatively insensitive to the circumferential pressure variation known to exist in the turbine discharge. The flow field is shown to be sensitive, however, to eccentricities of the exit gap between the aft-platform seal and the blade shanks. But in both cases, it is the centrifugal force which still dominates the flow pattern, such that any perturbation of the flow field or temperature field which results from either pressure changes or geometrical changes are, for the most part, confined by centrifugal force to the outer diameter of the cavity.

49

In addition to the above, the study also reveals that, for fixed flow through the blade shanks, the labyrinth flowrate is extremely sensitive to the exit area at the outer diameter of the aft-platform seal. While this result is somewhat misleading, since it is based on the unrealistic boundary condition of a fixed flowrate through the blade shanks, it nevertheless merits further consideration especially with regard to transient phenomena. Finally, as a related observation, the flowrate through the labyrinth seal is also sensitive to the eccentricity of the aft-platform seal clearance, even for a constant exit area. This sensitivity is something which has yet to be included in the current one-dimensional models of the flow through the pump's turbine section.

VII. CONCLUSIONS

The results of the study summarized above provide the following insight into the specific problems which initiated the study, i.e., ( 1 ) the cracking of the HPFTP second stage blades, and (2) the suspected hot gas leakage into the coolant cavity behind the aft-platform seal bolts.

As far as the blade cracking is concerned, the model has shown that the second stage blade shanks are subjected to varying degrees of thermal stress, both steady state and once per revolution. The severity of this gradient has been shown to be sensitive to asymmetries in the external pressure and to variations in the geometrical clearances. At the time of this writing, however, it is believed that the primary cause of cracking is not due to thermal effects but is the result of a very high mean mechanical stress coupled with the moderate thermal stress. The proposed solution to alleviate the cracking is to recontour the shank in the high stress area, to shot-peen the surface to reduce the surface mean operating stress, and to coat the shanks to reduce the thermal stress [8].

As for the variations in coolant liner pressure and temperature thought to be indicative of a leak into the coolant liner, they remain an enigma. In order to gain a clearer understanding of this problem, the fluid temperatures calculated in the current study will be used as an input to the thermal stress analysis of the hardware. Prior to this study it was believed that the temperatures in the cavity were on the order of 900"R hotter than predicted here [8]. With a better estimate of the fluid temperature, the thermal stress analysis will be better able to predict the deforma- tion of the aft-platform seal and the other components neighboring the aft-platform seal cavity. This will, in turn, generate improved estimates of the clearances and flowrates in the region.

The new flowrates estimated from the above will be fed back into the PHOENICS model for an improved analysis of the flow and temperature field in the cavity. Other changes which could be incorporated into the model would be to include the effect of heat transfer into the cavity and the viscous heating of the fluid itself, both of which will result in increases in the cavity temperature.

In addition to further analytical studies and improvements, there are plans to build a fuel pump that has pressure and temperature measurements built into the aft-platform seal, the labyrinth seal, the lift-off seal stack, and the coolant liner [8]. The test data from this instrumented pump, in conjunction with the computer model predic- tions should greatly increase the level of understanding of the operating environment of the high pressure fuel pump aft-platform seal cavity.

50

I

I REFERENCES

' 1 . Lockheed Missiles and Space Company, Inc.: Fluid Flow Analysis of the SSME High Pressure Fuel Turbo- pump Operating at Full Power Level. LMSC-MSFC TR D697954, May 1980.

~ 2. i 1

Spalding, D. B.: General Computer Program for Fluid Flow Heat Transfer and Chemical Reaction Process. International Finite Element Congress, Baden-Baden, West Germany, November 1980.

3.

I

4.

~ 5 .

I 6.

7.

8.

Keeton, L. W., Lowry, S. A., and Wayden, L.: Listings of Phoenics 2-D and 3-D Satellites and Ground Stations as Adapted for the SSME Aft-Platform Seal Cavity Flow Model. CHAM 404921, June 1985.

Rosten, H. I., Spalding, D. B., and Tatchell, D. G.: PHOENICS - An Instruction Manual. CHAM Report TW75, January 1982.

Morrison, G. L., et al.: Labyrinth Seals for Incompressible Flow. NAS8-34536 Final Report, April 1983, p. 63.

Conversation with Garry Lyles, EP26, Marshall Space Flight Center, NASA, January 1985.

Wineland, D. L. and Cramer, K. : HPFTP Instrumented Turbine Test Data. Rockwell International, September 12, 1982, p. F3.

Conversation with Rick Ryan, EP23, Marshall Space Flight Center, NASA, June 1985.

5 1

APPENDIX A: PHOENICS COMPUTER CODE SATELLITE AND GROUND ADAPTATIONS

FOR THE SPACE SHUTTLE MAIN ENGINE HPFTP AFT-PLATFORM SEAL CAVITY 3-D MODEL

N VI u n ' . - W m

2 3

N P o w - m v ) > a a

I-

d a a

a c LL

K N K U -I- z - - m e a a w -

> - V I < a I A m a - x a . a x a - z x - - -x .- . X

m - - x W Y N X e a - x

VVICJX - - e x - - z x

a a - x a a a x

X

x w X Z x o XI- xv)

x a

XI c i -

a a a

.-e m - m

a a o - c - a x a - m w

--- z c c

0 m - a

z W c I- v)

a LL

- a O K g: Q

- N m

m W -I m a c.(

.- -I- \-

z > I-

-1a x --le - c

e- w z w 0

z - > i z w o

U

c '"

V Y 2 0 a - U

I..- - + C Y

i

5

u a a u -1a

a 3

n u urn e - > 3 a O W w e a a

a -

u a a

w -

v 0 L

I..

O 0 -1

-1+

3 x + w

i

z W In m a

\ N U U \

- N - 0 I

- m Y * 0

f - - e m - m w c xm

0 g o c I/

I- -

a x + w o v - 23 a a-1 4-1 w w J U 0 z U +

n a a x

a w

1-0 a & a z - O K Z V I '

0 Z V O w CT -1

w i c + c J 3 - r ~ a

Y 4 L W 0 > . w c m

of C Z g -

+ t

t +

I: t

1 t t t t t

t t

W 1) 3 ' + K - w t c -3 t . - a t

l l l l e t -I. t ~ - I I t m - t - 0 - t U T I - 0 - t w z t 0 0 3 + -1na t

o * v

c

+

t

+ t

+ t

I)

1 + t t

t

(I

t t 0 0 t

1

-

a a n c

Y U 0 2 m z

c w -1

t n -1 0 U

F c

a - t - L

I n a U - w U a a -1

-. -. -. c

w z n z a

n a 0

w a

3 0 2 w

W

W 3

I

a

rn v) z 0 - c -1 a 3 V -1

V > a a

a

a

a c X Y c

I

m z

c :n

LL I

u

L t t

In I- 3 n

t t t

w w c u Z N Z W . v z

x x x x x x x x x x x x x x x x x x x x X % x x x x % X % X % X x x x x x x x x x x x x x x x x x x

4 - 4 0 D \ l Y U a - a w

- W I n a o- L In z

3 a z 3 z a 0

a

U

a a n e

w 5

w c w t - 2 - 1 - G U U

V a l n u n \ a w a -

w . w . z 2 z 2

4. z z n QI- a a u

J v z c . W I n

v +I

f U '

N Y O . a o w a O w

a a > 2

a 0 w a aInaIn

w o w 0 > v > v a + a *

r -

I- a w > a c W I 0

W W

t t +

o c o c + - e -

x x - w - w - w m w ---- c ,I L ,I

+ + I 1 + t *

I I I x x v v v u u u v u u v u V u u v v v u u u v u v u u u u v v u U U U L ' U U V v v v v u v u u u V

I

e X e x C X e x + x e ~ w w ~ w w w w w w w w w , lnalnolnwlnuowlno

c W -I

5 c 0 - W

E

3 5:

L- a - - a a a w

W

2 J J W a 0 I W 5 W m ln W W a o W 0

0 (D m u. 0 Z o c.l

c 0 4

LL a

c W -I

5 c 0 I

w &

c c a h

t 0 ln

a W

v

a a

: K 0

u 4 LL

I

I

W

:

+ + + + + +

ln W

c w

n

a a

n 0 K

+ . +

c W 2

t s 0 U

w I c I- a

I -1 \ 3 c

- m

m

a

- > 2 a I t z w

W X e

c W -1

t c 0 I w E I- a

I m -1 \ 3 c m

a a

- > d

E 5

E w

e X W

W

I

5 0

3 I-

W

I-

U

a

I

0

I m -I \ 3 c m

a a

- > -1

I c 5 W

e I

c W J z D J C 0

W

I

E c a

e LL

3 V \

z 5 - > c ln z W 0

W

I

E

z E I- z c

l n w l n -,w z c w o z v w ; ; ;ut c i z m 3 - K c y 3

V + w o 1-10 e v w

I- L L Z o w w II

z c 3 0 a -0 I - el-ln a z I -u- 0 2 K - a

n c LLELL o o a

V I Z V o a w - 2 c w > Urn0 w - c P 3 W - Y Y o w v

0 0 w - l u I W cciz

W - ,. c w z o C J W oo ln Z U 3

D l n -0

InZ (Do m a D m

-. -. -.

L v) a - Lu E 3 ln VI W E

+ + + * + * . + 0 O Q O O O O 0 c 0 c 0 c 0 c 0 c O c O c O ~ 0

m CY 3 + W o a P L > a W

E . . . .

c x c x c x c x c x c x I - x + x c w w w w w w w w w w u w w w w w w lnalnavlalnalnalnalnalna ln

+ + + + + + + + + + + + + + + +

' V 0 u v v v v v v v v u v v v u v v v v v 0 v u v v v u v u v v v v v v v v u v v u v v u v u v u v v u u v v v v v

m I- O 0 a

a m

- a - a x u w z

W C

-1

VI c a c . m m C O

0 m - e

Z V o w - V I c v o U K

o

V I + 3 V I

x x x x x x x x x x x x x x x x x x x x x x x x x x x x

a

ziz

m a a z w a

m >

3 <

a i

w

c

X W

w

"

5 U r 3 c

W

e c

I

a z C c V

a U

VI VI

I 0 N I

I

u

a

a

w c

c w -1 z

c 0

r A

w c e a z El c V

K U

VI VI

I 0 N I

a

a

W

E

c W

--1 V I 2 \- I m o

e w

- 2

\- V I 2

I 9, - g w c w

K C 3 O C

a r

d a U

VI m 2 2

c 0 e w

a

E N

lnc N

I 11 0

W - w v u o w e w w u r n

m o

. -

t t

t

t c c

+ t t

+ + I + a * a c u c r + z * 3 u r n + a c m

c

c

t

*

+ I)

*

t

+ I t *

t

+ t

I

I- a VI w m VI 0 -1

a 0 U

I-

w z I

2 U. U w 0 u VI VI 2

. . 0 ' -1 x x w o m a

x x ~ - w a v a m x x - 1 - - 0 m x ~ ( 3 a ~ u m x x a v o o . 2

0

0 U

c

-

m z 2 c o z 0 V

u

> a 0 2 3 0

a

m

c t c c

U

VI VI a I -1

I-

c

a 0

Y * c 0 c

X X u - . A h 0 x x w o o '

P

Q In m O

I)

0 0 0 7

11

c X W

a M

a

z x x x x x x i; X X O - K X O - c w I

+ a a a a a a a a a r m 0 0 0 0

1 + c t

* * * * . * x x x x x x 1 1 f

v u u * v v v u v u v v v v v u -... v v v u v v u v v 0 v v u v v v

W I X I

a

a L 2

5, .. aEl P O

UIK

-3

v) 3 0 a > I-

> V I a u v x

V a z z- 3

c1

a

-- w -

5 m wec x n 1-4

a o a a u =

- 0 (0 N

0 6

I , 1 ,

> I-

VI I

> , I-- -v)

I >w v u z , u -

a x

.. E- 0 C

W 9- u - K o a

I ,

v v v u v

wo

u o X - c m I C O I

-I W

a 0 a

U L .

UJ V W

I-

-I-

5 z m 0 O m

N I1 m

11 . N N

z * + V v u

- 0 0 m 0

5 1 0 6 0

I .

I ,

v v v

59

> -I z o z

W

w e m

w O K wa Z m

a u w wa ma 3

W - v o a

c r

W

0; 11 -1 . w wv 3 -1m a a > . - e '0

W K - w - N

---- N N N N Z Z Z Z . . . . > > > > z. ? 2.2.

--e-

N N N N

z. 2. ? z. > > > > Z Z Z Z

2. 2. ? ? . - - .

x x x x

m a m m " C 1 N

N N N - N N N N

N N N N

. . . .

. . . .

- N Z

> ? z: X

N c

N - rY

0

X

?

?

- m

u

K V -I 2

V

a W

a

x x x x Z Z Z Z

. . . . oooz N N N . . . .

. . . . c - c -

. . . . NNCYN

. . . . N N N N

. . . . m m o m . _ . _ o m o m

. . . . m m o m

m m m m . . . .

C C r t

. . . . . - c c c

. . . . x x x x Z.???

0-000 C - L C

. . . .

. . . . N N N N

. . . . x x x x z . . z z. 2- c - c -

. - . . X X X X x x x x

? z. ? ? - - c c

0006 0 0 0 0

. . . . . . . . C??? - 9009 I???? 0

E???? o v z w x m a n n a a a m a

w w w w w

- v v v v

???? v z w x a n o n K K K K w - w w

v v v v J J J J - 1 2 2 2 a a a a v v v v

3 2 2 J - I 0-1-12-1

v v v v ~ a a a a

.. 2-13-1 JJ2-1 - 1 - 1 2 - 1 3 2 A - 1 J

v v v v P v v v v a a a a o a a a a i 3 J J - 1 i A 3

v v v v a a a a 2-1-12

3 J d 2 - 1 o a a a a K V V V V

t L + * * c

w u v v v v v v v v v u v v v u v v v v u V V V v u v V

60

ORlGlNArL PAGE IS OF POOR QUALITY

4---

N N N N Z Z Z Z . . . .

--- e-- ---- ---e N N N N

z. ? ? ? 2. 2. ? z. ? z. z. ?

> > > >

x x x x

m m m m N N N N

NP.CY.- "PIN

o m o m

_ . . _ . . . . . _ - .

o m 0 0

x x x x Z Z Z Z

. . . .

. . _ . c c c c

d d d d 0 0 0 0 . . V Z W I n n n n K U K P W W W Y

v v v v J J A J

--e-

N N N N

? ? z. ? > > > > Z Z Z Z

' N N N ' Z Z Z

N N N N

?Z.? ? N N N Z Z t

N N N N

z. 2. 2. 2.

2.2. ? ? > > > >

X X X X Z Z Z Z

. . . . . . > > > z z. 2

z. z. ? . .

x x x

CIl"

. . - c c c

. _ . I D C D C D

CDCDW - . .

' > > > z z z > > > > z. ? z. ?

> > > > ? ? ? z.

> > > > ? Z . f ? x x x x Z Z Z Z

> > > > ? ? z. z.

> > > > z. ? 2. ? . . .

x x x

CYmm z. 2. ?

' -. -. -. N N N C C L

x x x x 2. ? ? ? n o o m c c c - . . . . o m m m C C r C

x x x x ???? o n m m . . . .

- c . - c

. _ . . O P P P

. . . . O O O O

. . . . m m m m N N N N

P . P . t - ( D . . . .

- c . - c

. . _ . c c c c

. . . . O P P O

m m m m

rn tnrnrn

x x x x

. . . .

. . . .

. . . . z. 2. 2. z. .-cc.-

. . . . .-.--o N N N N

m m m m

m m t n l n

. . . .

. . . . . . .

o m o

n m m . - . . . -

. . _ . m m m o

o o o o

x x x x

. . . .

. . . . z. ? z. z.

0 0 0 0 ;i;i

c c c c . . . . 0000

a n o n a a a a ---I

v v v v J J d J

. . . . P P O P

. . . . P P P P

. . . . V P P t

x x x x . . . .

Z.z.z.2 C - C r

0 0 0 6 . . .

. - . . * P O 0

. . . . P O P 0

x x x x . . . .

????

9 9 9 9 9 9 9 9

c . - c c . . . .

V Z W I a n o n a a a a - w w w

v v v v -1AJ - I A A J J

v v v v a a a a

. . . . r n m m m

. _ . . . . . . . . - x x x Z Z Z

x x x x ? z. 2.2.

&Odd G i W X

cc . - . -

0 0 + 0

a n o n a a a a v v v v

0 J A J J

3 A J J A

a v v v v o a a a a

*

x x x x Z.Z.Z.Z.

y o - ' :

c c c c . . . . tnrnrnrn

. . . . V Z W I a n o n a a a a - w - w

v v v v J A J J A A A J

v v v v a a a a

x x x x 2.2. ? 2.

x x x x 2.2. ? z.

x x x x 2.2. ? 2. c c c -

x x x z. 2. z. - c -

. . . --.- . - . 9 4 9 ' -99 Z U f a n n a a a v v v J J J A d d

v u v

- -w

a a a

r r C C

0066 . . . . . . . 6006 . . . .

. . . 000

9 6 6 v z w a n n

9 9 9 9 V Z W I

E a K K v v v v

a n o n - w - -

0 0 0-0 _ _ - 0 0 0. 0- G i W I n n n n a a a a - - -w

v v v v

. . . . G i W I V Z W I n o a n a n o n a a a a a a a ~ w - - w ---- v v v v v v v v rn

J A J J

v v v v A A J A a a a a A J J L

3 v v v a a a a

li t

u V v u V V V v u V V V w w

- N z > ? ? X

N

W

‘9

X z. 0

i

c

0

n - K V

d d

V a

---- N N N N

Z.t.z.5 ? ? ? ? ? 2. z. 2.

> > > >

x x x x

---- N N N N

*.??? N N N N

2. z. z.? N N N N

2. ? 2. ? > > > > ? 2. ? ?

> > > > 2. ? ? ? ? ? ? ? r t x x x

N N N N

. . . . c c - c

. - . . r r - r - r - . . . .

r - r - r - r -

> > > >

x x x x ? ? ? ? ? ? ? ?

> > > > ? ? ? ? z. z . - z- z x x x x

N N N N

> > > > > > > > ? 2. ? ? ? ? ? ? z z 2- z- 2- 2- z- z- x x x x x x x x . _

> > > > > > > >

x x x x x x x x ? ? ? ? Z.???

x x x x

m m m m ?Z.?? X X X X

N N N N Z.???

m m m c c m m m m . . . . - - - o N N N N

C D w w w . . . .

m m m m (YNNN

.--.-o N N N N

r r - r -b

. . . .

. - - .

. . - . r-I-I-r-

m m m m N N N N . . . .

m m m m . . . .

N N N N

. . . . m m m m

m m m a . . . .

. . . . -.--o N N N m m a m a . . - - . . . .

a m m a

x x x x . . . .

? 2.2. ? 0 6 6 0 - - c c

0 0 0 0 v z w r o n n n K K K K v v v v d J d d d d d d a a a a v v v v

---I

. . . . N N N N

O l m m m . . . . . . . .

w w w w

W w W W . . . . . . . .

IDCDww . . . .

m m m a . . . . m m m m

x x x x . . . . z. z z- z . . -.-.-.-

. . . . x x x x ????

. . . . . . . . . . . . . . . . x x x x x x x x r t x x x x x x x ? ? ? ? ???? z.??? ??Z.?

c c - c c c c c

. . . . x x x x x x x x z. ? ? ? 2. ? ? 2. - c c c ---.-

. . _ . x x x x ? Z . ? ? c c - c c c - c - - . .

9 9 9 9 9 9 9 9 V Z W I a n o n ~ K K K Y W W W

v v v v I-

. . . . g?,? . . - a ? -

6 6 6 0 0 0 0 0 V Z W I n n n o ~ K K K e w u -

v v v v d d d d d d d d

v v v v a a a a

. . . . 9 9 9 9 9 9 9 9 V Z W I n a L a - w u -

& & d o ? :?-. .I- . .

V Z W I

K Z K K v v v v d d d d -122-1

v v u v

a n o n u u w -

a a a a

6006 0 o q q V - i W I w w - u a n n o K K ~ K v v v v d d d d J d d d

v v v u a a a a

. . . . 8 9 8 9

;i;i nnnn Y W W U

K K K K v v v v

. . . . U Z W I

K K K a v v u v J d d d A d d d

v v v v

n a n n “ w - u

d a a a

K K ~ K v v v v

m

. -

--e- N N N N

2. ? 2. ? ? ? 2. 2. > > > >

---- N N N N ???? 2. ? ? ? > > > >

-e-- ---- N N N N Z.Z.??

e--

N N N 2. 2. 2.

2. ? ? ? ? 2.

> > >

x x x

0 0 0 N N N

P P P N N N

. . .

. . . c c c

c c c . . . c c c

c c r . . . x x x ? 2. ? T O O

?-.E

c c c . . - . .

v z w a n n a a a --w

v v v J J d A d d

v v v a a a

N N N N z z z z N N N N ??Z.?

N N N N ? ? 2.2. . . . .

> > > > ?Z.??

???? x x x x

m o m m

> > > > z z z z > > > >

x x x x ? 2. ? ? Z.Z.?? m m m m N N N N . . . .

> > > > ? 2. ? 2.

??Z.? x x x *

> > > > ???? Z.??? x x x x

o m o o

> > > > ???? ???? x x x x

0 0 0 0

. . . . x x x x

P O P 0 ????

x x x x Z.Z.??

x x x x

N N N N N N N N

N N N N N N N N

2. ? ? ? . - . .

~ m r i w . . . . c c c c

c c c c

N N N N . . . . c c c c N N N W

Q Q Q Q

- - . . N - N N

- . . - m o o o

. . - . P t P P

. . . . o m m m

. . . . P P P P

. . . . m o m o NN" . . . . c r c c c c - c . . . . C C L L

c c c c

o o m o N N N N . . . . $$$O Q Q Q Q

Q Q Q Q . . . . x x x x ????

Q Q Q Q . . . . 0 0 0 0 c c r c

Q Q Q Q

dm-60

. . - . x x x x z z 2- z-

???,

. . - . - r e

Q Q Q Q Q Q Q Q

0 6 0 i . . .

. _ _ . x x x x ???? c c c c

c c c .

Q Q Q Q 0666 c c c c . . - .

. . . . c c c c c c c c . . . . c c c c

. . . . c c - c

c c r r

. . . . c c c c c c c c . . . . r r r c c r c c

Q Q Q Q Q Q Q Q . . . . x x x x ??Z.? c c c c . . . . O O O O N + N - .C . .

. . . . $ Z O O

. . . . c c c c c c c r . . . . r - c c . . . . . . . .

X * X X Z.???

0000

c c c c . . . . . .

. . . . x x x x ????

. . . . x x x x z z 2. z. . .

. - . . x x x x ??Z.?

x x x x ????

x x x x ? 2. ? 2.

x x x x Z.???

c c c c c c c c r c c c L C C C c c c r

ioii :? 1

v z w x n n n n a a a a - w - w

v v v v

. . . . 0 9 0 9

. . . . 0 9 0 9

. . _ _ ?,or? ? :?? v z w r n n n n a a a a w w - w

v v v v

. c .o v z w x a n a n a a a a

. . . .

v v v v

9 9 9 9 v z w r nnnn amaa W I I I

v v v v

???? v z w x a n a n a a a a w w w -

- v v v v

. . . . v z w x n n n n a a a a w---

v v v v

. . . . v z w x n n n n a a a a ---- v v v v

. . . . v z w x n n n n a a a a w w w w

v v v v

. . . . v z w x n a n n a a a a W W W W

v v v v dJJ-1 JdJ-1 < a a a v v v v

c

d d d - 1 - 1 A A - 1

v u v v a a a a J A d d

3JJ-1-1 o a a a a a v v v v

d d d d

u v v v a a a a a a a a

u v v v

*

V V V U V 'J V v u V V V V

- N z > 2.

? X

P N

O N

c - - r

X 2 - 9 ? I

K V

-1 -1

u

a w

a

_e-- N N N N

? 2. ? z. ???? z-z z z

> > > >

x x x x

NNNCr . . .

--e-

N N N N

z. ? z. 2. ???? ????

> > > >

x x x x

N N N N

. . - _ c c c c

---- --e- ---- N N N N z z z z . . . .

N N N N

? z. z. 2.

??Z.?

??Z.?

> > > >

x x x x

N N N N

? ? z. 2. N N N N z z z z

> > > > z. ? 2. ?

> > > > ? ? z. z. ? z. z. ? x x x x

> > > > z. ? 2.2.

?Z.?? X X Y X

m m m m N N N N

> > > > ? ? ? ? z z z z x x x x . _ . .

CYNNN

> > > > ???? 2.2 z z x x x x

- . . m m m m N N N N

> > > > z. z z z-

z. ? z. ? . .

X X X Y

N N F I N

> > > > ? 2. z. ? z z 2.2 x x x x . . -

m m m m N N N N

x x x x z. z. ? 2. m a m a N N N N

X X X Y

m m m m z. ? ? ? N C 4 N N

m m m P N N N N

N N m N

. . - .

. . . . r c - c . . . . (YN" c - r c . . . . x x x x ? ? z. z. 9 0 9 9 0000

- - r c . _ . .

m m m m

m m o m

m m m o

N N N N . . . . N N N N . . . . _ C L C

V O V O N N N N

P P O P N N N N

. . . .

. . . . $$$C

. . . . m m m m N N N N . . . .

. . . . c c c c

. . . . m m m m CYNNN . . . . o m m m c c c c

. . . . c c c c

. . _ _ P P P O c c c c . . . . V P P P c c c c

. . . . c c - c

. . . . C Z Z Z . . . . m m m m c c c c

. . . . m m m m N N N N

m m m m . . . .

c c - c

. . . . " N N c c c c . . . . N W N N r c c c

. . . . m o m m r c c c . . . . n o m m r c c c

c c c c c c c c . . . . c c c c

c - c r . . . . Y X X X

??Z.? c c c c

6 0 0 0 0 0 0 0 . _ v z w x a a a a K a K K v v v v -1-1-1-1 -1-1-12

u u v u

v w - 1

a a a a

. . . . m m m m c c c c

. . . . m m o o - c c c

. . . . Z C C C . . . .

x x x x 2. ? ? ?

. . . . . . . . . . . . . . . . x x x x x x x x x x x x x x x x ?Z.Z.? ???? 5Z.z.z. ?Z.??

r c c c - - - c

. . . . . . . . x x x x x x x x 2- 2.2. ? 2. ? ? ?

c c c c

. . . . x x x x ??Z.?

c c c c

6000 9909

~ U W V W

V Z W I

K K O ! K

n n a n w - w -

c c c c . . . . 9 0 9 9 0000

c c c c . . . . m o m 0 0000

. . . . c c c c

0 6 6 0 ?6?? v z w x a a a a

~ V V V V

w w w - ~ K K K

i o $ - .o . .

0060 0 0 6 0 v z w x

K O ! = = v v v u -12-1-1 -1-1-12

u u u u

n a a a --vu

a a a a

0006 0000 G i w i a n a n w w w - K K K O ! v v v v -1-1-1-1 J J - 1 2

u u u u a a a a

. . . . v z u r a n a n K a a ~ -w--

v v v w -1-12-1 2 J 2 - 1

v v u v a a a a

. - - . . . . . v z w x

K K K O ! v v w u

a a a n V I W W

J-12-1 3 2 2 2 2 o a a a a K V V V U

-1-1-1-1 -1-12-1

v v v u a a a a

-1-1-1-1 3-1-1-1-1

P V V V V o a a a a

d-1-1-1 22-1-1

u u u u a a a a - 1 - 1 - 1 2

3-1-1-12

u v u v v o a a a a 2J-1-1

3-12-1-1

K V V V U o a a a a

t t

-e-- ---- N N N N

???? ? 2. ? ? ? ? ? ?

> > > >

x x x x

N O N N

. . . . - - - -

. . . . + + + + - c c c . . . . p . r+p . c c c c _ - . . x x x x ???? C r C C

0006 . . .

- N

2.

?

> z X

Ln (Y

m rl

m c

m ? X

c

In I-

V

K V -l d

V

a - a

N N N N

???? N N N N z z z z N N N N

2. ? ? ? > > > > z z z z

N N N N

? ? ? ? > > > > Z Z Z Z

. . . . > > > > 2.2. ? ? ???? x x x x

m m m m “(YN

> > > > ????

> > > >

x x x x

9 9 0 9

???? ? 2. 2. z.

> > > > ????

> > > > ? ? ? ? x x x x Z Z Z Z

. . . . . - . - x x x x

N N N N ????

x x x x

m u l m m N N N N

LnLnLnLn N N N N

???? . . . . . . . . 2$2?

x x x x z z z z x x x x

m m m m NN(Y(Y

LnLnLnLn N N N N

Z.???

. . _ .

. . _ _

. . . . m m m m N N N N

w w w w N N N N

. - . .

. . . . $2$?

. . . . m m m m N N N N

w w w w “(YN

. . . .

. . _ _

. . . . 0 0 9 0

9 O P P . . . . . . . .

L n i n v ) L n N N N N . . . . !c::‘” !!E:: . . . . . . . .

. - . . - c c c

. . . . O P P Q

. . . . t - + P - I - - c c c . . . . t - + + I -

x x x x

c c - c . . - . 2. ? ? 2.

9 3 9 9 c - c c . . . .

. . . . m m m m C r C C

. . . . z m z z ‘ D E ? ? +I-I-+

c c c c . . - . +++I-

x x x x -.--- . . . . z z 2- 2-

Y , o ’ ” ?

. . 7 - c - . . - -

+ r w b - - - r . . . . +++I-

x x x x - - - - . . . . ????

9 9 9 9 c - - c . . - .

. . . . ? E 2 2

? ? ? ? . - - -

x x x x

c c c c . . . .

. . . . 2220

??Z.?

9 9 9 9

. . . . x x x x

c-7.- . . . .

. . . . z z m c ???? . . . .

x x x x

C r C C

0 0 0 0 . .

. . - - zmmc . . . . . . . .

x x x * ? 2. ? ?

x x x x f ? ? ?

9 9 9 9 9 9 9 9

c c - c . . . .

v z w x a a a a a K a K v v v v w w - 1

x x x * Z Z Z Z

x x x * Z Z Z Z . . . .

- c c c - . . - 9 9 9 9

c c c c

LnOLnO -0- $‘,o?

:? 1- v z w i a a a a - w w w K K K K v v v v

9 9 9 9

$ V V V V

v z w x a a a a K a K K w w w w

9 9 9 9 v z w x a n a n K K K K v v u v w - w w

o ‘ * o o v z w x a a a a K K K K v v v u

. . - . - - -w

9 9 9 9 v z w x a n a n U U K K v v u v - - -w

9 9 9 9

m v v v v

v z w x a n a n K K K U W Y W W

. - . . v z w x a a a a w - w w K K K U v v v v J-l-l-l J d d - 1 -1-1-l-l

3J- ld - l

K V V V V o a a a a

J-IJJ J d J d

v v u v a a a a -1d-12 -1d-l-I

v v u v a a a a d d d d

3 - l d - I d o a a a a a v v v u

2 A d d a a a a u v u v

2 - l A - l

v v u v a a a - x

* c

+ c c

c +

---e N N N N

? 2. ? 2.

Z.??? > > > >

--e-

N N N N

? ? 2. 2.

z z. z z.

2. I ? ?

> > > > . .

x x x x

m m m m N N N N

m m m m N N N N

. . . .

. - . . ? ? ? ?

- - -A

N N N N

???Z.

? z. 2.2.

? ? 2. ?

> > > >

x x x x

m m m m N N N N

w w w w N N N N

. . - .

. . . . ???C ? ? ? ?

? 2. ? ?

. . . . _ . _ _

x x x x

C - L C

0 0 0 0

++?? V Z W I nnnn K K K K v v v v -1JJ-l JJJ-l

v u u v

W V W I

a a a a

- - A -

N N N N

? ? ? ? ?Z.??

2. z. 2.2.

> > > >

x x x x

m m m m . - . .

c c c c

+$$ 0 0 0 0

Z.???

N N N N

x x x x . . . .

c c c c

0 0 0 0 . . .

--e-

N N N N

? ? ? ? N N N N

? ? ? z. N N N z z 2-

2. ? ? . .

> > > > > > > ???? x x x x z z z z

> > > > 2. ? z. ? x x x x z z z z

> > > > 2. ? 2.2.

> > > > 2. ? ? ?

> > > >

x x x x Z.Z.2. z. z z z. 2- . _

x x x 2. ? 2.

x x x * Z Z Z Z

x x x x ? ? ? z.

x x x x 2. ? ? ? m m man N N N N . . . .

x x x x 2.2. ? 2. m m m m N N N N

w w w w N N N N

. . . .

i$ii i$$i . . . . x x x x ? ? ? ? C C L C

. . . . m m m m N N N N

( D w w ( 0 N N N N

. . . . . _ . .

m m m m . - . .

c c c c

. . . . P P P f

. . - . * * P O

P P P V

+ . - .

P P O P

. . - . ? ? C ?

. . . . m m m m c c c c . . . . 2 2 2 5

. . . . ? E ? ? N N N N

x x x x . . . .

f ? ? ? c c r r

. . . . . . x x x ft .5

x x x x Z.???

x x x x Z.?? ?

x x x x Z.???

x x x x ? z. ? 2.

c c c c . . . . 9 0 9 9 v z w r n o n o ?6?? w - w w K K K U v v v v Jd-1-l -1J-I-l

V O V V a a a a

c c - c C L c C

0 0 0 0 e?? . . . .

9 9 9 9 9999 v z w x o n a n

C Y K K K K v v v v c - w w w

3 d J J d

m'm 'o l-r-

' .o 9999 v z w x nnnn ~ K K K o v v v v

PI JJ-1-1

3JJ-1-1 ~ a a a a ~ V V V V

w w - 1

???: V Z W I a n a n K K K K w w w w

v v v v

9999 v z w x w w w w nnnn K K K K v v v v J A J d JJ-1-1

v v v u u a a a

. t . C . . . . v z w x

K K K K v v u v -1dd-l A J - 1 2

v v v v

n o a n - - w w

a a a a

. . . z w x

K K K v v u

a n n w-w

v z w x

K K K K v v v v

a n a n w w w w

K K K K v v v v -1-1-1-1 dJ-13

J J J J

v v v v a a a c

J-1-l-1 a e a a v v v v

O J J - 1 - l ~ a a a a

V O V V

J d A A

v v u v a a a a

c t t

* t

V v u V V U u u U V u v u U

I

I

I

1

I

I I

I

I I I

I , I

1 - 1 2 ?

! z

, >

X

P

' . P

c

N

c

N

X

?

9

c

N

I a a w

V

-l -l

V a

- - A n

N N N N

?Z.?Z. ? 2. ? 2. > > > >

nnnn N N N N

2. ? z. ? ? ? ? z. ? ? ? z.

> > > >

x x x x

P O 0 0 N N N N

P O P 0 N N N N

N N N N

N N N N

. . . .

. . . . c c c c

. . . . v r c c

. - - .

e--- N N N N ?z.z.? Z.Z.? = > > > >

x x x x Z Z Z Z

I -

N N

2. ? ? ? ? ?

> >

x x

c c

N N . . c c

N N

N N N N

N N N N

x x

. .

. .

. . ? ?

? ? 9 9

c c . .

w z a n a a w w

v u d-l d A

v u a a

-_-- N N N N ??*.z. ??Z.?

????

> > > >

x x x x

m m m m N m N N

m m m m N N N N

N N N N

N N N N

- * . . . . . .

- c c c

. . . . _ c c c

---e N N N N

? 2. ? 2.

???? ? ? ? ?

> > > >

X X K X

m L 7 m m . . . .

c c c c

. . . . N N N N N N N N

N N N N N N N N

x x x x

. . . .

. . . . ???? - - - e

N N N N

? ? 2.2. N N N N

2. ? ? ? > > > >

x x x x

w w w w

w w w w

???? ? ? ? ? . . . .

> > > > ????

> > > > ???? ???? x x x x

I - I -P-r -

> > > > ? ? ? 2.

x x x x

l n l n m m ? ? ? ?

x x x x z . . z z. z. m m m m

x x x x ???? gNogg 0000 NNNCY . . . .

. . . . m m m m N N N N

w w w w N N N N

. . . .

. . - . - c - c

. . . . m m m m

. . . . w w w w

. . . . r- r - r - r -

. . . . m m m m

CVNNN N m N N N N N N N N N N

x x x x

. . . . - - . . . . . .

??Z.Z.

"?? c c c c . . . .

. . . . c - c c

N N N N

N N N N

x x x x

. . . . c c c c

. . . . ????

:? :?

c c c c . . . . m 0 U ) r - P .o . V Z W I n a n n a a a a w - w w

v v v v J2-l- l JJ2- l a a a a v v v v

. . . . c c c c

N N N N

N N N N

X Y X X

. . . . --.-e

. . . . 2 z z. z. . . c c c c

&6++ . .

. - . . N N N N N N N N

N N N N N N N N

x x x x

. . . .

. . . . z z z- z. . . - c c t . . . .

. . . . N N N N NCVNN

N N N N N N N N

x x x x

. . . .

. . . . ???? - e - - . . . . P O P W . . . .

N N N N N N N N

N N N N N N N N

x x x x

. . . .

. - . . t. ? ? ?

? ? ? ? - c c c . . . .

N N N N . . . . c c c c

NNC(N

x x x x . . . .

? 2. ? ?

9 9 9 9 9 9 9 9

c c - c . . . .

V Z W I a n a n a ~ ~ a -w--

v v v v A - l - 4 2 A - l - l - 4

v v v v a a a a

x x x x Z.???

x x x x ????

c c c c . . . . 0 0 0 0

- r c c . - . . '99'90 0000

c c c c . . . . m o m o 9 0 9 , 999-.

V Z W I a n n o a a a a w w u w

v v v v

9 9 9 9

N N V V V V

V Z W I a n a n a a a o v - w u

0000

o n a n a a a a

;i;i - - w w

v v v v

. . . . V Z W f n n n n a K a K v v v v A - l - l - l -l-l-l-l

v v v v

- - w w

a a a a

. . . . V Z W I n n a n a a a a U W I I

v u v v .. d-l-l-l

3 - l - l d - l o a a a a a v u v v

-1-l-i-l

v v v v a a a a A d J - l

v v v v a a a a

---- e--- ---- N N N N Z Z Z Z

N N N N

Z.Z.?? N N N N Z.?? 2.

N N z z . . > > ? ? ? 2. x x

c c

N N . - .-.- N N

N N N N

N N N N

x x

. .

. .

. _ z. ?

99

c c . . N - . . U I a n a a -- v u J-l -Jd

v u a a

. . . . > > > > ? ?Z.?

> > > > ? ? = ?

> > > > Z Z Z Z

> > > >

x x x x z z z- z-

? 2. ? 2.

... m m m m N . a N N

.-.--o N N N N

m m m m N N N N

m m m m N N N N

. . . .

. . . .

. . - .

> > > > ? z. ? 2.

zzzz- x x x x

m m m m . . - . . . . c--.-

. . . . V V V P N N N N

O V O V N N N N

x x x x

. . . .

. . . . z. ? 2. ?

9 9 9 9 0000 a a n a U Z W I

a ~ a a

c c c - . . . .

. - - w---

o u u u u N

J- ld- l 3-l-l- l- l o a a a a a u u u u t

c

> > > > Z.???

?Z.?? $?25?5?

x x x x

Q Q Q Q . _ . .

P V P P N N F l r J . . . .

. . . . x x x x z. z. z. 2.

x x x x 2.2. 2.

x x x x Z.??? N N N N N N N N

N N N N N N N N

. . . .

% * X I z z- 2- z m m m n

m m m m N N N N . . . . N N N N

. . . . m m m m

m m m m

m m m m C I N N N

m m m m N N N N

- - . . . _ . . . . - . - . . .

. . . . - r - -

. . . . r C - C

. . . . m m m m N N N N

m m m m N N N N

. . . .

. . . .

. . . . lnlnlnln N N C Y N

l n l n l n m N N N N

. . . . . . . .

rn ln ln rn N N N n

rn lnrnln N N N N

. . . . . . - -

N N N N N N N N

N N N N N N W N

. . _ .

. . _ _

V V P V N N N N

V P O V N N N N

x x x x

. . . .

. . . . ? 2. ? 2.

9 9 9 9

c c c -

0000

V Z U I o n o n a a a u w - w w

u v v u J J A J - l -J - ld

v u u v a a a a

V V V P N N N N

x x x x . . . .

2.2. ? ?

m ? ? ? o o o -

c - c c . . . .

. . . . . . . x x x x Z Z Z Z

x x x x Z Z Z Z

% X X X

2.2. ? ? * x x x ? ? 2.2.

x x x x z.??? . . - . - - c c

6600 66?? U Z W I a a n n a a a a - w - w

r n u v u u N

c c - c

0 0 0 0 C - C r . . _ . Y?Y? 9997 U Z U I a n o n

- - - c

0 0 0 6 9 9 9 9 U Z U I n n n a

. - . . ? ? ? ? . . . .

9 9 9 0 U Z W I a n o n

999': U Z W I a n a n --w- u a ~ a u u u u

9 9 9 9 V Z W X a n n n a a a a --w-

u u u u - w w - a a a a u u u u

2 - l - l - l

v v v u a a a a

c

I

i I

t

t I

1

I

I

4-

N N z.? ? 2.

? 2.

2 0

> >

x x

m m N N

N N

Inrn N N

o m N N

x x

. .

. . - . 2.2.

9 9 ??

c c . .

w x a n U P v u d d 2 d

u u

-- a a

-_-- N N N N

??f? I--- ----

N N N N z z z. z.

z. 2.2. f 2. 2. f z.

. . > > > >

x x x x

N N N N z. 2. z. 2.

2. 2. ? z. ? z. ? z.

> > > >

x x x x

N N N N

? ? 2.2. N N N N z.??? N N N N

? 2. z. 2. > > > > ?Z.??

> > > > Z.Z.2.Z.

? ? 2. 2. x x x x

> > > >

x x x x 2. ? ? z. z .z .Z.5 1 s r 1 . . . . m m m m c c c c . . . . P-Pr-r- NNCYN . . . . r - r - r - N N N N

x g x x . . . .

2.2. z. z. IiOd6 c c c c

> > > >

x x x x z.Z.??

Z.?? z. m m m m & N N N

- - y o N m N N

r - r - r - r N N N N

r - r ~ r NCY"

x x x x

. . . .

. . . .

. . . .

. . . . 2. z. z. ?

9 9 9 9 9 9 9 9

C L C C . . . .

U Z W I n o a n K U K K v o v v 2 J J d 2 J J d

u u v u

w 4 w -

a a a a

> > > > ? ? 2.2.

z.5z.Z. x x x x

m m m m N P l N N

- - - o . . . . N N N N . . . .

> > > > ? x x x x ? ? 2. ???? P P P P c c c c . . . . c - - r

X X Z K Z Z Z Z

x x x x ????

x x x x ? z. ? ? . . . .

c c c c c c c c . . . . c c c c

N N N N - c c c

m m m m N N N N . . . . c c - 0 N N N N

w w w w N N N N

. . . .

N N N N c c c c . . - . c - c c

. - . . l- l-br- N N N N

l-l-I-r- N N N N

X X K X

. . - .

. . . . z. ? z. ?

9 9 9 9 E???

c c - c . . . .

U Z W I a n a n K K U U ww--

r - v v v v N

d d d d

o m m m - c c c . . _ . c c c c

. . . . m m m m

m m m m N N N N

N N N N

x x x x

. . . .

. . . . z. ? 2.2. -I--

0 0 0 6 ?6?6

P P P P - c c - . . . . P P P P r c - c . . . .

. . . . N N N N c c - c . . . .

w w w w N N N N

w w w w N N N N

. . . . . - . . W W U W N N N N

w w w w N N N N

x x x x

. . . .

. . . . z. z z z- _ .

. . . . m m m m

m m m m N N N N . . . . N N N N

x x x x . - . .

m m m m

m m m m N N N N

N N N N

x x x x

. . . .

. . . . 2. z. 2. ?

4 9 4 9 9 9 9 ,

c c c c . . . .

O Z U X

K K U K u v v v 2-1-1-1 22-12

u u v u

a a n a - - -w

a a a a

m m m m N N N N . . . .

w m w w N N N F (

x x x x . . . . 2. z. 2.2. c c c c

. . . . m m m m N N N N

x x x x . . . . f 2. ? ?

c c c c

0 0 0 0

d o ? ? v z w x a n a a w - w w U U K K

w v v v v CJ

c c c c

0 0 0 0 + ? d ; O u z u x n a a n K K U U - -w-

u v v u - 1 - 1 2 - 1

v u v u -1-1-12 a a a a

. . . . ???-. U Z W X a n a n V Z W I a a a n w - w w K K U K v v u v dd-1-1

K K K U ~ V V V V N -1-1-12

3d-1-12 o a a a u a u u u u -1d-1d

u u v u a a a a 3 - I d A - I

K V V V U o a a a a d d 2 J

v u u u a a a a

t t

c

69

I- o m I

A

--- N N N

? 2.2.

z. z. ? 2.2. z.

> > >

x x x

m m m N N N . _ . - 7 . -

N N N

N N N m o m N N N m m m x x x

. . .

. . .

. . . ? ? ? " 0

a. a. -.

c c c - . . a a

a n n a a a

v z w w u -

v v v A - l A A - l A

u v v a a a

---- N N N N

???? e--- ----

N N N N

? z. ? 2.

e--- ---_ NN" z z z z h N N N

???? N N N N

2. ? z. 2. N N N N

2.5 2.5 z. ? z. z.

2. ? ? 2.

> > > >

x x x x

r - l l r - r c - c - . . . . r - l l - r - - c c - . . . . N N N N m m o o N N N N m m o o x x x x

. . . .

. . . . Z . ? ? ? - r r -

. . . . > > > >

x x x x z. ? 2.2.

? 2. f ? E z z ?

> > > % > > > >

x x x x x x x x ?Z.?? ??Z.?

z. ? 2. ? 2. z. ? ?

> > > >

x x x x ? ? 2. z.

z. z. 2.2.

z z z z z ? z z . . . . . . . .

r C - C

m m ( 7 m . . . . c - c r

x x x x ? ? ? ? ? z z z . - - . - c c c

. . . . N N N N m m m m N N N N

x x x x

. . . . n m m o . . - . ? ? ? Z .

0000

c c c c

6600

x x x x ???? m m m m fi N,N N

m m m m N N N N

.-.--o NCYNN

. . . . tn tnmtn c - c c . . . . y c c c

. - . . c - - c

m m m n

m m m m . . . .

- c c c

. . . . x x x x ????

0 0 9 0 0090 G Z W I

- c - c . . . .

nnnn a a a a - w w -

- v v v v m

- l - l A d 3 A - 1 A - l

K V V V V o a a a a

. . . . c c c c

. . . .

6606 m m o m $ s i . . . . x x x x ????

o m m m . . . . x x x x x x x x

? ? ? ? x x x x ????

c c - c . . . . 0 0 0 9 9 9 9 9 V Z W I a n a n a a a a Y I I W

v v v u A - l A A A A A A

u v v v a a a a

c c - c c - c -

0066 0600

. - . . 9 9 9 9 0909 V Z W I nnnn

6000 0000 V Z W I a n a n

?':?-. V Z W I a n a n Y I I V

a a a a v v u v

. . . . V Z W I n n a n a a a a -w--

N V V V V m

W I I I

a a a a o v v v v - a a a a

v v v v VI

J2-l-l 3-1-l-l-l

K V V V V o a a a a

1

V V U V V v u V V v u V

70

I

1 I I

I

I I I I

t

f

1 I

I

1 I 1 - I ?

, N

>

Q N

N m N (1

X ? c -. c 4 a X a a W

V

-1 -1 4 0

ORIGINAL PAGE 1s OF POOR QUALITY

n--- N N N N

???? -I-- N N N N

Z.???

? 2. ? ? > > > >

ne-- N N N N

???? ? ? ? ? > > > >

---e N N N N

Z.???

2. ? ? ? ? ? ? ?

> > > >

x x x x

r - r - r - r - c c c c . . . . r-r-r-r- c - r -

-_-e N N N N

???? .???? ? ? ? ?

> > > >

x x x x

Q a Q Q N N N N

---o NNCYN

P P O P o m m m P P P P

x x x x

. . . .

. . . .

. . . . m m m m - - . . ???? c c c c . - . . c c o c 4 4 . 4 a a - a GiLi a n a n aaarr w w - w

v v v u 2-l-1-l 2-12-l

V V V V a a a a

e--- N N N N

Z.???

2. ? ? ? 2. ? ? ?

> > > >

x x x x

I - I - r - r - c c c r . . . . r- r - r - r - c c c c

---- N N N N

?Z.??

? ? ? ? > > > >

ne--

N N N N

? ? ? ? ? ? 2. ? > > > >

N N N N

???? ? ? ? 2.

2. ? ? ?

> > > >

x x x x

> > > > Z Z Z Z

> > > > z z z z > > > > ? 2. ? ? . . . .

x x x x

r - r - I - r - Z.???

c c c c

. - . x x x x

Q Q Q Q N N N N

-.-.-o

???? . . . .

NCYNN

x x x x ???? ? ? ? E

x x x x ? ? 2.2. E ? ? ?

x x x x Z Z Z Z

x x x x z z z z . . . . Q Q Q Q N N N N .---o N N N N

m m m m m m m m

. . . . - . - . _ _ _ _

. . - . p.r - t - r - c - c c . . . . m m m m m m m o

. * . . - c c c

. . . . v o w 0 m m m m * P O P

x x x x

. . . . m m m m . . . . ????

9 9 9 9 .--e- . . - .

. . . . c c c c

. - . . m m m v ) m m m m m m m m m m m m

. - . .

. - . .

. . . . m m m m m m m m

o m o m x x x x

. . . . m m m m . . . . ??Z.? c c c c

. . . . m m l n m m m m m

. . . . m m m m m m m m m m m m m m m m x x x x

. . . .

. . . . ???? c c - r

. . . . . . . . ( D w ( D w w w ( D ( D m m m m m m m m w w w w ( D W W W . . . . . . . .

. . . . P O O P m o m m P O O P m m m m x x x x

. - . .

. . . . ????

YYY? c - - c . . . .

. . - . m m m o

x x x x m m m m . . . . ? ? ? ?

. . . . m m m m

x x x x m m m m . . . . 2. ? ? 2.

"-340 c c c c . . . . go?-. v z w x a n a n a ~ ~ a W I I I

v v v v 2 J 2 - l 2 J 2 - l 4 4 4 4 U V U V

m m m m m m m m . . . . . . . .

x x x x Z Z Z Z

x x x x ???? r c c c

0 0 0 6 ??66 U Z W f a a a a aaKK

w v v v v 0

w w w w

x x x x Z.??? . . . .

C C L t

6000

v z w r a n a a

. . . ?6?? v w w w a K K a

m v v v v 0 -122-1

3-122-1 0 4 4 4 4 K V V V V

c c

c c - c C r C C . - . . c c o c 4 4 * a aKCK v z w x o n a n K K a a v v v u 2 - l A 2

. . . . w w w -

L C t C

m'm-0 '0 0 0 0 -

. - - . 9 9 9 9 9 9 9 9 v z w x n a a a a a a a w w w w

m v v v v (3 -1-12-1

3-1-1-l-l 0 4 4 4 4 K V V V V i

0000 009-. v z w x a n a n a a a a w--w

u v v v

. . . . . . . . v z w x v z w r a n a n a n a n a ~ a ~ a a ~ a - w w - www-

v v v v o u v v v m

. . . . v z w r a a a a a a r x ~ - w - w

v v v v

. . . . v z w r n n a a K K K ~ w w w -

v v u v -1A-1-1 22-12 4 4 4 4 V V O V

-1-122 -1-122 4 4 4 4 V V V V

2-1-1-1 % A 2 2 2 0 4 4 4 4 K V V V V

-1-12-1 3 - 1 2 2 2 0 4 4 4 4 a v v v v

e c c c

V U V V v u V V U U V V U V U

71

---- N N N N

2. ? 2.2.

?f?? > > > >

---- - A * -

N N N N

2. z. ? 2.

2. 2. ? 2.

Z Z Z Z -

> > > >

x x x x _ . . m m a m N N N N . . . . --.-o N N N N

N N N N z. z- z z - . > > > > Z Z Z Z

N N N N

z.5Z.z. > > > > Z Z Z Z

> > > >

x x x x Z Z Z Z

? 2. ? 2. > > > >

x x x x 2. ? 2. z. 2.2. 2. 2.

. . . . - r - -

. . . . bl-l - l - m m n n

m m n n . . . .

l-l-l-l-

x x x x . . . .

2.2. ? ?

9 9 9 9

C r C -

0 0 0 0

V Z W I npnn a a ~ a e---

P - v u u v 0 A d d d

3 d J d d

K V V U U o a a a a

> > > > Z.Z.2. ? ???? x x x x

m m m m N N N N

> > > > z z z. z.

???? - .

x x x x - . . . . . . . . . .

x x x x ???? m m m m N N N N _ . _ . r c - 0 N N N N

l-l-l-l- n n n n r r rb n m n n

. . - .

. . . .

. . - . x x x x ??Z.? c c c c . . . .

x x x x ????

x x x x Z Z Z Z

x x x x Z Z Z Z - . . -

x x x x . . . . m m a m

- - - Q C Y N N N

( D w w ( D m n n o ( D ( D ( D W n n n n

N N N N . . . . . . . . . . . . _ _ . _

. . . . m m m m N N l "

P'Pl-r- C C C r . . . . 7 - c c

. _ . . m m m m m o o n . . . .

. . . . .---o N N N N . . . . m m m m o c n m m m m m m n o n . . . . . . . .

0 0 0 0 0 0 9 0

m m m m n n n m . . . . m m m m n n n n . . . .

6 0 6 6 - 3 9 9 9

x x x x . . - .

m m m m n m n n . . . . . _ . x x x x Z.Z.Z.Z.

. . * . x x x x Z:? 2. 2.

x x x x Z Z Z Z

x x x x z. ? ? z.

x x x x ? ?Z.? - . - .

c c c c cy.-- . . . . 9 9 9 9 .-OF'

v z w x o n o n . . . .

-w-- a K K K v v v v

d d d d 2 d d d

v v v v a a a a

c - c c c - - r . . . . c c c c

6 6 0 0 . . . 0 0 0 0 . . . . . . . .

t-cor- a a ' 4 :

a c t - a

c c o t - a a . a a = - a _ _ _ _ 990.9 G i l i i v z w x nnnn n n a n K K P ~ K K K K v v v v g v o v v - - w w w---

. - . . V Z U I n o o n ~ a c t a ---- u v v v d A d d d d d d

u u v v a a a a

. . . . u z w x

K a K K v u v u

a n n n Y W I W

. . . . v z w x nnnn a a a a - v w w

v v v v

Viii a n a n ~ a a ~ - w w -

u v v v d d d d d d d d

v v v v a a a a

.. A 2 d d

3 d d d d

a v v v v o a a a a A d d 2 d d d d

v v v v a a a a

c c

V v u U V v u V v u W v u u v V

72

CXGBNWL PAGE IS OF POOR QUALITY

I

I - . I I . . . m

U W . . . . . . . . . .

I ,

I ,

. , v v u u u v u

V

a w "

+ . * * * * * * * * , + * . * * * * * * + I

I , . * * * * * * * * * I

0 W -

A 0 W

0 -

? e " IO E a. -

- P v u O h r .

- h V I--- a v o W N I a o a o x m. a a a - A

v v a - - 3 0 0 O I I a a a u - a

W

f Y z a K (I, W W a

0

0 2

2

a

I- W W bl.

m 0 3 -1 m

U

A m

z e(

W K

m I-

z 3

a

C(

+ + +

0 - A U

0 z 3 0 K u z 0

M

W I-

J 3 V J

v m

a

a

I

> c

I , , I v v u v , I u v v u u v u v v u I I

v v v v u u u u u u v v u u u u v o v v v v u v v v u v

73

? I V

0

0 K u z 0 W I- a J 3 V J a V

v)

3

I

z E c V

K U

-r -r

- a 3 K 0 L L

(D N

N

m

a W u c

In I- \ K e I a X

' V 0 Z Z

0 c n o - w

x a v m z m

VI 0

K . 0

W J K 1-4

a w at-

0 z > K I

W

: E' v)

0

a I

i- z + \x + a a a m u 3 0 z J m u- \

A -

+ >

W

a -

EL - v

11 - K

-I- c z Z W w v Z V O W w - m + K W W J

- 2 W W Cv)

K V 31 0- At- L L I

- a-u

a

a 0

o n w a c-u

-v) W - 3-

53:

W I

a a .. > a

a

0 v)c - 0

- I -

< t z -I a--u J O Y

n . I - w

2 N -

J a w W I lnc

> u rno a -I> c

U m W K m r r w z 0

V J m v c a m 0 \ c w m

w u w m 3 I-J c m

a ? >

a z w

w - t-- I V

I C -uz - 3 w CnQV W K V V I W - +

a

0l-0 t - w e 2

w - w v ) c V I d W - o >

V Z - 0 v w v O X w t - z u c x

a c

- C Z O

o a

w w

4

L W J w v a ~ a > a d o

a v t- I I1 11 11 I1 It II

: - r r m v m w w v v v v v v

o a n a o a K W W W W W W 0 U U U U L U U

U

a d d m a a W J J

3 v v 4 + 4 * . ! &

- 4 c * 0 I D ,

74

ORIGINAL FAQE fS OF POOR Q U A L m

N - e W

\ c

-1

e z -..

u t '? c

c 2 v u z z u u - - LL c c z z w w u u V U w w t t -- . . o m \ \

mf- e(-

+ *

. . * I

o n . .

N N

VlVl 0 0

VlVl

U V

ow

v v w w w w uu. 11 I1

-- Y Y

s t - c

u- r +

o n

PlQ V U 0 0 w w u u

W [3 i i-

v z 0 c t.

W E w n c a A a u

n -1 OVl v u 3

Y -1 I V l I-\ U

e a

-e a i

rn T U 20

e W i

5 3 0 u W

4

E c a

c U

3 V \ 0 3 A

+ u! - t 0 -1 0 V

W

E LL 0

c U w

. . Ln - - - .

N . -

w - l - l - 1 - l - l - l - l v a a a a a a a a > > > > > > > -l0000000 o u u u v v o u

-ld-lJ-l-l-l-l -1-1-l-l-lJ-l-l

u u v u u u v v a a a a a a a a -1- l iJ- l - ld- l - l - l A J - l i i - l

v v u u u u v v a a a a a a a a

1 * . + + t t

75

- cy

V

X O

- K + -

Z t z a

- a a

z5;

- N +

V

X O

-e I--

-Iu

Z 5 z a

- a + s o c 0 - m t

+

I- W d

5 0 I ea N I

N

I- O I

+

> u a > u J Z

i m u a o a x w I V I - i m - I - d 0 \ m z z m I - u m u w w u -

W O O I + u m 3 L . w I - w >

- d d I 'e E m m L - . -

c Ly d z m

I- c I

I-

I- a

I- LL

3 V \ U 3

I-

W z

+ I w u 2 W f"

CL c w

- W I - I U I-

@ I - m a - - a zu - E a 0

+ W

t c 0 I

W

E c a

+ W - t. r I- O

I- a I-

5 z U W

*.

t c - * * c . + * * . t *

z E c a c m m O

C

m

I

f W u z W d 3 a 3 I-

>

a

n

- I- m a d 13 X

I f n a a \ I

- 2 c y -

- + a * + I- t, - \ 3 l n + . c y - + ( I * * -- I - t u - Y 0- a + K m

u - 1- 0 , - m

:5 W U

v v v u u v u 0 u v u u V 0 V V V o u u v u v v u v v u v i: u v u u u

76

CRZiNAl, PAGE .IS OF POOR QUALITY

- I- - Y

rl

I 0

0 z 3 0 K (1

t 0 7 c w m

I

I

I

I ..

h

V

I-

-1

h

V w Y c A

- I - - -

C

I-

X w

- a h

m - > K

h

I 3 , - a

a a -10

+ a 7 X I - - L

. . . .

r r r

m e W A c 3 0

r + 4

w a a a a > v v v v

+ + I) *, +

77

h

I

v v v

L C L I

L L L L X W I

. . h h A

$ $ ? 4 ! W

w w . . . r - - , v v v Z Z Z --I IIt- 0 3 0 III a w a . . . A h h

$ $ $ w w w . . . v v v x x x IIC 030 IIf

a a x

~ w a . . . h h h 000 - - - W I I

. w w e . . , c c

I . , 1 - 1 A

V e

- z I a a ? A c "

z 0 I z. m a a n

m a

c

?

3 c

-I

t- o I m o

. Y - I .u - I w - a 1 u o x

I . I h

w ' C a + - J . e L L V V

C W o v r m

78

WiGINA‘S. PACE 1s OF POOR QUALITY

C C L L L L L C

b b b b b b b b w w w w w w w w I w w w w w w w w I

> a a 0 z 3 0 m c z a - . . w I - -I&

E W w z a - OI- a z a - a -a

m W E - a K U I-- w = a w a a

0 0 u m c 2: t’

*

I , w w v w v v v v w v ‘J v w w

w

Y - a i m 3 w I .

. ! z , -3 - > a

- . A m a

rn * Z

v u

79

80

i

m - I - a a m a a

a >

OREGlNkl PAGE 15 OF POOR QUALITY

- N 0 0 m o m o

m -. a t a

I- t w a n

81

v) e X W

N

c-

- -1 W 3 0 I

0 0

m 4 a

a

a c U

I W m VI

K 0 U

> A (3

f 0 V V

X 4

a

5 I- W m d

-1 W

v) K W c a a n r

w v c v ) W

V c W

+ + + L

e + + > w w w c v ) u o u - w

-1- A A d U K -1-r-1ue a a a w z v u u - 1 w + + + + *

. I

w w I -1 m I .. .. I

. , . . . . . . . . . . . . . . . . . . . . , w w w w w w w w w w ' z z z z z z z z z z - 1 ~ 0 O 0 3 Q 0 0 0 0 0 i I z z z z z z z z z z a I

,. ,. . . 0 0

w w w w w w w w w w U 7 0 0 0 0 0 ~ 0 0 0 3

z z z z z z z z z z Z Z Z Z Z Z Z Z Z Z

h

. . . . . . . . . . . . . . . . . . . . . ,

I c

> -

W > c c

a m i am

rn a +

I Z

m z :$ I-3 - -

c m -J a

d

P c 8 0 >

m e - P P LL e a

I - a

-.3 3 m G - - a 3 1 1 - -

- 3 P . ' U 0 0 m o c W A Z -

- u z 3zu-1 u- l 3 - e - v: - o w

W J Z

5 - 0 w c w a Q ~ O O

m n r n

w ~ n a u a -1 - z a n > - w x K K I - w 7 z 11 men-

Z K W o o n z z u i z o w 2 2 3 a m -

I U 3 - l ' * * + I - +

> c z 0 A P

z - O N

I -x

n o

. ? w a z a > . Z W

- l >

' X U N Z 0 -

K Z 2 0 2 2 = m

0 m E

I - z z V

A i 0 -1

N W U - U Q

a - w

0

2 0 -I m e z - w c

W

I? n - a

a -

0 >I C Y 9 12:

> w 5 5 z

c w u c 0 -

x 0 z z c

OWGINAL PAGE I2 OF POOR Q~JALITY

C 0 U

v, a

- 2 w a z C Z W i->

r n i w

= a - -PI- m e r -

a uc7+ 0 - W ,

w m - w - l >

E;?

N

m e

r . - - -1 a 3

r 3 J 0 > 3 0

W

i

;z -3

V r Q - N

N > C Y .

- - ;E P >

x 0 P N - 3

v -

. .

u. 0, (13 u - - 0

N C U P . . G O

i x a

-9. r Y

N

- w - u I W -c

u - ea a z

84 I .

m

m W I-

s - - .. VI a " z m a - a

zv) 0

w r

I-

-1 . r

w 3- a z m 5 o.:

-3 , m N u * a - O W

0 9

0 -

I . K O W o --I

- - a m

.- m u .I s z

- 0 A * c-

- -1 -1 W c V I -

I - O b - a - u x

W

a - c

n o & c 1 K - w o u o V I - z a a a a n

a - a -

a m + a--

n x a z w o a u - u- w -I- = - - 3 m - m - a m e a - w - n K X V ) a w a

a - c 0 w M U x z z W O W

- t i a m a 3 2 >

u - 0 m a w

c v ) - a - c - - -- *L;

e::

\ 0

? . . A I v ) m

\ 0 m Ln 3

o n N N

w 53 -J

> z VI z w

a

z z n

n d

W z v) E 5 t n

a

a

a

VI > D! K

4 c

? 2 i

V w v) n

w

VI w U z W J

> 3

a I

2 In > K 0

a

a

v) W -1 m a

a - K > v,

I v) W U

d

"

a a K w v) 3

_ _ 9 -

- 0 a 9 W - - K -u

e-

a?, :s *. g

-- - > 3 > - - m - 0- 9 3 - x > 3

- m

- a

- m m 09 - a - a

a. L - 0

9 -

3, u >

0- \ ?

i 6 - t m

39 1- z a o a I > I U 0 0 V

I a

n a n 0

n x w 4-1 x m n

I n - a u n

a u u

a - . -n J

a o a m -

-trio

> Z I -I- u x m x

w - - - e N

:I.=

E o ; - v -

Q X X z w w

!-am VI N Q.

w *i

699 Q O N > - a - 0 . a - 0

Q - - > a 0 K ~ K w w w w w w ++I-

o u 0 Z Z Z l c l l

- 3 N

. . mnN - 0 . . c I - -

o- o m -

I N V

5 I- - N U

II- t

- w - 0 I -z

W a i o > e 3

- --

w a w -

5 2 U - a a K a a a u + e w I -cc o a a v) a a a -1 Q O 3 n n n

a a a a a a n a a c c c

V U

a Y 1 _. ~. x x * . x x r * * * + x x X X t t b t

u v v v u u v u u u v v u v u u u v

+ * . t t t t

v v v v

85

z E! C

V 0 A

X

a

I

C m -l

c m a LL

a

I

t a

n L 0

z 0 I

-c 5 : 3

o v m A - a

V a \ w z 0- a w a m 0 t-r x o

A'?

0 - v w o . at-In I-z . o w - *e z\

W A C N a w w x 2

a - V I . - . - u N

w x a w a -

I- a 2 L1

x C N w - e

c x - o z w x z w a o - w u . o v a

a w w o uvac? a m a a o m a o - 1 z o

w - 3 0 m u m y

L L V )

0 0 Y - l

I e m .I-

c c

m C a c

' m m O N z u 0

a

Y Z t

- p z z v a

v n w a w m a x

w a u

o w z - m w C K

O C K Z V)I-0

=m. x x 5 x x a x x x x . x x e x x w x x m

- a m a ? a a

a

0 m K

m I z. a a P

m c(

0 cy

0 I- - m 3 0 a u a 0 LA.

. a

a

w t - m a

cy P

0 C

In cy

VI a 5 a u (Y 0 U

a a n c

c m A w

- L z K W

a

L

- cy

a n n m a - m a a

- 0 m a

a7

0 m a

+ - 0

3 m

- - a a

a - a - n a 1 - 0

O K

5: u 2 0 -

d

- Ln - - a C n VI a - t - - a n Q

m a

m a n a a

Y

m

c

m m a 0 n m (Y

-

P m - a a o m 0:

VI 0

0 m

v a a a

V V

v v v v v i j i j v v v v v v v v v v v v v ~ v v

5 - - cy

x VI N \

a a 0

I- 3

t-

a f U 0 > K

L I 3 In

a

I cy

X In cy \ \ \ \ \

m . w

- 2 m - -I a ' V 3 - z c m - u-z .I w . v - L Y -223

u m

E Z Z L a ~ - m

- a a u - - 0

A w - v u - W m m n o

r .

24-3 a v o z w - - l O N U U

n z

a x

z a

O C - 1 -

w w m

c

a o m L L W c m a m A 0 a A

u a a z - A

2 t c w - 3 n c o

- K w . o v - u z m

w a m

2:s a v w w z - A - V V I -

U

\

2 m - A

W m a

E f LL >

a A

c

m

a 0 t

t a w c

cy I

0 U

> A a a E In z w

VI X N

m

- ( Y N N

m F(

86

u

a W I-

E

t

i2 W

X d

I 0 N I + N I L 0

W I--

K - 3 c O U

a .

a W rn

z f K > m a d

I u 3 0 0

f

&

I-

o

(L W I- z W

N I W

E U 0 > I-

rn - t FJ n W

z W W 3 c W m u t

5

2

a W I-

W K

X

I 0 N I

- + N I W z

5 5 z j , I

I W n o a ~m

N U

N - 1

+ z - 3

W

L. L u w

E m o f

u U

> * w I- . - - - z v v w o w d r n 0 W \ N >+I U

I--u w o

m a 3 I-

t W 3 I- W m o

(L W I- z W

W

2 I- X

I 0 N I

5

a

I

+ cy I

a m a

I

>

-1

W

I-

I- a

N

k

*

A. I-

U 0 > c cI

u 0 -1 W > I- W -1

f - 0

-1 - A 0 3 0 I I-

‘0 - 2 I- ( C K i w \I- 3 2 I - w

- N I

W U W K I

v u V I 0

a

m

at- - - I

n E w a z a m o -3 K i 3 U

n

. a W

.I-

- v r n 0 2’

I- C - V I *

I- - 0 - c z x - 0 w x u

z c O Z I w v r n W I

r n 3

-- L

w w n r n

v 3 :

c a I - a

aw z a o

u a - 1

e >

- a m - 2 u 3 w u 0 u 0 U Z V

d I

r n I - 0 0 0 - i + x I I N N - - m V N

w w

W d I

m a 0

. .-

W

f m a 3 I-

u

I-

X W

0 N I

0 z 0 I- V

t. -

U

a a LL

v) v) 4 I I W 0

X N

0

N

W c

5 . .

- 0 c 0 0 0 0 3 0 n 8 ’ ” . . m o m o n . . . . . . . . . . . . . .

- w v m r - m g = I - - I - I - I -cc- a a a a a a a a -------- C e L I - I - C c + W w w w L u W Y w w w w w w w w w uuuuLuuLL -----_-- W W W W W W W L I . v v u v u v u v z z z z z z z z a a a a a a a a ~ ~ a a a a a a a c a a a a a a w w w w w w w w d - 1 A i d d i - 1 v u u W u v u v n n a a n a a n a a a a a a a a o u u o u u o u C C I - I - I - C I - e

x x x x x x K x w w w w w w w w X I I X X I I I D o m o m m ~ P o m o o o m o m x x x x x x x x N N N N N N N N

m m o o o o o o N N N N N N N N

c I - ~ c ( - c I - c I

. . - . . . - . _ . - . . . - . . . . . . . . .

v1

I G

f W rn 3 K 0 U

VI W

I-

I- z 4 3 0 0

- -

W

w u z > c - z a

VI m a d w e - + r e u 3 0 5

w w w w a a a a 3 3 3 3 rnrnrnrn rnrnrnVI

U 0 . w

-I- > C d

w w w w K ~ K K a n n o

y l w w w a a a a a c a n

CI-I-I-

x x x x w w w w I I I I r - r r - ~ ’ N N N N

x x x x N N N N

m n e o

--..1c(

. . . .

. . . _ x x - N N - -0 o o c . .

N N N W c c c -

u * - 1

+ u c + a W

CY 0 N

87

cL 4 > >

rd

u u -.

a w c v,

- 17 a c m > 5 hr-' m m w ZI a oc:

3 Y Z C u u . I a IC Y 3-1 u 4 o m 2 5 N r n . I

> a w

n n e .:z

W

w - - a U I W W

- u c w - 3 U u o

u w a 2 2 3-c3 o a w w - z m a w a > I- S K I -

a z o > w w o - r + z 2 z w a w n v o w z a o w z o o 3 w +.

O m V W K

" > 3 0-0 u m 0: w a Q + a n a > a a o x z o u-I-

X O W

w . a c ~ e m u - > 3 o u * +

Z c c 1U.I: UVI

88

V) W 0 J u? >

a

rn 0

I a a

W

U

t > i W c W J

I 0 U

(YV) 0- M J W

J O O U M

a

-

- X z > 2. a a > > K > U

I > I U

W J + J 4 w - 0 > * - A 3 >

a

(Y n

i 3 0 I-

t P

a 0 U

V) W

c V) 0 U v) -n > x

> > - 2 I - . u - w 3 U I

a

CI

w

w ?

:: 0. 7

a 2 W E c- q t - Iw 3 a U J J a d u a

a

V W > v)

X > N

f 2. 2. ~. 3>3- X > N > O O Q Z

3 > 3 u - X > N O o o a

c c c c w w w w a o o o J J J J J J J J

u v v v a a a a

+ + + ln + U W U U w

2 E 2 c

P U

J J

3 W c J 3 0 J

0

a

a

a

+ + +

3 3 > a . w

000 ' w w w o

. .

3 > 3 w ' X

0 0 0 ' w w v o

. .

n

+ 4 4

I

v)

J d

3 u

- a

5 I-

c 0

a

a c + I

P O K o - a M a > u > 0 t u * c I K

o a 0 u > u - u x - - e s - 3 >

2; ~

' - u w 0 -

w z , N C

0 A 3 w - 0 N O - V )

+ c

0 P

- c

0. ;; 2. c. > 2.

?a a a a > w > z - O K . > x u

0 ) l n - -z O U U d . N J era

- w - v

u a

5 ;

- c

. - o x - z O >

o u

u > 5 u u > - u I - u 5 - > i - -I

.~

: z. - a

a >

o a w -

n u w - N J - A

- w - u . .

- - c c c -

. I . - c _- . . - .

o x

O r .z : z. ?;

? < ' Z

? S ? X z 2 F o > c

- 2 - . . . ? Z ?

'2

a c 0 - .K a > Y > 5 - O K

- > x u I - - 3 z o > I n - - E

n u -

u a

w - - N J LD

U

> > a

a > c

o a o

u > u 5 u I u > 0 - u - i . I

w - w K > V

a

I > I V - c y

' - 0 -I A w i Y I - w n . o u . N C N l

, - v w n - J W I

, Ne Q -3 w -0 N O m m

-I Lri 3 u 0 w - rn N i

0 1: R -.I.

- 2 - w 0

m u r - 0 . . . - m m

. . 2 '". L". 8 m x m z - - 0 - x - r - Z 0

- * c1 O K - , u a . o > K Y

- 0 a . a z > n w r o w W w Z N x-1- - u-1w - I -30 I-

z . a o .

o u a o

~- o m - - z xln

c

. - o x

O > . z O Y

u > z a u > - v x -

C Y 3 > 0 - W I - 0 0 - w r N C -3 -0

: ? . . - a a >

o a w -

c

9;: ' 2. 9,

? a .z

a a u > tu> I - O K - >

3 L O > V I - 4

o v W . N - l - 2

' W

o a

!z?

.. m - - - 0 0 2 - z - I- e x 0 - x -zt-z

x z o x -

0 . - . * O K - O K -

:: c

m

- - . -

o x

C > - z o a - a a > u >

:? . -

cy 8 . c

m V

. - os

x - o x - - 0 2 - z - -c - x o - x

m m V

m m

. I . - 0.2 o x

- 2

2 m

V

c

* - +r'

m v u

* . f f i

m t -

v u

91

- c

. - o x - z O r .z O K - 4

a > u > I P 0 4 c > - 0

3 > c - m I - v

W I r u I - - 3 - 0

m m

. .

. -

w .

E;

n .

C .

- c

0 - - x ' z. 0,

? & . z

4 4 u > w > I -

u 4 - > I V e - 3 H 0 5 m - -I

w - N 2 - 2

' W

o a

n u

z 0.

0 0 c N

. - o x .z 0 2 - 2 . .

? 5 N a 5 N u > Ln w -

I K 0 0 4 c a >

- v 0

a > - 0 - - Z I P - v N

W N e

u E ;

E x 2 - e ' - 0 n - z

w N .

- R

N u > m w -

I K 0 0 4 c u > - u 0 I - u I-I

K > - 0 - - Z I N - v N 0 -

' W I Y NI- 2 -(I ' - 0

n N Z

. 0 * P I m m U

- Ln N m 0

O L9

I-

- - m N

w 2

n w N - 0 z a - N

Y w 0 0 o w

N 0- e- Ne O 0 z v ) . 3- 0 u (L - . + u .. N N N * N N cy N . N

Ln Ln m m m In Ln m m m m m v v v u u v u v u V v u V u v V

A I - lv w - u c m v ) u a

x o * W e .

W P

- N In

d

W m 2 0 c7

9

9

0

C U W z 0 u I- v) W 3 0 W N w

0 P

cy m X ?

- Z

2. > z a a > > K > V a

5 - 1 I -1v w - V I -

u w 0 3

Y O I P 3 -

- N > m

m v r

- .

I .

N P N .. .. ..

In In m In m In m In m In V V v v V v v v v v v V V

.. .. .. m (7 D In In In In

U

h " r 0 0 Lo ( o w 1 5 t 1 5 m m m 0 n m o + m m o m

In. In In l n M M Ln I n M M

U

U 0 v u

0 : OI (? M L7

U

94

01 P In

. > . *

0- a- 0 K - a . a . > ? Z ?? 3 3 I O f a > A > 2 -

-I 0 2 o v - w v W -

0 N - N-1 In -3 - A

- Q - W

. .

0 c

0 a . .

n z w u N - o -3

In - 0 In r--l

c - + 8 c y S 0 In- -

o m - N u x -

0

W 2

a W N I

0 z a n

- 0 ' W

0 '3 03 W J

W -

0

W W

0 W N U - c

- 0 - 2 3- 0 u K -

3 w - 0 u u K - W

J W d J

v - U V a u a a

95

-- ' X

- 2

' Z

. a O > - > O K - a ' > o u 3I 0 > 2 . -I o u

w - N J

- w

I . - >

? &

. .

. .

r-v -F'

N - o- - m - N m X - z - 0 - X I - -z 0 K - > K

w - 0 W - Z Y J - 0,-

e 3 0

F .

. . u a - -0 a n z >

. a n o u a

u a a f,, - A ,

- V U

-- X

- - z

' Z 0 . -12 ' <

I .

- >

92

9 U.

. . O K - a ' >

3 I 0 >

-I o u w - N- l - J

J -

- C

0

0

9

z. 2

7 - e K 0

N - 0 N

. a o a - > ' a

' > ?; ? & . a

C > - u 3 - 0I J >

' > ? &

0 ? Z zi

m . a

0 U > u 0 - Z I - -0 - o -

v W I . N C

, > (D 0 - m - c L m . a

O > 0 - u I- I - ?I

0 K > u 0 - Z I - - u - 0 -

m w x , N C

- 0

c L > 0 - Z H - u 0 -

:i - 0 I - +I K > 0 - ZCI - u 0 - w I N C

r- - a m -c m m z

fi a >. 0 - Z I . v o -

w x N Y C

m - 0 ln - 2 m - K

_ _ o x w u N -

N - 3 m - 0 m O J

w - c L J - 0 . - 2

- 2 - w

m u N .

- N . o N -

w o - - N P - N - ( V - m N - m N . . m -.-m .C

a c - x . - 2 c - 7

Q z o z ~ x . Q z X N m X N .

I)

+ - 2 *In m

m s I ) 0 L? (0

- m m 0

(D r- m m m m

n r- a2

m m LTI

m 0 ..

0

m m m m m m m m m m

m m m m '" In

U u v U V u u U

V N F)

U

X z Z

K > > K > V

2-

a

a

P z V

2 2 W V

0 N

0 cy

X

2. 7

a a > 0 t w 0 0

d d

V a

0 m 5 m Ln m Ln

V

n .e

- x I. f O?

?: ?&

.a

' >

. a O >

01 8 m W

. > ? & . a O >

0 0

- V

K > 0 - Z I - v

E;

U N m

* v 0 - u x

Is) N C ID - E m -0 n F;?

P- (D In

0 C O o w a m -0 -I- t- - 0 .a 0 ' >

0 W - l N W

0 - > +-a N e e 0 0 0

Z Z v ) . . 3-- 0 u u e - -

P-

w -

r -03

. .

+ 0 + * m o q g + m

W W W W W + W W W W * W m m m m m Ln m m m m m l n m (ZI

V V V V v u v v

3 > 0 - tnL

vr c X w

c

- a v) W m tn 0 -1

I 3 I-

t I 0 I K 0 Y

I- z 3 0 u U a

I - a Y m

E 0 Y L

a 2 G

c LL a ~m

N

0ln

.o o c 0 0 -0 In-

- X

?

+ 4 0 t r -

- * w e x > - w - W t,?:: -1 e- a

- - t n - tn--zu

- -0 l -u w - --z2- -I 3 x u u x z u a - -

? Y 3 K l - I -- t- m W -1 -1 a z m U 5, c

+ d * r - m

0 * m 0, + a (0 * ( D (D

ln Ln ln ln ln ln v u v u u u u u u u u u u u u u u v u u 0 v

- X

3 - - m m - u

w

I u 5 m - m - w X K

K I C - 1 c

v - X

L'U u u u v

2z * c- - - - a x

a - B K - w u 3 m K a 1 - 4 - n

- a w - w r w

e c a w e 3 I + - - u - m - > K X K - Y

- I-

O N u .

0 C ' w v

K ?

3 % U w l n O F - \

Y K m e

- X

z. ? >

K

> > K c > V I > I V -1 2 W V

+ > + 7

>

a

I .

t -

- Z ? I- 2. w x

w -

W 2 c)

0:

>

a

a

u

W

E 2 K

w c a

a a a a a W 3 d

>

- K

0

a

U 0

3 0

n - 1 , u 0 m L L : m

E 2 x 3

0-1 a w u . a as-z W O K 0 -1v- o u m m z z w 3 a - a u

u x O W -!\ wa > w a -1a

- - 1 a

- 1 2

W I I v 3 0 m z K X : wa n u . \O a - II

-1--

w x -

v 3 I->

- a - I a 3*-

- X

r- m In

0 c 0 0 - - ? Y -1

U N W

E; 3

Z b 'In '0

w e W '0 + u u

I N + w -

x - e- * - 3 w x v z

n am -

n w K w

E m z 0 V u 3 w m a

a

a

1

1

a

w K

w V U

: -1

I-

E

. w u . - X - 0 K 0 - w u 0

- V > N 3 0

0

=.:wz

-z m a - - a 0 > u

x 2.

2.

:?

'1 : : 9

>

I X

I> U Z

I 3

X - e 3 +I- --

-1 N w X

? U N W X n 3 11

X

7 0 m ln

0 0

c(

c z

X -I 3 3

, I Iv )

10-0 c 11 3

w o x z

C N * m * * + * + o

+ $ * o * m

K m - a z u - m

ln ln -

v u v v u v v v v v v v v v v v v u u v u In v u v _. v v v u v v

99

N W I I Y Z - - N x w - i c z K 3 z m v m , W

I-+ I- A - - I - x x 4 3 m - w

w w Q >

3 m 2 4

I-

X W

W

- E I- a L W

0 5 W Y c

I. c 3

>

2

c

-

1 0 A L

m m Z X 3 n-l

a

w u

K O v u m v l w 3 LYU n u a n

a

m~ a o

I-

0 u o w w > N 0

u L

W A m a w - I at- a > - a w w I* +I-

W

0 L o arn

, t - +It, a +

~ . a - > x z z w

O K alL0 L O U C. a c m

2 4 w a 2 3 W A H a z a > o m

C C W P

w o a a I - - x u X X A A

I N > > C I I O U

!I II II I t

x x x x 3 3 7 3 1 - w -

- . . . > > > > - 3 3 3 3 -3111

X

2. > Z

a > > a

a a > V

I > I U

-I A W V

> 3

> 3

* A * N + w

I

L(

* -

5 3

S U ari w w

r- m m 0

0 0

I-

- 3 0 W

a a n >

f,

5 0

n N W t - Y -3 X V I w - l

+ * + * H 3 m m m

L n rn v v v v u v v v v v v v v v

A A a 3 B I

3 W LY w U CY w c

I- 1

5 C

z 0 n > W

t- I

I- a E 0 m I > x > o o

? ? d o 4 - - It I I

d m W m m - >

1

3

0 2 a 0

0

0

- w

Y N 1 - LY 0

: 0 w > Y 1

*I- 0 o, m t m

m v v u U ' J V V U

U m A w >

" t

x x 3 3

> > 3 J 3 - w - A > - - 3 0 3 * 0 a d + * u r n A > ' w 3 0 > + + 32- - w x + > - r n > 7

- - I -

. . - 1

. + -

100

W c a K

1 2 LL c 3 0

Ln Ln a I - J

c c

a 0 -

I

X W

n z 0 V W ln

c a W I- C K

3 0 -I U e 3 0 ln

W C i

2 X

I a

- W r a 3

z L5 0 a

-. w

n > - a.

U 0

> c ln 0 V v, > n z a

I

- > e 3

9 8 n - a c -0

30 C Q

w - 1-m a a x x J L - - 3 v ) X U u s II I1

z I W

c I c a Y a

n w -v , +\a a -I

0 z a 0 0 W

2 0

n 3 v) 3

a W I -.lo I L W I a v I

I , , .. I

W 3 z

I f - W 3 E c z 0 u

L

i 0 u

I . I . t .

. . . I ! + , . I

O r

u . v u v u v v V v v v v v v

101

N

- 0 2

> -1 n a : z W

W P 3 C X

I 0

I

W C a -1 3 V - d m a a V J

VI 11 0 .

uz a x

W a 3 I-- am a a W J G V )

C X

a >

X N -I

D w o V N 3 1 w o

E=. w z. ? ? E a

o a

3, z n 0 0 K - u - u : a.

5 2

I C

c c I o VI-

o a

a d ~a

a

o W C J

3 u V J

L'

X c a W a

0

Y X - e

y ? > ? ;;e 5;

$ ? =--

a >

I--

o z

a - w -

3 X

I-

0 0 W T I--

- 1 W 3 m V 2-1

0

a +

a 2 v a

a m m c.

cy c

I V

f c Ly

VI - W

5 d

I- X

I I

W

- 2 u x O Z > >

m I C a a o J 2- m u I w m . x - C e > Q 0 f E-1 - (-1 u -14 - 3 u u

-I

u a

c

X I

m w > 3 - c x > - a -

a z z c

a II I I - = x > x w r n n t

Z C C Y

00; =oo- m - - u --I

m - I Q O C 0 0 0

W Z W

'E,

X

=. : 2.

E ?

>

K - I c - w3 I

> Y Z v -

X

a - m -

z 2. w - m - O I VIJ

> 3 1

3W c m X - 2 I J o v W c- J I 3s d J

U U

v a I.-

W - a c

a

a 3

u a a -

V v u v v u u v v v v v v u v v v v v u u u v v v u

102

W J m a

a

a

- K

>

0 U

V

+ I * I + I

I K 0 Li

-1 a w z ; c c > N * 2 + 0 a + - ZCI < + a

N + W I C I C + C + 0

C N U + I o m m I+ m m o C + N - C U I U I V v 11 w I1 0 >" K I I 3II V

- Y 0 v)

I

3 0 J W

- r

m z z C

-1 a

n a 0

E! X W

c 3 - a

K O

O W w 0 10 " 0

3: O N - 0 A O C

I 4 c w o

v) 001 W U F P V 0 - 11

I1 c a .. w .. w . . 0 O . . O U N N Z N N W I I a x x K a

t m e

I - I I C 8 I C I I - I v v ~ v u v ~ v v v v v v v v v v u v v v v v v v u 0 V V V V V CJ V V V V V V V V V V V V V V

€03

\ w

UlD- i a m w m n m o - c c o \ w

\ m !2 G 6 ? m .i v VI r $ c

m m

m

VI

0

VI

m W D W c) IC , N VI VI 0 N p1

m

m m

I- In

P

n I-

I 3 > a

m O

W v) 3

0 I-

vr c

U

W >

- a P l c 7 --lo-

W Ln 3 0 I-

m I-

U

W > a 3 U

u

n

\ m a +

L W e ?

5 0

> - 0 d o u t - 0

c7 Z U -

- 0 z ' 0 w f f i o U O N 30 ' K > a .

W '

nc-1 I . a a s Q I W 3 w c + I - , '0

w o z I z a. e a u . m

. . 000 -0

u . . *

Y e W

0 : m c

O N " a I I I 3 v m a 0 I1 II 11

u m a w u m a >

. . . . v u v V v u v u U V U V v u

104

Q m 3 Ln

z vr I-

U

W > Q 3 u U 0

0

a- u *

-

W

5

0 - I P -

Y

0

0 z 0

Z 3 0 0 Z 0

c

- c -I 0

0 c X w c 3

a

a a

rn -

I -- 1-0 I X N

n I Z I - a z 3 0 K CI

- 0

N W W 0 U

e

0 N I

0 > c m

0

z

5

a

U

I

5

2 w

LD

I-

0 N

*

5 0

c -0 Cvm *I + m a m - + I +

I- Q Z O 0-1N u r x

N U A I + a a m u + c a c > z * 2 w o a c w z-I a z o

-I+ w + c I N + e x 0 U + I O N W I+

m o o t W N - P I L L X U

w I I a > N N Q I I x c i a u

I w * " c c

. I - c

rncv

w 11

vr -1 -1 W V

w Y u 0 -I

a

rn > 2 -I 3 U

E 0 a W N 0

0 ' + ? 3 3 a n

e t c

m N

. I - c c 4

0- c u u 0 , o a - a a - + a

0 3 m a W n

3 I-

c m - . m \ w * -

L A L L

c(

> I Y - I- o

W 3 E c Z 0 u

105

W f N

w m 3 m

I- U \ L -1

E - K

n

8 P

Y 0 a

4

W z

a a - - N W w U

I- 4

a

.

0 I-

\ w VI I-

U I W

P - W > K 3 V

O + c LL \ c? 3 -1 m

2 0

I- * \ I m -1

I- K w > z 0 V

a\o O P C cr- v - - a .o U N r - o- z\ *

O K N - > m cnz u 0 I1 w v > o 0 e I1 V U

O K I - > - 2 z o 3 v

z a

a m I O : E - W

K

2 2 2 2 N. I- * - + e a a - Q 1- LL w o I-N c? *I z

m 2 i W 0

0, Y V 0 -1 m > i -1 3 U

E 5 >

e 0 c

-3 I et N I-- w - r t Y

X I

- a t 0 x 1 w

- e + - 4 u - 0 >

O \ + N c o w x VI

o w I- + e a

I Y e a * + - 2 : 0

0 e

-. N

Z X c?+ O N K X

w m

n u

c W

m:

106

u w ln

K

E ln

, t v,

I c U - . -

x u ' Z 0 -

N

g : - a U

0 c c

c w

W > CL 3 U

L 0 a

I W

a z 3 0 O N n x

w u 2 a e -

c

0 m m *I - u P + I C 4 u t e + + a

r a c I C U I " x w a + * O N 0 L * N * T

- 1 P O

-+I

J X + t m o Z + N a a x

I W w w + x c o + I N

N I uxu. 00-

m m x

LL I1 0 N N u x x

> 3 3 a 1 1 3 w w u

a x + v w a c * w > N e

+ a

Li5Y

I c K

W a

I 0 a U

c 3 0 I-

f o a m a O K e o

a i c

x x x > - - z z - -

v) W V

K "

w .. 0 U N N w x x K

t a ln

107

0 r lc I-

O N I 0 + N t

a I O C Y * .

- n -

V m

U

N w w K LL

3 0

5

i w m

a m I w I-

CY 0 LL

u t N W w CY U

I- < W 3 2

> m I-

a

- 0 C

> I-

v)LL 0 3 V 2 v)3 -I > W

K O W N

- I1

C I a 3

a a

3I w

K cp. -

w u I-z < - I-0: - a i m -

a z v Z - - - -1z

Z W I 3 0 a x I Z O v a

c z w i 3

CY 7 I-

CY 0 U - K > > a

z1 - K

> > 0 z am z - 0 K .- a + > u v z 0 3

a

I 1 u -l

K - l

> 3 V

K

I -z 3-

0 2 V 0 o z

- a a

u a

a s s a

e a 1

w v) 0 E T)

n

n

0

D '3

VI- ZL, 0 2

W

- u I-

i w a z 3 W u m a + i

u w > i w c 0 0 0 2 I m -I 3

n a

2 0

la N U

a u O Z u - > 0 0 -I i

> K

I-0 0 U

I Z z Z - - 0 v ) w - 1 - e - l > v a o - 3 a (run c m . . w - N CY

L L -

a a

;v)0

W I - . 0 - 2

H p a W K I U 4 c z a a u r n -13 a m a -I K C a e > a

Y d

2 5 - 0

w w d' m m z i a m u am w c u =

C Y 0

c Y

U C f C Y

a -i 0 -

Y m V w w I C Y u < m - l m - l K 3 V U

w z a C 0 - 2 2 - - 0 0 0 a v U

0 O Z + a

5 5

o a

e-

N

108

-1

a 3

0 LL

v: c

t

ln W c O > a

Z E W

0-1

- 0 ) a d v 2 m w 2 w x a 0-3

0

z a

- a 2 w a a m v -

Z Z W - 0 t > z c A 7 3 w o o > a a e 3 v z m w - m.0 c w w a a o

o w -e m

z m m

I - a

- o m

o o

a

a a z 2 W L

i m ? o w l - A d a m w v a ~

- a -1a a a o V > Z

m Z ( r w w o x >

I-- - 0 w . N W m 4 > w N O I

2 W a

, m a w a z

v v v

2

- > - - 1 a w o a ( r o z s a x 2 - 1 K K W

\ a o z w m Z W I Z > > c w

o w u o O W C K E a 2 2

v v v v

t - N z 3 N 0 o 0

-

f

> z a

> C #

- *. >

m a

t m W

03 W d v a z > W -la a - > W -I 3 c 0 w c w x

- e o x m m 2 7 - - z - 3 > i m - 1 o > w

a e W

> W > a

a m e

o a z a a v a n a m

0 . o a z a a > a

- 4 0

z v a

>

* > Z -o

> ( r z w u -I> a + w a > -0 > z e a c z 4 - 7

o a o

SEf 0 > O O V O K

v u 0

> Z r n C ' - 0 0 . u u w

: w a v o 3 w

: 0U-r

o a - z a ac23

a w i u a x c - 1 ~ t- m a a

w 7 3 a

z d a w m t z w i G G l urn a v 3 3 w o

A u a w -1K m-1 a o I 2 3 u a a a 0 K V

+ v v v v v v

109

W 2 m a

a

e K > Y I e

a a o

-1un a u a a u z I

- n e w w z s m a -

i w e

w > I J K

rn 01

n d o z n d z a w a z c m v o m a

a z m m u

t a 5 21 J c o a

w e

w w o -

m - c v -3 0

w v w LI m v e w - - a m z a w z w a v - x d m c c 3 w a 0 I O m a - VI w o w . a

w . w

-e>'?- e m 6 - m w m m o a - z

- n w w r

a t z r o a w a v

z w m w a a m - v

I 3 I a z

o z v a m w : m (re > 3 e x o 1032

z m w w - w > - l

d w o c m 3 J

w m .-I c r v > o

m rn

110

z z C V z 3 u o w

Z X - ~ a +

m 3 A

> A J

3 W

c1 0 d

X

\

a

a

Y

+

a - - I1 m 3 -1

3 a

a

-1 -1

3 a

I 0 a I&

-I W -1 0 a a 3 - 2 - I W O -I C O X -1

I 3 O O K a

m o 7 a - a a a a c w

o z \ 7

00- z -lL c

> m

5 > O

z W I I-

m

-

I

m VI w K I- m K

w X m d -1 a 3

a

C

5 -I 3 K 7 c W

I-

m

I

> A I-

m w r a :E. E z i p, , w

- x

K > Z K A U w w w u > m w - a c c

v o > w z

m 7 -1

3 \ In 3 - r V d m o - r > r V I 3 w 0 + - o x 3

a

a w w e >

-1 c

3 3 3 I- a

C Z U '

a a Z Z O &

O Z f Z 3 o o w

w w o 3 1 x 0 0 C e v x

m 7 -1

> a 0

0 5 W

I 1

VI

a - O

m c o a -0 e m a + 00 m z \ >

. w - a I1 11 m m 33 - I 2

7 >

-v

a n

z w

. A .

- - w w

W W > N EL"O a

l? 0

- w w 0 v z a - 1 z --0 -1aua VI n e z a z g x a A

0

W K

0 a 11 m u

5 -

u a c

m

m a

I 0 K LL

z Y

W A

I-

W

a

a a

C

W

I

t- In N m

3 ? m .

- o w Z V a z

w - K P W t - L L

. w m a

a 0 m w a a

I1 - w >

w w X I 3I-

c - w l s u - 2 0 C A a w u I>

Z r a u

V

111

4

m cy

N-l

- 4 3

w w A I .e 0 0 z c

. . n-r

m a n c A W 0 2 E 0 w w K I -3

m m m W K V l # - - m

W K J a n W L L I0 m a a

m a

-I - 1 D 4 c 311

V K O 4- l

$ 2 J a g: 2 4

W

u-l

0

I v 0

W w m

LL m

0 4 W C .

> a 8 W Z N

K W J W K 4 u 3 W W Z 2 x 4 <+I K c u m

4 I W

0 - e a

W

I I C

K -

- + \ . o . - . N - t . .

W

> n

LL w - L

3 0 n Y I-

W t I I 0 y W w - 0 W m m W

cy . z a a K

K 3

m II 0 3 A W J 3 I -1 0 0 4 IZ3

JI- 03 LLZ

-1 1 4 3

2 1 W J 1 4 K a a > I-

u 0 -l Y > W > c

W J a u c 30 Y Z

4

“J - 2 - 4 -3

- L -0

N K - u N W - v -z - e

- V I r J -

-

*

a

> a

.-

m a

- n w

2 2 0 L a a

a

W I- W

n Y > 0 J

4 Y

v) r - -

2 0 e 1

c

1 0 4

w 0 I

3

112

p.

0 I-

O u

c

c

X a I Y

5

113

,I I

v, K W 0 2 U rn a x I

a

a c lJl -

- a -1 > V

W a 4

lJl N > I

K K a

a I

I

0 (7

3 0 H I -

a a W K I- K lJl E I

I -

C - a n d n -

c. - X + z 9 a - a > 0- c

z 0 L E- m I I -

+

V c L T - 0 : g m t m

v

114

0 - w m I-

o m m m Y NFlNN P

APPENDIX B: PROPERTY CURVE FITS

The individual enthalpy curves for water and hydrogen have been combined in order to calculate a mixture ~ enthalpy, Enthalpymix, defined as:

ENTHALPY OF WATER'

CURVE F I T I ( 49 2<T<9 7 5 R

H (Btu/ lbm) = -424.5938 + .82414T + 1 . 3 0 6 7 ~ l O - ~ T ~

CURVE F IT I I (9 75 R<T< I I 84.6R)

H = 2289.552 - 4.577089T + 2.81'j249x10-3T2

CURVE F I T I l l H = -7363.69 + 6.913T ( 1 184.6R<T<1223.3~)

CURVE F IT I V (1223.3R<T<1281.4R) - H = 599.5881 - 1.27177T + 1.369267x10-3T2

CURVE F I T V H = -307.5449 + 1.190721T ( 128 1 .4R<Tz1400R) -

STANDARD ERROR = 4.08 Btu/ lbm

'These curves were f i t to da ta taken from Thermodynamic P r o p e r t i e s o f - Steam, Joseph Keenan and F r e d e r i c k Keyes, (New York: W i ley and Sons, 1936) PP. 72-75.

ENTHALPY OF WATER

I 1 1 I 1100 1200 1300 1400 4 0 0 ' 500 ' ' 600 ' I 700 ' ' 800 ' I 900 ' I 1 O W ' ' '

TEMPERATURE I'R)

Figure B-1 .

DENS I TY OF WATER^

CURVE F I T I (490R<T<l I8OR)

CURVE FIT I1 ( I 180R<T< 1250R) -

CURVE F I T I l l ( 1 25OR~T~1400R)

d e n s i t y ( I bm/ f t3 ) = -82.117 + .62353T - 6 . 7 7 6 9 3 ~ 1 0 - ~ T ~ -3.41207~ IO-7T3 + 9 . 2 3 4 0 6 ~ 10-1°T4 - 3 . 9 6 8 8 ~ 10-1 3T5

d e n s i t y = 119.1372 - 4.7 0 3 5 7 ~ 1 0 - ~ T - 1 . 0 0 6 9 4 ~ 1 0 - ~ T ~ l! + 5.516186xlo- T3

STANDARD ERROR = 0.61 Ibm/ f t3

'These curves a r e f i t t o data taken from Keenan. p p . 72-75.

DENS I TY OF WATER

, I I I , , , , , , , , 1 700 800 900 1000 1100 1200 1300 1400 400 500 600

TEMPERATURE P R )

Figure B-2.

119

V I SCOS I TY OF WATER3

STANDARD ERROR ( x l O 3 ) = 0 . 0 0 6 6 I b / f t - s e c

3This curve i s f i t t o d a t a taken from Steam T a b l e s , Joseph Keenan, e t a l . , (New York: W i l e y and Sons, I n c . , 1969) p . 1 1 3 .

VISCOSITY OF WATER

TEMPERATURE (OR1

Figure B-3.

4 ENTHALPY OF HYDROGEN

7000

6000

5000

- I

t

3

4000

m

3000

2000

1000

0

CURVE F IT I

CURVE F IT I I (508R<T<2000R)

STANDARD ERROR = 4.39 Btu/lbm

4These curves a r e f i t ' t o data taken from the Hydrogen Techno log ica l Survey - Therrnophysical P r o p e r t i e s , Robert D. McCarty, (Washington, D.C . : NASA S c i e n t i f i c and Technica l I n fo rma t ion O f f i c e , 1975) p. 472.

ENTHALPY OF H Y D R O G E N

I l i l r l I I 1 1 1 '

2w 400 600 600 1000 1200 1400 1600 1800 2000

TEMPERATURE I'RI

Figure B-4.

121

DENSITY OF HYDROGEN'

3.5

3.0

2.5

- I?

f 2.0 3 - > v) +z

1.5

1 .o

0.5

e

where H i s the en tha lpy o f hydrogen.

3 STANDARD ERROR = .0189 I b m / f t

'This cu rve i s f i t to da ta taken f rom McCarty, p. 472.

DENSITY OF HYDROGEN

200 400 ' 660 ' 800 ' 1000 ' 1200 ' ' 1400 ' ' 1MX) ' I 1800 ' ' 2000 ' TEMPERATURE ('At

Figure B-5.

v I scos I TY OF HY D R O C E N ~

V I S C . ( I b m / f t - s e c x 1 0 5 ) = .4989 - 5.4575xl0-5T

+ 5 . 1824xl0-7T2 - 1 . 4 9 4 8 ~ 1 0 - ~ 0 T 3

STANDARD ERROR (xIO5) = 0.00047 l b m / f t - s e c

6 T h i s c u r v e i s f i t t o d a t a t a k e n from the Hydrogen T e c h n o l o g i c a l Survey - Thermophysical P r o p e r t i e s , Robert D . McCarty, (Washington, D . C . : S c i e n t i f i c and T e c h n i c a l I n f o r m a t i o n O f f i c e , NASA, 1975) p. 473.

VISCOSITY 0 F HYDROGEN

TEMPERATURE I'R)

Figure B-6.

123

APPENDIX C: CONVERGENCE CHARACTERISTICS 1 I The insensitivity of the model to the initial values chosen for temperature and velocity is demonstrated by the hdution sets given below. There were four different test cases run using identical boundary conditions but with the different guesses of velocity and temperature listed in the following table. Cases 1 to 3 were run to test the sensitivity of the solution to the initial temperature guess, and case 4 was run to check the sensitivity of the solution to the initial lchoice of the velocity field. I

INITIAL FIELD VALUES

Temperature Theta Velocity

CASE I Hot Guess Omega = 0.4 Disk Omega

CASE 2 Best Guess Same as above

CASE 3 Cold Guess Same as above

CASE 4 Best Guess Omega = 0

(Omega in radiandsec)

After 500 sweeps, the values of velocity, temperature, and pressure, at a reference point in the middle of the cavity, have converged to the extent shown in Figures C-1 to C-8. By 500 sweeps the constant pressure lines (Fig. C-2) for all four cases are very similar, as are the streamlines (Fig. C-8), and to a lesser extent the temperature profiles (Figs. C-1 and C-7). Of the three, temperature is the slowest to recover from a poor initial guess. However, even with an initial temperature estimate 1OOO"R off the final values, the temperature at the monitoring point has converged to within 200" of the final value after 200 sweeps, and to within 75" of the final value after 400 sweeps. This is an acceptable convergence rate for our application, especially since the magnitude of the error is readily apparent from the slope of the temperature convergence curve (Fig. C-1). In the event that greater accuracy were required, the solution could be bracketed or else extended the necessary number of sweeps.

125

'1400 -,

1200

a00

2 f 600

480 i 800 I I

I I I i 1

0 1 00 499 - '* SWEEPS

420

400 0 P c 2.380

368

mrx CASE 2 (BEST

GUESS)

FLEFERENCE LINE f I 350'- - - - - - - - - - - _ _ - -- I I L 1 I

0 100 290 300 400 500

300-

858 -

200-

TXIX ( COLD GUESS)

CASE 3

0 100 . 800 788 400 500 SYEEPS

Figure C-1 . Temperature convergence, cases I to 3.

126

6.=+@5 -- -- - - - - - - - - - - - P S P E R E ~ C E L I N E -

P I (HOT GUESS)

6.0E+05

5. M+05

CASE 1

6.4E+05 &6:: li - -- - -- - REFERENCE LINE - - - - - - - - - -_

P I (BEST GUESS FOR R l l X ANU UI) 5.8E+05

CASE 2

1 5.4E+05

5.ZE+05 1 I I I I 1 0 188 400 500 SCTEPS 300 z

P1 (COLD GUESS) s . ~ E + ~ s I

- - - - - - - - - - - - REFERENCE LINE - - - i- 6.28+05 .

6.0E+05

5 . a ~ + o 5

5.6E+05

5.4E+05

CASE 3

1 I 1 1 1 400 500 0 i0e i?OO SWEEPS 300

Figure C-2. Pressure convergence, cases 1 to 3.

127

CASE 1

-10 I I 1 1 I 1 8 100 i?ea SWEEPS = 489 -

"1

-4 ' I I I I 1 0 100 408 508 288 SWEEPS 380

W 1 (COLD GUESS) l0 1

~~

Figure C-3. Circumferential velocity convergence, cases 1 to 3.

128

I

15

10

-5

CASE

I 1 i i i

15 -B

10 ut

(BEST GUESS k l x , til)

CASE 2

U i

(BEST GUESS k l x , til)

CASE 2

I I I I I 400 588 0 1" - SYEEPS 308

15 -, U1 (COLD GUESS)

s ' I I 1 I 1

0 100 488 588 200 SMEPS 388

Figure C-4. Radial velocity convergence, cases 1 to 3.

129

6 - 4E+05 6.2E+05

6.0E+05

5 - 8E- 5 - 6E+%5 5.4€+0!5

5 - X+05

REFERENCE LINE - - - - - - _ - _ - -

CASE 4

P1 (INITIAL META VELOCITY (U1 =) 0)

I 1 1 I I

408 500 8 188 tee SWEEPS 308

MIX (BEST GUESS FOR TEXP) (INITIAL ANGULAR VELOCITY - 0)

400 - I -

3 CASE 4 I ? -

2 380 - - L. - -

360 I I I I 1 fa 100 300 400 500 200 SUEEPS

Figure C-5. Pressure and temperature convergence, case 4.

130

Figure C-6. Temperature fields, cases 1 to 3.

131

ORlGlMAI,, PAGE ES OC: KKlR QUALITY

Figure C-7. Velocity field and streamlines, cases 1 to 4.

132

1. REPORT NO. 12. G O V E R N M M ACCESSION NO. 13. RECIPIENT'S CATALOG NO.

NASA TP-2685 I 4. TITLE AND SUBTITLE 5. REPORT DATE

Space Shuttle Main Engine High Pressure Fuel Pump Aft Platform Seal Cavity Flow Analysis

7. AUTHOR(S) S. k Lowry and L. W. Keeton*

9. PERFORMING ORGMIZATION NAME AND ADDRESS

George C. Marshall Space Flight Center Marshall Space Flight Center, Alabama 358 12

~

12. SPONSORING AGENCY NAME AND ADDRESS

6. PERFORMING ORGANIZATION COO Y 8. PERFORMlNG ORGANIZATION REPOI

( 0 . WORK UNIT NO.

M-544 11. CONTRACT OR GRANT NO.

13. TYPE OF REPORi' & PERIOD COVE

Technical Paper

7. KEY WORDS

Space Shuttle Main Engine High Pressure Fuel Turbopump Aft Platform Seal Cavity Computational Fluid Dynamics Turbine Blades

National Aeronautics and Space Administration Washington, D.C. 20546

18. DISTRIBUTION STATEMENT

Unclassified - Unlimited

S u b j e c t Category 34

3 1.3. SPONSORING AGENCY CODE

9. SECURITY CLASSIF. (d thln roeart)

Unclassified

I 15. SUPPLEMENTARY NOTES

20. SECURITY CLASSIF. (of thin pago) 21. NO. OF PAGES 22. PRICE

Unclassified 137 A 0 7

Prepared by Structures and Propulsion Laboratory, Science and Engineering Directorate.

*L. W. Keeton is employed by CHAM (NA) Inc., Huntsville, Alabama 16, ABSTRACT

A general purpose, three-dimensional computational fluid dynamics code named PHOENICS, developed by CHAM Inc., is used to model the flow in the aft-platform seal cavity in the high pressure fuel pump of the space shuttle main engine. The model is used to predict the temperatures, velocities, and pressures in the cavity for six different sets of boundary conditions. The results are presented as input for further analysis of two known problems in the region, specifically: erratic pressures and temperatures in the adjacent coolant liner cavity and cracks in the blade shanks near the outer diameter of the aft-platform seal.

For sale by N a t i o d Tecwcll Information Service, Springfield. Virginia 221'

NASA-Langley, 19