s.a contractor nasa crs.a contractor r e p 0 r t * nasa cr.73z23,m¢ 9¢ gpo price $ cfsti price(s)...
TRANSCRIPT
S.A CONTRACTORR E P 0 R T * NASA CR.73Z23
,m¢
9¢
GPO PRICE $
CFSTI PRICE(S) $
Hard copy (HC)
Microfiche (M F)
ff 653 July 65
RADIATION TRANSPORT FOR BLUNT-BODYFLOWS INCLUDING THE EFFECTSOF LINES,AND ABLATION LAYER
by Jin H. Chin
• (THRU)
(NASA CR O_R TMX OR AD NUMBER) ..........
Preparedby
LOCKHEED MISSILES & SPACE COMPANYSunnyvale, Califarniafor Ames Research Center
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. December 1967
https://ntrs.nasa.gov/search.jsp?R=19680014234 2020-04-03T04:04:02+00:00Z
NASACR- 7 5
RADIATION TRANSPORT FOR BLUNT-BODY FLOWS
INCLUDING THE EFFECTS OF LINES AND ABLATION LAYER
By Jin H. "Chin
Distribution of this report is provided in the interest of
information exchange. Responsibility for the contentsresides in the author or organization that prepared it.
Prepared under Contract No. NAS 2-4219 byLOCKHEED MISSILES & SPACE COMPANY
Sunnyvale, California
for Ames Research Center
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
For sale by the Clearinghouse for Federal Scientific and Technical hfforllmtion:Springfield, Virginia 22151 - Prices
FOREWORD
The work described in this report was completedfor the National Aeronautics and SpaceAdministration,Ames Research Center, under Contract No. NAS2-4219,with N. Vojvodich as Technical Monitor.
The author extends his appreciation to L. F.Hearne and K. H. Wilson for their help and discussionduring this study.
ii
CONTENTS
FOREWORD
SUMMARY
INTRODUCTION
1 NOMENCLATURE
FLOW FIE LD ANALYSIS2.1 Governing Equations2.2 StagnationFlow Field
2.2.1 Air Layer2.2.2 Ablation Layer
2.3 Streamtube Formulation
3 RADIATION TRANSPORTANALYSIS3. 1 Basic Radiative Equations3.2 Equations for a Finite Number of S.ublayers3.3 Treatment of Continuum Contributions
3.3.1 The Continuum Absorption Coefficient3.3.2 The Multi-Band Model
3.4 Treatment of Line Contributions3.4. 1 The Line Absorption Coefficient3.4.2 The Line Transmittance and Equivalent Width3.4.3 Absorption Coefficient Notation3.4. 4 Line Calculation Limiting Cases
4 NUMERICAL METHODS4.1 Division of Continuum Bands and Line Groups4.2 Schemeof Flow Field Iteration4.3 Schemeof Line Calculation
5
ii
1
1
3
7789
1013
15
15
17
19
19
20
20
20
22
24
27
3737
3738
RESULTS 43
5. 1 Effects of Exponential Kernal Approximation 435.2 Effects of Continuum Band Models 435.3 Effects of Line Model 43
5.4 Effects of Ablation Layers 445.5 Effects of the Number of Sublayers 445.6 Effects of Environmental Variables 45
5.7 Effects of Perturbation of Radiative Properties 455.8 Effects of Precursor Heating 465.9 Equivalent Width Results 465.10 STRADS Results 47
6 CONCLUDING REMARKS 47
iii
CONTENTS (Continued)
Appendix
A THERMODYNAMIC PROPERTIES OF AIR AND ABLATION VAPOR
B RADIATIVE PROPERTIES OF AIR AND ABLATION SPECIES
B. 1 Absorption Cross Sections of Air SpeciesB. 1.1 Continuum Absorption Cross SectionsB. 1.2 Line Absorption Cross Sections
B. 2 Absorption Cross Sections of Ablation Species
C THE STAGRADS CODE
C. 1 STAGRADS ListingC. 2 Input Data Listing
C. 3 Output Data Listing, Including Details of Line Data Used
REFERENCES
LIST OF TABLES
LIST OF ILLUSTRATIONSi
LISTING OF STAGRADS AND INPUT DATA
!OUTPUT DATA LISTING, INCLUDING DETAILS OF LINE DATA USED
49
5151515152
53535558
65
67
75
99
163
iv
RADIATION TRANSPORT FOR BLUNT-BODY FLOWS
INCLUDING THE EFFECTS OF LINES AND ABLATION LAYER
By Jin H. Chin
Lockheed Missiles & Space Company
SU_CIMARY
Efficient numerical methods are developed for fully radiation-coupled inviseidblunt-body flows. The spectral nature of radiation transport is treated in detail,using as many as 21 spectral regions and accounting for air and ablation vapor con-tinua and 75 discrete atomic lines. A concept of line-group equivalent width andaverage transmittance is introduced in a finite-difference formulation of radiationtransport. This concept is used to account for line overlaps and to formulate the
numerical procedures.
The application of the numerical methods is described. The effects of environ-mental variables, of the numerical methods used, and of the uncertainties in radiativeproperties are discussed. The results demonstrate that the self-absorption and energyloss effects decrease the sensitivity of the heat fluxes to the changes in environmentalvariables and to the uncertainties in radiative properties.
INTRODUCTION
This report presents the analyses and results of a study of radiation transportfor blunt vehicles returning from space missions at superorbital velocities. Thestudy is an extension of the author's previous investigation (refs. 1 and 2). In refs. 1
and 2, the effects of non-grey, self-absorption and energy loss for inviscid blunt-bodyflows were considered. The air continuum spectral absorption coefficients were r_p-
resented approximately by a 2-band (level) and a 6-band model. The contributions ofdiscrete atomic line radiation were not taken into account. The solutions of the flow
field for the stagnation and nonstagnation regions were formulated separately. Asimilarity solution was obtained for the stagnation region and a streamtube methodwas used for the nonstagnation region. The formulation of the present investigationincludes the effects of discrete atomic-line radiation and the presence of an inviscid
ablation layer adjacent to the wall. Due to time limitations, however, these effectsare incorporated in the numerical calculations for the stagnation region only. Anewer set of air continuum absorption coefficients is used for both the stagnation and
nonstagnation regions.
The purpose of this investigation is to develop economical numerical methodsfor fully radiation-coupled, inviscid, blunt-body flows, and to assess the effects ofenvironmental variables, of the numerical procedures used, and of the uncertainties
in radiative properties.
_" The flow-field analysis is reviewed in Section 2. The flow field is assumedinviscid with an interface between the ablation products and the shock layer air.The stagnation region air layer and ablation layer are analyzed using similaritytransformations, with continuity of pressure across the interface. The streamtubeformulation for the nonstagnationregion is very briefly described, as only an exam-ple result (fig. 23) is given.
The detailed analysis of radiation transport is given in Section 3. The diver-genceof the radiative flux is calculated by one-dimensional approximations. Forthe integration of the energy equation, the shock layer and ablation layer are dividedinto a number of sublayers, each with constantproperties. The integrals of the radia-tive divergence over the individual sublayers are used in the finite-difference energybalance. The various terms in the expression bf these integrals are of the samebasic functional form. Studies of the mathematical properties of this basic functionalform lead to a theory of calculating the continuumand discrete (atomic lines) radia-tion contributions. A concept of line-group equivalent width and average transmittanceis used to calculate the contribution of a group of lines, accounting for line overlaps.Readers not interested in the details of radiation transport calculations may skip mostof Section 3.
Section 4 describes the numerical methods, including the division of continuum
bands and line groups, the scheme of flow fielditeration, and the scheme of line
calculations. Some of the equations [e.g., Eqs. (129)- 156)] presented are used
in the computer program and may be omitted by readers not interested in thenumerical details.
The results of calculations are given in Section 5. Except fig. 23 which showsan example of the heat flux distribution for a blunt vehicle, all results are for the
stagnation region obtained using the STAGRADS code.
The air and ablation vapor thermodynamic properties, the radiative properties
of air and ablation species, and the description of the STAGRADS code are given inthe appendix. "
a
A
Ail, i2,m
b
b'
bnn, (v)
B
C
d ,nil
e
E
E n (_)
f
f ,nn
F
g
gn
h
H
I
k
K
_s ' _y
L
m
1 NOMENC LATURE
constant in exponential kernal approximation E 3 (_)a = 4 is used.
1 +Ky
function defined by Eq. (118)
constant in exponential kernal approximation E 3 ( _ )b = 2 is used
b ( 7re2/mc 2 )
line shape factor , Eq. (81)
Planck distribution function
speed of light
line-center shift
electron charge
energy of electronic state above ground state
n-th exponential integral
band system f-number; or A L+I pv [ RN/( L + 1) u r
absorption f-number for transition n --n!
pV/PsV s or A L+I pV/PwV w
h/h s or h/h w
fkgnk , a constant for each line
statistical weight of the n-th state
static enthalpy or Planck's quantum constant
total enthalpy, or enthalpy ratio (Appendix A)
specific intensity
Boltzmann' s constant
body curvature , = 1/R N at stagnation point
directional cosines of vector _2 , Eq. (54)
0 for two-dimensional, 1 for axisymmetric
electron mass
-b_:~ (b/a) e ,
-b_~ (b/a) e .
1/2, Eq. (40)
_e
mw
n
N
Ne
P
q
%qr ' qr
s yQ
r
ro
R
R M
RO
R*
8
g
S
T
Trans
U
Ur
v
Width
Y
Z
Z
"ge
wall blowing mass flux
number of sublayers; state index
k-th line lower state index, see Section 3.4.3
species particle number density
electron particle number density
static pressure
net radiative power gain per unit volume
radiative flux vector
components of radiative flux in s and y directions
radiative power gain for a finite sublayer; partition function
distance from plane or axis of symmetry i
value of r at streamline entry point
ratio of equivalent width to line-group width for isolated lines
maximum radius, fig. 22
nose radius
distance from stagnation point to effective axis of symmetry
ratio of equivalent width to line-group width by integration
distance along body from stagnation point
distance along body from point of symmetry, fig. 22
line strength, Eq. (110)
absolute temperature
line-group transmittance
velocity parallel to body
value of ablation layer u at interface , Ue = Ur (S/RN)
velocity normal to body
line-group equivalent width
normal distance from body surface
normal distance from shock wave
compressibility factor
line effective half-width
k-th line half-width due to 1 electron per unit volume
4
w
Ynn v
5v k
A
AHV
AY i
AZ.i
AY m
AT.]
V
Va
Vo, nn t
P
PSL
(7
T
¢
f_
Subscripts
a
ab
c
d
e
i
i I, i 2
for n-_ n' transition
integration interval for k-th line
total thickness between shock and wall
latent heat of vaporization
Y i+l - Yi
zi+ 1 -zi; Az i = Ay i
integration interval for m-th line group
/zj Az. J
transformed normal distance; dummy variable
transformed normal distance, Eq. (34)
linear absorption coefficient
frequency
average frequency within line group
frequency corresponding to unperturbed transition
transformed distance from stagnation point, Eq.variable
density
sea-level density
cross section; Stefan-Boltzmann constant
optical thickness
angle between flight vector and normal
stream function
solid angle
air layer
ablation layer; with ablation layer effect
continuum contribution
discrete contribution
at interface between air layer and ablation layer
species index; dummy index for sublayer
specific sublayers
n ..-_ n !
(33) ; dummy
5
J
k
nn t
S
sat
w
v
oO
dummy sublayer index; dummy index for frequency integration
dummy index for lines
for transition n --* n'
immediately behind shock; toward shock
at satellite velocity
at wall conditions
at frequency
at ambient conditions
2 FLOW FIELD ANALYSIS
2.1 Governing Equations
To simplify the formulation of the problem, the flow field is assumedquasi-steady; inviscid, non-conducting, and in thermodynamic equilibrium. An interfaceexists between the ablation products and the shock-layer air. The wall is assumedto be at the equilibrium sublimation temperature corresponding to the surfacepressure.
In body-oriented Coordinates (s , y) shownin fig. 1, the conservation equa-tions for a two-dimensional or axisymmetric, inviscid flow may be written as follows-
continuity
0.r Lpu + 8Ar Lpv = 0 (1)as 8y
s-momentum
au au _apu_-_+ Ave+ Kuv = - P as (2)
y-momentum
0v _v A apu_-+ Ave- - Ku 2 = --- (3)p ay
energy
OH OH AU_" + Av_, = _-q (4)
The net radiative power gain per unit volume is related to the divergence of theradiative flux:
q = - div_r = - [ ay + _J (5)
The equations-of-state of equilibrium air and equilibrium ablation vapor maybe expressed in the following form:
p , T , N i , . . - function (p, h) (6)
The boundery conditions are as follows:
At the wall,
.
u = u = 0 (7)w
4v = vw = ;-nw/P w = (qw - O'Tw)/PwAHv (8)
h = h w = h w(pw) (9)
The wall is assumed to be black.
At the shock wave,
u = u s = uoo [sin _sC°S (_bw - _s ) + ¢ cos _s sin(_w - Cs ) ] (10)
v = v s = u [sin_bsSin(_ w - _bs) - EeOSCsCOS(_ w i_bs)l (ii)!
oouP = Ps = p oo (1 - ¢) cos 2 (_s + P_ (12)*
At the interface,
Pa, e = Pab, e (14)
V ,e ab,eUa, e Uab, e
(15)
These equations will be approximated for different regions of the flow field'.
2.2 Stagnation Flow Field
For the stagnation region, s/R_ << 1 , the conservation equations may besimplified. The kinetic energy change" is of a smaller order than the enthalpy changeand the pressure variation normal to the wall is small. Equations (3) and (4) may
then be approximated by Eqs. (16) and (17), respectively.
8p = 0 or P = Ps (16)8y
8
*In superorbital velocities, Poo and hoo are negligible compared to Ps and hs, re-spectively. Precursor heating may increase hoo . However, for the velocities of
interest in this study, the precursor heating is small, as will be discussed inSection 3.1.
Oh _h A
u + Av --Oy= p q (17)
For this study, the shockwave and the interface are further assumed to be con-
centric with the body surface so that _bs = Ce = _w "
2.2.1 Air layer. - For the stagnation air layer, Eqs. (10) to (13) reduce to:
S
u s = u -- (18)
= - ¢ u (19)VS ¢o
Ps = P_ u2_ (1 - _) - + p_ i (20)
1 2 2)h s = _Uoo (1 - + h (21)
For a concentric interface,
v e = 0 (22)
For inviscid flow, the interface conditions for u and h cannot be specified;
they are part of the solution to be obtained.
Since there are more h,own boundary conditions at the shockwave, it.is more
convenient for the air layer solution to use the shockwave as the datum for the nor-mal distance. Using transformations similar to the Lees-Dorodnitsyn transforma-tions for compressible boundary layers and assuming similarity, the following re-sults were obtained in ref. 1":
_p_ d___z (23)d_ - (L + 1) Ps RN
F - - pv = __pv (24)p_ u Ps vco S
_ d F = u (25)d_ u
S
*The analysis of the stagnation flow field for the ablation layer is similar to that forthe air layer. Consequently, only the analysis for the ablation layer is presented
in detail in the following subsection.9
10
h
S
(26)
F d2Fi Ida)2d_2 = (L + 1) - 2E (1
-E)Ps
(27)
d_ (L + i) pu(28)
or
2F dg - 3 ( q dz)
P_ u_o
(28a)
where z is the distance from the shockwave toward the interface. In Eqs. (23) to(28), the shock layer has been assumed thin compared to the nose radius so thatA_I.
The boundary conditions are:
At the shockwave
dF= 0 , F - d_ - g = 1 (29a)
at the interface,
where g is the value offrom theeolution.
= _e ' F = 0 (29b)
at the interface. The value of _e must be determined
The numerical integration of Eqs. (27) and (28a) is described in Section 4. 2.
2.2.2 Ablation layer. - For the stagnation ablation layer, Eqs. (16) and (20)indicate-
2P = Pe = P_oU (1 - c) + p_ (30)
Ue
Since the layer edge is a streamline and is concentric with the wall, using
= u r (s/RN) one obtains:
dPe _ dUe 2 s
ds Pe Ue ds - PeUrlt 2N
u2 s__
- 2p_ _ (1 - e) R_q
(31)
"Hence,
1/2
Ur = [2P_(1-E)]uoo Pe(32)
where Pe is not known before the solution is obtained.
Equations (1), (2), and (17) may be simplified by introducing the followingtransformations:
S2L
UedS
0
(33)
ii
y
dy (34,_-,/_0
8_ 85 L_yy = -purL ' _ss = Apvr (35)
f(_,77) = (36)
hg(_'_) - h (37)
W
If similarity is assumed so that all dependent variables are functions of 7? only,the transformed equations may be shown as follows:
f d2f 1- df 2 PooU_o K (L + 1)R N fdf
d_?---2= (L + 1) - 2 2 AL+lp u _.pu r r
(38)
f dg = ARNdU (L + 1)PUrh w q
(39)
The transformation also yields the following relations:
f jR. ]AL+lpv (L + 1)u r
1/2
(40)
11
d_ [(L+ l)Ur]
_/2
ALp
Further simplification of Eqs. (38) and (39) is possible by introducing thefollowing:
F
f A L+I_ pv
PwVw "(L + 1)u r
1/2v w
l 2" 1/2
1d_ = 1/2 dq =
OwVw "(L + 1)u r
2P_o p A L dy(L + i)
\pwVw / Pw RN
Equations (38) and (39)become, respectively,
F
fd2F = 1 ;{dF] 2
d_2 (L + i) [\d_/[ 2)iJ2Pw 1 + 1 _PwVw F
P AL+I \2Poo u2oo
1 A L+I q dyF dg = PwVwh w
For Aab/R N << 1, A _ 1.
For conditions with
v 2 /p u 2Pww/_oo
<< 1
one may neglect the last term in Eq. (45).
(41)
(42)
(43)
(44)
(.45)
(46)
12
F d2F 1 "r/dF_ 2 Pw
d_2 (L + 1) [\d_] p(47)
The boundary conditions are as follows:
at _= y= 0
1/2
= - Pe ](48a)
at _ = _e or y = Aab
i/2
dF (Pw_ 2F = 0 , d--_ = \Pe] ' P = p u (1 - e) + p_ (48b)
Because of the last term in Eq. (47), the momentum equation is coupled to theenergy equation. The surface blowing rate and the layer thickness are not knownbefore the solution is obtained.
The numerical integration of Eqs. (46) and (47) is described in Section 4.2.
2.3 Streamtube Formulation
For the nonstagnation region, the streamtube formulation described in ref. 1 is
used. For the inviscid air layer, the conservation equations for an infinitesmalstreamtube may be written as follows:
continuity
L dy _ L dropur ds - P_u_oro d---s (49)
momentum
du dp (50)Pu _-_ = - ds
energy
d +
pu ds = q (51)
13
" where r o is the radial distance at which the streamtube enters the shock. Theentry conditions for the streamtube are given by Eqs. (10) to (13). The applicationof the streamtube method during this study has been limited to air-layer calculationswith continuum radiation only. The integration of Eqs. (49) to (51) follows the pro-cedures described in ref. 1 and will not be discussed further in this report. The
streamtube method may also be applied to the ablation-layer calculations.
14
3 RADIATION TRANSPORTANALYSIS
3.1 Basic Radiative Equations
The net radiative power gain per unit.volume by the gas in local thermodynamicequilibrium is given by
f f - qq(s,y) = dv #v(S,y)[Iv(S,y,_,¢2 ) - Bv(S,y d_2
v _=47r
(52)
where I v is the monochromatic specific intensity (power per unit solid angle per unitarea normal to the direction of propagation), By the Planck distribution function, and#v is the spectral (monochromatic) absorption coefficient accounting for induced emis-sion. The monochromatic specific intensity is governed by the following equation ofradiative transfer.
In body-oriented coordinates, Eq. (58) becomes
where _s and ty are the directional cosines of the vector _ with respect to thecoordinate axes. The flow field is assumed either two-dimensional or axisymmetricso that the gradient of the specific intensity in the circumferential direction vanishes.
The net radiative flux crossing a unit area with unit normal -_, is given by
Then,.
P_= 4r
f •qr = _s Iv (-_) d_2 (56a)
v,s _=4_
r
qr = J ty Iv (_) d_ (56b)v,y _2= 4_r
.15
Integrating Eq. (54) over all solid angles and using Eqs. (52), (56a), and (56b), one
obtains the monochromatic form of Eq. (5)
qr 0 qrv, s + v, y
Aas 0y - divqr = -qv(S'Y) (57)V
Under conditions that the temperature gradients in the s-direction are small com-
pared to that in the y-direction, one may neglect fs[a Iv(_)/A as] in Eq. (54) and
(O qrv, s/A as)
in Eq. (57) to obtain the "tangent slab" or one-dimensional approximation for radia-
tion transport, i
For the environmental condition of interest in this study (uoo< 60,000 ft/sec,
RN < 10 ft) the effectof precursor radiation heating of the ambient air ahead of the
bow shock is to increase the wall heat flux by the order of 10 percent or less (refs. 3and 4). Thus, itdoes not appear that the neglect of precursor heating in the basic
•radiative calculations contributes significantlyto the uncertainty in magnitude of radia-
tion heating. For this investigation, the precursor heating effect and the shock-wave
reflectivityare neglected and the body surface is assumed black.
For one-dimensional radiation transfer, the monochromatic net radiative power
gain or the radiative flux divergence is given by refs. 1 and 5....
0 qrv, y
0y = qv (y)
-47r #v Bv + 27r #v BW,
f ' B' E 1#v p
OY I)yf, " dy"
#v dy' (58)
where E n(_) is the nth exponential integral and A is the thickness of the "tangentslab. "
The succeeding terms on the right-hand side of Eq. (58) can be recognized as (i)
radiative loss, (2) absorption of radiation from body surface attenuated by the medium
between the body and the local point, and (3)absorption of radiation emitted by medi-
um on both sides of the local point.
16
In terms of z = A - y , the monochromatic optical thickness is given by
Z
T v(z) = f #v(Z') dz'
O
(59)
A
_'A,v = f #v(z') dz' (60)
O
Equation (58) may then be rewritten as
T_, Y
qv(rv) = -4_PvBv + 2_#vBw. vE2(TA, v - TV) + 2_.p f Bv(t)EI([T v - tI)dt0
(61)
The monochromatic heat fluxes incident to the wall and leaving the shockwave to-ward the ambient air are given by Eqs. (62) and (63), respectively,
TA, V
I "qw, v = 2_ B v(t) E 2(TA,v - t) dt (62)
O
T&, V
f "qs, v = 27r B v (t) E 2 (t) dt+ 27r Bw, v E3 (TA, V) (63)
O
Integration of Eq. (61) over all frequencies then yields the q required in Eqs. (28),(28a), and (46). Equations (62) and (63) may be integrated over all frequencies to ob-tain the total heat fluxes.
3.2 Equations for a Finite Number of Sublayers
In order to reduce the computation time required for integration of the conservationand radiative transport equations, the shock layer and ablation layer may be dividedinto a number of sublayers. Across the width of the sublaycr, constant properties
are assumed, but radiation traversing through is attenuated by each of the differentialwidths within the sublayer. Equation (61) may be integrated across the width of thei th sublayer from the shockwave (fig. 2).
17
Zi+l
Qi(v) ---f qv(z) dz (64)
Z.I
Carrying the integration, one obtains
Qi(u) = 2n BwIE3(Tn+ 1 - Ti+l) - E 3 (Tn+ 1
n
j=l
For convenience, the subscript v has been omitted in Eq. (65) and the equations
following. Equations relating monochromatic quantities may be recognized by the
presence of at least one term with argument v. The optical thickness for the sub-layers are given by
h_-i(v) = _i Azi (66a)
TI (v) = 0 (66b)
i
7"i+l(v) = _ A'r i , i = 1, 2,..., n (66c)
j=l
The monochromatic radiative heat flux to the wall, qw(V), is given by Eq. (67), ob-tained from Eq. (62).
n
qw(V) = 2_ _ Bi[E3(Tn+ 1 -Ti+l)- E3(Tn+ 1 - Ti)] (67)
i=l
The monochromatic radiative heat flux leaving the shock front may be similarlyderived.
n
qs(V) = 2v _ Bi[E3(_-i)- E3(Ti+I) 1
i=1
+ 2_ Bw E 3 (Tn+l) (68)
The total energy gain or heat fluxes may be obtained by integration of Eqs. (65),(67), and (68) over all frequencies. The spectral integration for the continuum radia-
tion and the line radiation will be discussed separately in the following subsections.
!8
The third exponential integral, E3 (_) , may be calculated using numerical correla-tions and series. However, considerable computation time may be savedby using the"exponential kernal approximation" for the exponential integrals. For instance,
E3(_) ~ b e-b_ (69)a
Matching the values of the approximate function and its slope at zero argument with theexact values E3(0) and El(0 ) =-E2(0) , respectively, one obtains b = 2 and a = 4so that
E 3(_) ~ 0.5e -2_ (70)
The individual terms of Eqs. (65), (67), and (68) are of the general form
Iij k(v) = B kE3([T i - Tj[) r
(71)*
One then can study the spectral integration of Eq. (71) as a basic step for performingthe spectral integration of the more _'nrnplin_t_d _q_ [c_l 1_7_ ._n,_ t_c_x v+ ...-11 ,.^shown in the Section 3.4.2 that the exponential kernal approximation enables a con-venient formulation of line radiation transport.
3.3 Treatment of Continuum Contribution
3.3.1 The continuum absorption coefficient. - The spectral absorption coefficients
and the optical thicknesses may be separated into two parts - the continuum (subscriptc) and the discrete (subscript d).
#(v) = ]ze + #d (72)
= + (73)T rc Td
The discrete part varies with frequency much more rapidly than the continuum part.For the present study, the discrete part corresponds to the contribution due to theatomic lines. The continuum part corresponds to the contribution of molecular bandsystems, free-free, and photo-absorption processes.
*With b = 1 and a = 1 , Eq. (71) and many of the following equations may be usedfor radiation transport calculations for a pencil of radiation.
19
3.3.2 The multi-band model. - For the continuum radiation, the multi-band modelis used. The continuum spectral absorption coefficient is assumed constant within the
individual spectral bands or intervals. The Planck intensity function is integrated using
appropriate series expansions. For instance, consider spectral integration of Eq. (71)over a band with v1_< v<_ v2.
v2 v2
f Iijk (v) dv = E3(17" i - "rjlc) /J Bk(V) dv
vI vI
v2
, ,o)fvI
B k (v) dv (74)
The value of the spectral absorption Coefficient for a given band is calculated byappropriate averages. For bands which are "expected to be optically thin for mostcalculations, the partial Planck-mean defined by Eq. (75) may be used.
(T, v1 , v2) =
/2 o/.d
fv1
/.t(T, v) B (T, v) dv
v2
f (T, v)B dv
v1
(75)
3.4 Treatment of Line Contributions
3.4.1 The line absorption coefficient. - The absorption cross-section of a line(bound-bound) may be written for the transition n--* n' as follows:
Crnn,(V) = (re2-_ 1 ( -fnn' bnn' (v) 1 e -hvnn'/k
\mc /(76)
where
e, m = electron charge and mass, respectively
c = velocity of light
2O
k
fnn' =
Vnn' =
bnn ,(v) =
Boltzmann's constant
absorption f-number for transition n-- n'
line center frequency for transition n--, n'
line shape factor normalized according to
The factor (1 - e -hvnn'/kT)
oO
f bnn,(V ) dv = 1
0
accounts for induced emission.
(77)
To obtain the spectral absorption coefficient, the cross section is multiplied by theoccupation number, N n , of state n (the number density of a given species in state
n). For a monatomic gas in local thermodynamic equilibrium, the occupation numberis given by the Boltzmann formula
where
gn --
Q=
E =n
N =
gn" -(E a/kT)
N n = -_-Ne
• ,
!
statistical weight of the n th state
electronic partition function of monatomic species
energy of n th electronic state above ground state
total number of particles of the given species per unit volume
(78)
The partition function is given by
- (En/kT)
Q = _ gn e (79)
n
The energy levels, partition functions, and statistical weights for nitrogen and oxygenatoms and ions are given in ref. 6.
From Eqs. (76) and (78), the bound-bound absorption coefficient for transitionn _ n' is then given by
2 / gn -En/kT [ (-h Vnn,_ I#nn' : \mc(_re, frm,_.Ne bnn,(V) I - exp_ "k-T ]l(80)
21
For electron-impact broadening, the lines follow a Lorentz shapewith the half-width proportional to the electron particle density:
1 Tnn,bnn, (v) = -
v - (v - dnn,) +O, m'l v TnnV
(81)
Tnn, = Tnn, Ne (82).
where
Vo, nn'
dnn , =
Tnn,-
Nei
= frequency corresponding to unperturbed transition n--* n'
line-center shift
half-width due to 1 electron per unit volume for
electron particle number density
n-_ n I
The normalized half-width 7nn' , as calculated by R. Johnston, for carbon,
nitrogen, and oxygen atoms and ions are tabulated in ref. 7.
Excited electronic states having the same core configuration, principal and orbitalquantum numbers but with different spin multiplicity and total orbital angular momen-tum for L-S c6upling have nearly the same En. Therefore, it is convenient to com-
bine these States into a single group so that only the occupation nmnber of this groupof states need be calculated. However, the f-numbers of the individual transitionsmust be multiplied by the ratio of the statistical weight of the individual lower state to
¢
that of the group of states. In other words, one may consider in Eq. (80) that gn isthe statistical weight of a group of lower states of similar energies and fnn' is aneffective f-number which is equal to the f-number of the particular transition multi-
plied by a statistical-weight ratio.
The total bound-bound absorption coefficient is the sum of all #nn' due to differentlower-state groups of different species.
3.4.2 The line transmittance and equivalent width. - Consider the spectral integra-tion of Eq. (71) over a finite frequency interval Av across which the Planck intensityfunction and the continuum contribution may be approximated by constant values.Then,
Iij k =_ f Iijk(V)dv
AV
-blTi-rjl
_ - B,_ ea
Av
_Bk(Va)a exp[-blTi rJl c, va AV
-bl i- jlc d
e dv
dv
(83)
22
where va is an average frequency (e.g., center frequency) within Av.
Define the average transmittance as follows:
f _b] Ti__.jTTransij -= A----V e
Av
Then,
-b[ ri-Tj[b
Iij k -_ _ Bk(Va) e
c, vaAp
Transij
Alternately:,ii
2_v
!i
i
Cijk
one can write Eq. (83) as follows:
c bf Bk ee dv -
Av
-b[ Ti-Tj ]
_ b Bk (Va) ea
C,Pa
f (1 .. e-bITi-TJld)dvAv
where
cij k aAv
-b Iri-Tj]C d vB k e
is the continuum contribution.
Define the equivalent width over Av as follows:
Widthii = - e du
Av
Then, Eq. (86) may be rewritten as
(84)
(85)
(86)
(87)
(88)
• 23
-b] _'i-_'j ]b c, v a
Iij k _ Icijk -_Bk(Va)e Widthij (897
The use of Eqs. (87) and (88) permits different spectral divisions for the calculation
of the two terms on the right-hand side of Eq. (89).
From Eqs. (84) and (88)
Widthij = Av(i - Transij ) (90)
The study of line transport then reduces to the calculation of Trans.. or Width...Ij Ij
3.4.3 Absorption coefficient notation. - In order to avoid using too many indices
to describe the variables, the following system of notation will be used: i
i = dummy index for sublayers
i1 , i 2 = specific sublayers
k = dummy index for lines (except in product kT7
m = dummy index for line frequency regions
n. = kth line lower state iridex (The species index is eliminated and the
value of n k specifies the species. The term Nnk will be used todenote the total particle number density for the species specified bynk. )
Thus,
(v) = 2tF gnk -Enk/kTi (i -hUk/kTi) .'7re -- N. e
(_c2]_.Z fk Qi, n k 1, n k - e bi, k (v) (91)
k
represents the spectral absorption coefficient of the i th sublayer at v. For many
situations, the line-center frequency vk in [1 - exp (-hvk/kT)] may be approximatedby vm , an average frequency for the mth frequency region.' It should be noted that
lines outside the m th frequency region may contribute to the value of #i(v) at vwithin the mth frequency region.
In terms of the new notation, one has the following:
1 Ti, k-- (927
bi, k (v7 = 7r 2 2
[v- (v k - di, k) ] + 'Yi, k
24
if[ii 2Transil, i2, m - Avm exp -bi
Avm
Widthil, i2, m = AVm(1 - Transil, i2, m )
J_i(v) Az i dv
)x_I [-b il' i2
1 - exo
L L
The sum in Eq. (93) represents the following operation:
(93)
d_
(94)
"0 , i 2 = i I
11 , 12
i
_i =
12-I
i--t 1
_i ' i2 > il (95)
iI-I
i= i2
i2 < iI
In order to simplify the calculations, the effects of line shiftwill be neglected. For
the conditions of interest, it is reasonable to consider only lines due to neutral nitrogen
atoms with an effectivedensity equal to the sum of the actual NI and OI values.*
With this approximation, Ni, nk may be replaced by N i and Qi, nk by Qi. The
numerical computation is thus considerably simplified. One may now write
*One may also neglect the contribution of the OI vacuum ultravioletlines.
25
Width.11, i2, m = - exp - b' _ 1
Avm
AZi _ gfk exp (-
k
- exp
%kTi/ Yi, k k}/2 : y2
(v- Vk) + i,
where
b' = b(_e2/mc 2)
gfk = fkgnk ' a constant for each line
The expression for the average transmittance now becomes:
fili2{I ]7 exp -b' NiTransi l,i2,m - Avm i Qii
Avm
/
exp1 _/i, k
(v - Vk)2 +
dv
(96)
dv
(97)
26
where
i1, i2
/7i
I-
_i =
i2-i
/7
i 1-1
/7i2
, i 2 = i 1
_i ' i2>il
_i ' i2 < _il
(98)
In the conditions of interest, itis reasonable to assume that _'nn'-__i,k is aslow-varying function of temperature. Then, the following approximation may beused:
Ti,k _ Tk Nei (99)
3.4.4 Line calculation limiting cases. - The study of line transport now reduces to
the calculation of Widthil, i2,m or Transil, i2,m according to Eqs. (96) and (97).
In general, the integral in Eqs. (96)or (97)can not be expressed in terms of simple
combinations of analytic functions. Although numerical integrations can always be
used to evaluate these integrals, the computation time required may become exces-
sive. One should then look for appropriate approximations applicable to different"
asymptotic situations. For situations where numerical integrations must be used,the numerical schemes should be selected with a consideration for the minimizationof computation time.
The approximations for different asymptotic situations are discussed below:
(1) Thin-line approximation. -
For _ <<1, e-_ -_ 1 - _. Now,
27
b'
iI,i2
i
- exp
- kTi/j azi gfk exp - kTi) _ (v - Vk)2 + T 2i, k
-< b'
i1,i2
i_-ex_ - G)]_z_ _*kOX_- - _ _.
k kTi] 7r 0 + yi,2 k 11'i2'm
(1oo)
where the identitydefines Til'i2,m , an effectiveoptical thickness evaluated at the
line centers. Then, for Til,i2,m << 1 :
Widthil, i2, m
iI,i2
I : [ (- i__ _, _ _ _I _"_ _,_2
AV m i Qi 1 exp kTi/JAzi k gfkexp kTi/_ (v - Vk)2 + 7i, k
dv
iI,i2
i Qii 1- exp - kTi]jAz i gfk exp kTi/ 7r (v Vk)2 2k Av - + 3_i, km
dy
iI,i2
i_ Nil ( hVl:n_] ( Enk_= b' Qii 1 -exp - kTi]jAzi _ gfk exp • kTi]G(Vum• k
, V_m, vk, 7i, k ) (101)
where
=i[ ru .: t=-, <io ,G<Vum,V_m,Vk,Yi, k ) _ tan -1 ('_m-_Q]\ Ti, k \ _i_k ]J
with rum and V_m as the upper and lower limits, respectively, of the mth frequencyinterval.
28
i
When the contributing lines are well within the frequency region (notnear the twoboundaries), Eq. (102) may be approximated by:
G (rUm, V£m , Vk, Ti ' k) = i (103)
With the above approximation, the equivalent width then becomes the sum of the so-called strengths* of the contributing lines.
Equation (101) is valid for both isolated and overlapped lines, provided
Til,i2,m << 1.
(2) Isolated-line approximation
When Av m is large compared with 7i k and the contributing lines within thefrequency region may be considered isolat'ed, the integral in Eq. (96) may be approxi-mated by the sum of integrals for the individual line. Thus,
W{A+_.
.....Ii,i2,mI { if'i2 [
N i hVm_ ]
1 - exp -b' 1 - exp (- k-- iJ]ahk 6v k i
gfk exp - kTi/Tr })(v - Vk)2 + T 2i, k
dv (104)
where 6v k is the integration interval for the kth line.
The integral in Eq. (104) for a single value of the index k represents the equivalentwidth of a single line for a nonisothermal path. The use of an effective half-width
(ref. 8) may enable representation of the equivalent width in terms of the Ladenburg-Reiche function (ref. 9).
Consider the following integral:
I = ; [1- exp /-1 _ Si--Ti- ._]
oO
s[, /-'-'-_ - exp 2 e27r_ +,y_ °
d_ (105)
*The strength of a single line over an isothermal path (L)
(Ni/Qi) [1 - exp (- hvk/kTi) ] Lgf k exp (- Enk/kWi)
may be defined as
29
where Te is an effective half-width.later. )
(Thedetermination of Te
Equation (105)may be integrated to yield:
will be discussed
(106)
where
f(_) = _ e-_ [Io(_) + 11(_)] (1o7)
is the Ladenburg-Reiche function with Ifunctions. _ o
Two asymptotic expressions of f(_ )
! f(} ) -_
(_) and I 1 (})
exist:
, _ <<I
, _ >>1
being the modified Bessel
(lO8)
2at } = -
"/r'the two asymptotic expressions become equal.
Since the lines are considered isolated, the interval of integration 6k may be ex-tended to (-oo , oo ). Therefore, one may rewrite Eq. (104) as follows:
i 'i2 1
• _ Si, k, mTi,
X "- 2_ Tekf 1 ____ . (109)Widthi 1 , i 2 , m k 2_e k
where
t hVm__ Eak
5 --qi - kT---:S. = b' I
I, k,m Qi - e ZXzi gfk e (110)
30
4
Now, consider the determination of "ye. Comparing Eqs. (101) and (109) and usingthe first of Eq. (108) for isolated thin lines, one obtains:
i 1 , i 2
Si,k,mTi,ki
Te k = il ,i2 (111)
Si k mi ' '
which is mean weighted according to the line strengths of the individual sub-layers.
Using the second of Eq. (108) for isolated strong lines, one obtains the followingequation:
172i 1 0i 2
Widthi 1,i 2,m _- _ 2_r_/e kk
Si,k, m _/i, k2 i
27r _/ek
. . 1/2
= 2 Si k m "Yi k[isolated strong line
' \square-root approximation]
(112)
The effective half-width does not appear in Eq. (112)• Therefore, it is reasonable to
assume that the effective half-width defined by Eq. (111) may be used, as an approxi-mation, in Eq. (109) for situations between the thin and strong limits for isolatedlines. *
For numerical calculation of the Ladenburg-Reiche function, the following approxi-mations are useful:
0.685 }3/4
-< (0.685) 4
, -(0.685) 4 < _ <1
- 27r (0.685) 4. (113)
4 1
2 )4(0.685
*This may be compared with the Curtis-Godson approximation (ref. 10).
31
(3) Effective isothermal region. Extending the idea of the effective half-width tonon-isolated lines, one may rewrite Eq. (96) as follows:
Width.11 , i 2 , m
Avn
1 - exp _ Ni-b' Qi - e A_. i
i
(114)
E
kTi 1 vi, k- '2 _dv
gfk e 7r(v _ Vk)2+ Ye k
k
!
where 7e k is an effective half-width of the k th line.
Under certain situations, a single lower state or a small number of lower states ofnearly the same electronic energy contribute to the mth frequency region. One may
then introduce the following approximations:
E N E (115)_k m
Yi k _- Yk Ne. (116), 1
Equation (114) may then be written as follows:
Width.11 , i 2 , m
z_
• 1 - exp
m
-b, o- e- Az i Ne i
i
gfk_-(v - Vk)2 + ye k
(I17)
32
The two summations in Eq. (117) may be performed separately, thus saving aconsiderable amount of numerical calculations. Let
iI,i2
Ail'i2'm = Qi 1 - exp/--_i)JAz iexp - _ Ne ii
(118)
TJhen, Eq. (117) becomes
Widthi 1,i2,m -_ I
AP m
i - exp_ b' 1-All, i2,m gfkk (P - Vk)2 + "Ye
dl)
(119)
The fact that All ' i2, m is a sum, independent of the dummy sublayer index i ,implies that the equivalent width given by Eq. (119) may be considered to be that due to
an effective isothermal region. -
Equation (119) may be further simplified, if the thin-line or the isolated strong-lineapproximation is used.
From Eq. (100),
i 1 , i 2
[ ( hVrn/] _ ( Enk_l 1Ni 1 exp Azi gfk exp kTi/_'il, i2, m = b' - -i Qii kTi]J k 7r 7i, k
i I , i 2
i k _ 7i' k
Then, for Til,i2,m <<I , Eq. (I01) yields with G = 1:
iI,i2
[ ( ( Em )Widthil, i2, m __ _ _i 1 - exp - kTi]jzkziexp _b' gfk• i
(120)
(121)
33
Now, consider Eq. (119) for isolated thin lines, one has in analogy to Eq. (109).
Widthil, i2, m -_k
All, 12,m27r7e k ---,_2
2,__ek
X" _kA
i 1,i2,m Z b'g_k ek
(122)
Subtracting Eq. (121) from Eq. (122), one obtains the following
i 1, i 2
yu' k 1-k i
= 0 (123)
For arbitrary variation of gfk and of the number of k-terms, an expression of theeffective half-width satisfying Eq. (123) is given below:
• o
hVm_ E m
• "2 hVm_ ] Tk (124)
Equation (124) may also be derived directly from Eq. (111), using Eqs. (115) and(116).
The equivalent width for isolated strong lines for the effective isothermal region,
according to the square-root approximation, may be obtained from Eq. (112).
34
2 A1/2 mi I , i 2,
k
1/2
'_' gfk _;
(b' gfk Tk )i/2 (125)
The effective half-width does not appear in Eq. (125). From the above consideration,it is assumed that the effective half-width defined by Eq. (111) (or Eq. (124) as a specialcase) may be used as an approximation along the whole "curve of growth. "
(4) Criteria for line isolation. From the above discussions, itis apparent that the
calculations are much simplified ifthe isolated-line approximation may be made. With,
the Lorentz line shape, the wings of the lines extend to +_ and -_. Therefore, the
lines always overlap to a certain degree. One needs then to qualify the description "iso-latedlines" and to establish suitable criteria for line isolationfor numerical calcula-tions.
Let
Then, if
quantity calculated using the isolated-line approximations
quantity calculated by numerical integration without the isolated-line
approximations
IZ Z*] < £ << 1 ,]
Z l(126)
the isolated-line approximations may be used.
NOW,
1Trans. - Trans.* -11, i2, m _1' i2' m A_ m (Widthil, i2,m (127)
35
The following criterion is more stringent than Eq. (126):
Transil, i2, m - Trans*l, i2, m
Transil Widthil, i2, m)Smaller of i2, m' Av m
-<_ <<1 (128)
Unfortunately, the usefulness of the isolated-line approximations is lost if
Trans*il, i 2 , m or (Width*J1, i2, m/Avm) must be calculated for application in
Eq. (128). Therefore, other more convenient approaches should be used.
Consider the calculation of the equivalent width of two overlapping lines, as shownin fig. 3a. The values of the integrand in Eq. (96), calculated by summing the twoisolated-line contributions exceed that based on two overlapping lines. For the profilesin fig. 3a
R -= Av---'m Widthk, m ~ 0.3 t
1R* -- _ Width*
Avm m
R - R*ZR - R* ~ 0. i
In fig. 3b, the line-center separation is three times that in fig.Z R is much smaller than 0.1 for the same value of R.
3a and the value of
Qualitatively, the value of R may indicate the degree of overlapping. As the valueof R increases, the value of R* approaches unity, as illustrated in fig. 4: At
small values of R, the values of Z R are small so that the lines may be considered
isolated. For a given group of lines (hence a particular pattern of line spacing) withinAv m , it may be possible to obtain an approximate relation R* = F (R) by correlatingthe numerical results of some typical distributions of temperature and species con-
centration (with or without using the approximation of effective isothermal region).This approximate relation may then be used in radiation-coupled flow field calcula-
tions. In view of the uncertainties in f-number and half-width values, the aboveapproach appears reasonable.
IThe value of R is inversely proportional to Avm.
2_
4. NUMERICAL-METHODS
The division of the continuum bands and line groups used in the computer code
STAGRADS (STAGnation point, RADiation-coupled, with external Source*) is brieflydescribed in Section 4.1. Details of the spectral divisions are given in AppendixesB and C. The schemes of flow field iteration and of line calculations are discussed
in Sections 4.2 and 4.3, respectively.
4.1 Division of Continuum Bands and Line Groups
For the continuum air radiation, eight different models are provided before thelines are incorporated. These models consist of from two to nine spectral bands
(see Table 1). The first option (LINEUP = 1) of radiation transport calculationsaccounts for only the air continuum contributions using one of the eight models. Forcalculations with lines, two options are provided. In one option (LINEqSP = 2), the
air continuum contribution is first calculated using one of the eight models of banddivisions. The continuum contribution is then corrected for the presence of thelines [ see Eqs. (86) to (89)] . In another option (LINEq_P = 3), the spectral divi-sion nf the continuum bands is made _'_-* - "_ *_-_ um..... _,_,_,_,,_ w,_** _,,_u of .... line groups and the
total contribution is calculated according to Eq. (85).
A total of 21 line groups are used (see Appendix C). Three of the groups containonly a single line per group. Three of the groups contain no lines at all so that thelast option (LINEUP = 3) of transport calculations may be used.
For calculations with ablation layer, only the continuum contributions of the abla-tion species are included in order to reduce the requirement of computer storageand time. To account for the variation of the ablation species absorption coeffi-cients over a wide spectrum, the division of the continuum bands for these species
is made coincident with-that of the line groups. For calculations with ablation .layer, the last option (LINEfltP = 3) is used so that the contributions of the airatomic lines are included.
4.2 Scheme of Flow Field Iteration
The integration of the conservation equations is for the air layer from the shock-wave toward the interface and for the ablation layer from the body surface toward
the interface. For the integration of the radiative terms and the energy Eqs. (28a)and (46), the air layer and the ablation layer are divided into a number of sublayers.
The air layer is divided in equal increments of the actual distance, but the ablationlayer is divided in equal increments of the transformed distance [see Eq. (44)] . At
*In ref. 1, the radiation transport formulation includes the presence of external
radiation sources. For this study, no e_ernal radiation sources are considered.
37
_J
present, a maximum of ten air sublayers and ten ablation sublayers is allowed in
the computer code in order to reduce the requirement of computer storage and time.
The integration of the momentum Eqs. (27) and (47) requires much less com-
puter time and storage. Consequently, their integration may be performed with
finer increments than that for the energy equations. The density ratios in Eqs. '(27)
and (47) are calculated by interpolations from values obtained using the energyequations.
The flow field calculation begins with the integration of the air layer momentum
equation, using an assumed distribution of Ps/P _- h/hs • The integration isstopped when Eq. (29b) is satisfied at the interface, thus determining the air layerthickness. The air layer is then divided into a number of sublayers and the energyEq. (28a) integrated. Across a sublayer, an average value of the nondimensional
normal velocity F is used, and the net radiative energy gain is given by the spec-trally integrated Q [see Eq. (65)] . Iteration is made to obtain a consistent set
of velocity and enthalpy distributions.
For calculations with ablation layer, an initial enthalpy distribution, linear in
the transformed normal distance between the wail enthalpy and an enthalpy at theinterface estimated from the air layer results, is assumed. The density distribu-tion is calculated using the equations of state of the ablation vapor. The initial
blowing rate is estimated using the air layer results, a surface heat balance, andan estimate of the vapor enthalpy rise. The momentum and energy equations are
integrated in the same manner as for the air layer. The air layer temperaturedistribution is maintained unchanged during ablation layer calculations. Iterationon the wall heat flux is also made to obtain a consistent set of wall heat flux, blow-ing rate, and velocity and enthalpy distributions.
The effect of the presence of the ablation layer on the air layer enthalpy distri-bution is then checked. If necessary, the air layer calculation is repeated while •
the ablation temperature distribution is maintained unchanged. Further repetitionsof ablation layer and air layer calculation are then made, if required, to obtain a
consistent set of wall heat flux, blowing rate, and velocity and enthalpy distributionsfor both layers.
At present, the criteria for convergence used in STAGRADS are: enthalpy dis-tribution, 1 percent and heat flux to the wall, 2 percent. The convergence of theblowing rate is governed by that of the wall heat flux. The convergence of thevelocity distribution is governed by that of the enthalpy distribution.
4.3 Scheme of Line Calculation
When numerical integrations are required to evaluate Widthil i2,m orTransil, i2,m , the selection of the computation schemes should f_e carefully made
to minimize the computer time. For example, the calculation of exponential func-tions requires much more time than for performing multiplications. Therefore, itmay be more economical to calculate
38
exp _ il'i2 il1
according to
i 1 , i 2
ff exp (_i)
i
ff there are many variations of i I and i 2 . Computation time may also be re-duced by using common factors and by avoiding calculating the same variables manytimes (with increasing computer storage requirement, however), i
Consider the numerical integration of the integral in Eqs. (96) or (97). The
integrand is to be evaluated for many values of u, say v], within A_, m . Let jb'e the dummy hidex for _. In terms of the .... _-_ "_ _,_;_,,v _+_n _l_t_d th
important variables are summarized as follows: (Note Ay i = Azi)
Bgfk: b'gf k (129)
N.1
Dynqi = Qq AYi(130)
N°
1
Dynqei = Qii Ayi Nei = Dynqi Nei(131)
Dynqii, m : QqAy i 1- exp kTi/j = Dynq i 1- exp -k_-i/](132)
[_ 1 1 - exp Ne. = Dynqii, Ne.Dynqiei' m Qi Ayi -_i/J 1 m 1
(133)
Si,k, m = Q--_.Ay i - exp _i)]exp Enk b'gfk ,mk_i = Dynqii
= _ ( -Enk_s. = si • exp1,m ,k,m Dynqli,m
k k
exp _Enk _ .
(134)
(135)
39
_°
11,i2,k,m
iI ,i2 iI ,i2
= _ Si,k,m = Bgfk
i iDynqi i exp
,Ill
iI ,i2
Si],i2,m = _ Si = _ S.,//l
i k il'i2'k'm
?i,k = Nei'Y k
Si'k,------m _i_.L__kTi,k,m,j = _Tr
(vj - Vk)2 2+"Yi,k
(136)
(137)
(13s)
(139)
z,m,j _-i,k,m,jk
= .Si, k; m 1
1,k,m rr Yi,k
(14o)
(z4z)
_'i,m = _ Ti,k, m
k
i 1 , i 2
Ti1,i 2,k,m = _ Ti,k,rn
i
(142)
(143)
iI,i2
Til'i2'ra = _k "ril'i2'k'm = _'_ Ti'mi
SYi,k, m = Si,k,rn_/i,k
S_/i, rn = _ SVi, k
k,r/1
(144)
(145)
(146)
4O
SWil, i2 , k, m
iI ,i2
iSYi k m
9 (147)
SYi1 , i2 , m
Yei I , i2 ,k, m
11 , i 2 , k, m
Em'ri,k, m, j
Em'r.1,m,j
]h_.m T.
ll,12,m,j
Width.11,i2,k,m
= _ STil,i2,k,mk
SYiI,i 2 , k, m
S.
11,i2,k,m
SYil, i2,k,m
2_Ye 2ll,io,k,m
: exp (-_,k,m,j)
= exp (- Ti, )m,j
iI,i2
= H Erll T.
1,m,ji
= 2_rYeiI,i2 ,k,m f(}il,i2,k,m )
048)
(149)
(15o)
(151)
(152)
(153)
(154)
Widthil,i2, m : _ 27rnil,i2,k,mf(_i1,i2,k, m) : _ Widthil,kk 12,k, m
[For isolated lines; f(_) is given by Eq. (113).]
J- Emril,i2,m,j) 5v.J
(according to appropriate numerical integration scheme)
(155)
(156)
41
\
= F/-A_---- Widthil i2 m , LINE SPACING PATTERN)(157)1 Width_ 1 , i 2 m Vra ,/kv m , ,
(approximate, empirical-numerical correlation).
The numerical integration of the integral in Eq. (96) then corresponds to the
application of Eq. (156), with a suitable scheme of evaluating Emql ,i2,m,j accord-ing to Eq. (153).
In the numerical scheme of line calculations for each line group, the equivalent
width is first calculated assuming that the lines are isolated. If this value exceeds1 percent of the spectral width for the line group, Eq. (156) is then used to calculatethe equivalent width with line overlaps. The correlation discussed in connectionwith fig. 4 may be used for radiation-coupled flow field calculations. However, the
computer time for typical flow field cases is found to be of the order of only oneminute (UNIVAC 1108). Consequently, efforts in obtaining equivalent-width correla-tions appear unessential (though desirable) at present.
The numerical calculation of Eq. ,1_ [or " " .........._v! in_g_Liu_, of Eq. (96)] is based ona variable inte/'val Simpson's Rule, modified in sttoh a way as to yield results twoorders higher in accuracy than the basic Simpson's Rule. Depending upon the
pattern of line ispacings, the degree of overlap, and the accuracy criterion imposed,the number of evaluations of the integrand [of Eq. (96) or (1 - EmTil,i2,m,])] fora line group may vary over a wide range say from 20 to 800 for a single equivalent
i
width.
42
j
5. RESULTS
Results of numerical calculations are presented in this section. Almost all of
the results presented are obtained using STAGRADS. Only an example of the heat
flux distributions obtained using STRADS is given. Additional results from STRADS
may be found in refs. 11 and 12. A reference case (u_ = 15.24 km/sec, P_/PSL= 1.66 × 10 -4 , RN = 256 cm) is selected for uncertainty studies.
5.1 Effects of Exponential Kernal Approximation
In ref. 1, the exponential integrals used in the radiation transport calculations
were calculated using series representation and numerical correlations. InSection 3.2, the exponential kernal approximation, E3 ([) -_ 0.5 exp (- 2_), isintroduced. This approximation enables the formulation _f the concept of line
transmittance and equivalent width, as discussed in Section 3.4.2. The effectsof the exponential kernal approximation on the shock layer enthalpy distribution,the radiative flux to the wall and to the shock, and the air layer thickness are shownin fig. 5, for a representative environmental condition (the reference case). The
air continuum contributions alone are considered, using band Model No. 4 (5 bands)and 10 sublayers. The close agreement of the enthalpy, heat fluxes, and air layer
thickness can be seen. The enthalpies calculated with actual E 3 (_) are slightlyhigher than that with approximate E 3 (_). However, the optical thicknesses of thefour of the five bands with actual E 3 ([) are slightly lower than that with approxi-mate E 3 ( _ ) . The net results are slightly higher heat fluxes with the exponentialkernal approximation. The approximation E 3 (_) -_ 0.5 exp ( - 2_ ) is used in allresults presented below.
5.2 Effects of Continuum Band Models
The effects of air continuum band models on the radiative flux to the wall can be
seen in Table 1. Only air continuum contributions are considered. The resultsindicate that the two-band model does not account fully for the self-absorption effectin the vacuum ultraviolet. Band Model 4 (Planck-mean for 0.5 -< u -< 10.95 eV
plus 4 vacuum UV bands) provides results sufficiently close to that obtained withmore complicated band models and is adopted for "nominal calculations2' Figure 6shows further comparison of results obtained using Models 1 and 4, including linecontributions.
5.3 Effects of Line Model
Two options (LINEOP = 2 or 3) of line calculations are discussed in Sec-tions 3.4.2 and 4.1. When air layer calculations alone are considered, either ofthe two options mav be used. When the effects of ablation layer are examined, theoption LINEOP = 3 is used. Figure 7 shows the nondimensional enthalpy distri-
butions, heat fluxes, and air layer thicknesses for the two line calculation options.
43.
"The results are for the sameenvironmental condition (the reference case) as forfigs. 5 and 6. The two line options yield nearly the same results.
5.4 Effects of Ablation Layers
The presence of a relatively high density ablation vapor adjacent to the body sur-face further attenuates the radiative flux to the wall. In an inviscid analysis, thechemical reaction betweenthe air species and the wall is not considered. The wallis assumedto be at the equilibrium sublimation conditions corresponding to thesurface pressure. The surface material considered Ln_this study is 70/30 carbonphenolic (see Appendix A for its thermodynamic properties). A 21-band model ofcontinuum absorption coefficients, with the same spectral divisions as that for theair line groups, is used (see Appendix B. 2).
For the reference case, the air layer temperatures are increased slightly near
the interface when the effects of the ablation layer are included, as shown in fig. 8.The heat fluxito the shock is increased slightly (3 percent) whereas that to the wallis reduced g_eatly (41 percent). The results are for the nominal f numbers for the
molecular band systems given in Table.B-2, unless indicated otherwise. Figure9shows the temperature and tangential velocity distributions across the air and abla-tion layers, lit is seen that the ablation vapor is heated slowly at first near the walland very rapi_lly near the interface. The tangential velocity distribution is replotted
in terms of aitransformed "mass" distance in fig. 10. The ablation layer is seen tocarry about the same mass as the air layer, although it is physically thin compared
to the air layer.
The particle number densities in the air and ablation layers are given in fig. 11.*The continuum optical thicknesses are given in fig. 12. The ablation layer is opti-cally thicker than the air layer over all frequencies shown.
Figure 13 shows the spectral heat fluxes. Actually, the heat fluxes should be a
series of steps, being constant over a line group frequency region as shown for theoptical thickness in fig. 12. In order to show the effects of ablation layer in thesame figure, the spectral heat fluxes are "assigned" to a selected point of each linegroup and straight lines are drawn between adjacent points. The results indicate
that the ablation layer effectively blocks all radiation fluxes with frequencies greaterthan 10.95 eV, due to the photo absorption of C ground and excited states, C+ ex-
cited states, and H Lyman (see Table B-l!.
5.5 Effects of the Number of Sublayers
In order to test the accuracy of the finite-difference method of integrating theradiation transport and energy equations, the reference case is recalculated with
fewer sublayers, from 10 + 10 (10 subdivisions in both the air and ablation layers)to 6 + 6. The results are compared in fig. 14. It is seen that the effects on theheat fluxes and total layer thickness are small. The surface reradiation is of the
same order as the heat flux to the wall, causing a significant deviation in the non-dimensional surface blowing rate.
*Only species used in radiative property calculations are plotted (see Appendix B).
44
For the study of the effects of environmental variables, the combination of 6 airsublayers and 6 ablation sublayers will beconsidered adequate.
5.6 Effects of Environmental Variables
The effects of nose radius variation on the radiative fluxes, for constant velocityand altitude, are shownin fig. 15. Increasing the nose radius from I to 8.4 ft (30.48to 256 cm) only increases the heat flux to the wall by 68 percent without ablation layereffects and 24percent with ablation laver effects. Increasing the nose radius increasesthe heat loss to the ambient air more rapidly.
Further results for different enviror/mental variables are given in Table 2 and figs.16, 17, and 18. At the highest altitude (250kft) considered, the surface reradiationexceedsthe heat flux to the wall. The assumption that the surface is at the equilibruimsublimation temperature is not physically meaningful. However, the results may stillbe presented with the understanding that zero ablation is considered. The number of cal-culated points are insufficient to construct figs. 16, 17, and 18 accurately. Someof thepoints are obtained by "interpolation" of datapoints in other figures.
The results indicate that the heat flux to the wall is changedaccording to the fol-lowing:
(1) It increases as the velocity increases; the ablation layer tends to decreasethe velocity effect.
(2) It increases only slowly as the noseradius increases; the ablation layerreduces the nose-radius effect further.
(3) It decreases due to the presence of the ablation layer, more for loweraltitudes, larger nose radii, and higher velocity.
In general, it canbe stated that the ablation layer becomes more efficient in re-ducing the zero-ablation radiative flux to the wall as the latter increases. The analogybetween the ablation layer effectiveness in reducing radiative heat flux to the wall andthe convective blowing effectiveness is noted.
5.7 Effects of Perturbation of Radiative Properties
The effects of uncertainties of the radiative properties are illustrated in figs. 19 and20. In fig. 19, curve 3 corresponds to the reference case when nominal (N) values ofthe radiative properties are used. Curves 1, 2, 4, and 5 represent results withoutthe ablation layer. Curves i and 2 are results of calculation (LINEUP = 1) with aircontinuum contributions only, with nominal cross sections for curve 1 and 2 × thefirst-band cross sections (u < 10.95 eV) for curve 2. The effects of air lines may
then be seen by comparing curves 1 and 3 and the corresponding heat fluxeS. Doublingthe first-band cross sections alene is not sufficient to "match" the line effects.
For curve 4, the line half widths used are twice the nominal value, while for curve 5the line f-numbers are doubled. The effects are to increase both the heat fluxes to thewall and to the shock, but by less than 10 percent.
45
JIA
For curve 6, the line half-widths, line f-numbers, and the ablation species molec-ular band system f-numbers are twice the nominal values. Again the heat flux increaseis relatively small, being less than 17 percent.
The effects of increasing the ablation species molecular band system f-numbers
alone by a factor of two may be seen in fig. 20. The heat flux to the wall is onlyslightly reduced. Attempts were made to study the effects of increasing the f-numbersby a factor of the order of 10. However, because of the rapid temperature rise nearthe wall, convergence difficulties were encountered in iteration of the ablation layer
enthalpy distribution. Approximate energy conservation calculations for the enthalpyrise for a one-dimensional ablation vapor flow indicates, however, the enthalpy riseis nearly the maximum. Consequently, increasing the f-nmnbers further may notchange the heat flux to the wall appreciably, although the enthalpy distribution maychange significantly.
5.8 Effects of Precursor Heating
No detailed analysis of the precursor heating effects is performed in this study.However, using the results of spectral heat fluxes to the shock obtained from STAGRADSand the results of refs. 3, 4, and 13, a first order estimate of the preheating effect onthe radiative flux to the wall may be made. For instance, using fig. 11 of ref. 13 and
assuming all radiation leaving the shock with _ > 12.15 eV* (see Table 2) is absorbed
by the cold air near the shockwave, for u_ = 15.24 km/sec and po_/PSL = 1.66 x 10 -4,the increase in radiative flux to the wall may be estimated to be in the order of 5 per-cent for the nose radii considered in Table 2. For velocities higher than 60 kft/secthe preheating effects may become more significant, however.
5.9 Equivalent Width Results
Figure 21 illustrates the results of equivalent widths of the line groups, for isother-
mal air layers of different thickness. The temperature and density correspond to thatimmediately behind the normal shock for the reference case. As discussed in Section4.3, the equivalent width is first calculated assuming that the lines are isolated. " If
this value exceeds 1 percent of the spectral width of the line group, the equivalentwidth is recalculated by numerical calculation of Eq. (156) to account for line over-
laps. As the layer thickness increases, the equivalent widths of line groups withstrong line overlaps approach the group width asymptotically, for instance line groups19 and 15 in fig. 21. When the ratio of equivalent width to line-group width is much
less than unity, the slope of the line may indicate whether the lines are thin or withinthe strong-line, square-root regime (see Eq. (113)). For instance, the lines withingroup 1 are thin whereas that within group 13 are within the square-root regime, formost of the values of A in fig. 21.
*Cold N2 has a multitude of narrow molecular bands starting at 12.4 eV, but strongabsorption may not begin until the N 2 photo-ionization continua are reached at about
15.5 eV (ref. 14).
46
The results of fig. 21 indicate that line overlaps must be taken into account forblunt-body radiation transport calculations.
5.10 STRADSResults
Downstream of the stagnation region., the stream-tube formulation is used to calcu-late the flow field and radiative fluxes. In this study, the computer code STRADSde-veloped in ref. 1 is modified to incorporate the air continuum band models used inSTAGRADS. The effects of ablation layer are not included. The air line contributionsare "simulated" approximately by multiplying the first-band cross sections with anappropriate factor.
The configuration considered is a blunt vehicle flying at a 33 deg angle-of-attack,as shown in fig, 22. The formulation of STRADSis based on two-dimensional andaxisymmetric configurations. In order to simulate the actual attitude, the effectiveaxis of symmetry is displaced at a distance Ro from the stagnation point, as shownin fig. 22. !
The radiation heating distributions, normalized with respect to the stagnation radia-tive flux to the wall calculated by STAGRADS,are given in fig. 23. The dimensionalsymbols are defined in fig. 22. Varying the value of the first-band cross section mul-tipliers from 2 to 4 produces no significant changesin the normalized heating distri-butions. The radiation heating distribution is seen to exhibit a strong velocitydependency. Further results from STRADSmay be found in refs. 11 and 12.
6. CONCLUDINGREMARKS
An efficient numerical procedure for fully radiation-coupled blunt-body flows hasbeen formulated in this study. The effects of air continuum, atomic line, and ablationlayer radiations are taken into account for determining the stagnation point velocity andtemperature fields, the heat fluxes to the wall andto the shock, and the surface blowingrate. The efficiency of the numerical methods is due to the finite-difference integra-tion of the radiation-transport terms andthe energy equation, to the formulation of theconcept of line-group equivalent width andaverage transmittance, and to the procedureof calculating the equivalent width with line overlaps.
Using the computer code developed, the effects of environmental variables, of thenumerical methods used, and of the uncertainties in radiative properties have beendelineated. The results indicate the following conclusions:
(I) The exponential kernal approximation of the exponential integral yields results
for enthalpy distributions and heat fluxes very nearly the same as that with
actual exponential integrals.
(2) The two-band model of air continuum radiation underpredicts the self-absorptioneffects in the vacuum ultraviolet. Increasing the vacuum UV band numbers
from 1 to 4 (Model 4) provides results sufficiently close to that obtained with
more complicated band models.
47
(3) The two options of line calculations, LINEOP = 2 or 3, yield nearly thesame results.
(4) The ablation layer is very effective in reducing the heat flux to the wall and
thus the surface ablation rate. It practically blocks all radiative fluxes withfrequency greater than 10.95 eV. The ablation layer becomes more efficientin reducing the zero-ablation radiative flux to the wall as the latter increases.
(5) Reducing the number of sublayers from 10 + 10 to 6 + 6 still yields resultsadequate for the study of the effects of environmental variables.
(6) The self-absorption and energy loss effects decrease the sensitivity of theheat fluxes to the changes in environmental variables and to the uncertainties
in radiative properties. An order of magnitude change of the nose radius(from 1 to 10 ft) only increases the heat flux to the wall by 30 percent. Con-
sidering the change of convective heat rates with nose radius, blunt configu-rations may prove to be more effective than slender configurations in reducingthe total wall heat fluxes.
(7) For velocities less than 18 km/sec (_ 60 kft/sec), the effect of precursorheating on the radiative flux to the wall is small.
In future work, the following studies are recommended:
(1) The formulation of radiation transport may be extended to include the pre-cursor radiation and the spectral emissivity of the body surface.
(2) The contributions of additional radiative systems, such as the lines due to
nitrogen ions, and oxygen and carbon atoms and ions, may be inclu4ed.
(3) The formulation of the flow field may be extended to include the precurso_heating effect and the viscous effect (including conduction and diffusion).
(4) The radiation transport formulation developed may be used for calculationsaround the body.
(5) The accuracy of the one-dimensional radiation transport may be investigated,particularly when the temperature gradient in the direction parallel to thebody is large.
(6) The numerical procedures used in STAGRADS may be further improved toincrease the accuracy and to reduce the computation time.
48
APPENDIX A
THERMODYNAMIC PROPERTIESOF AIR AND ABLATION VAPOR
For a given composition of elementary species, two state variables are sufficientto specify the thermodynamic properties of a gaseous mixture in thermodynamic
equilibrium. The state properties may be put in a tabular form so that property cal-culations may be made by Lable look-up and interpolation. Alternately, curve fits of
one state variable versus a second at several levels of a third may be used. Thelatter method is particularly convenient when many calculations are made at a fixedlevel of a certain state variable. For instance, for abluntbody in hypersonic flow, theshock layer normal pressure gradient may be neglected. Curve fits or correlationsof temperature and other state propertiesversusenthalpy at several constant levelsof pressure will be very convenient for calculations along a body normal. Correla-
tionsof this kind for air may be found in refs. 15, 16, and 17.
For air calculations in the present study, correlations of refs. 15 to 17 may be
.used. However, for the purpose of reducing the computer time required, new cor-relations with a more suitable set of units are generated. The division of enthalpyranges is also selected in such a manner that-interpolations of the coefficients in the
correlations between adjacent pressure levels may be more conveniently made.
The units selected for correlation purposes are:
P , pressure in atm
T , temperature ineV, leV =11605.7 °K
H , enthalpy ratio, h/hsatellite , hsatellite = 12,484 Btu/Ib m
PV/VsL , in atm, VSL = sea-level (i atm, 288.16 °K) air specific volume
The correlation formula coefficients for air are given in Table A-1.
With the values of the compressibility factor available, the concentrations of airspecies may be readily calculated ff Hansen's (ref. 18) simplified model of air chem-istry is used. However, in this study, the particle densities (their logarithm) ofneutral nitrogen and oxygen atoms and elections are calculated by double interpola-tion with respect to temperature and the logarithm of density using the results of
Gilmore (ref. 19) and Moeckel and Weston (ref. 20).
The surface material considered in ablation layer calculations is 70/30 carbon-phenolic. The 70/30 carbon-phenolic is assumed to be at the equilibrium sublimationconditions corresponding to the surface pressure. The sublimation latent heat is
4,270; 4,220; 3,900 cal/grn at 0.1, 1, and 10 aim, respectively. If the material sen-sible heat is included in the energy balance, the effective heat of sublimation will behigher.
49
The thermodynamic properties (temperature, enthalpy, density, species particle
concentrations, and pressures) of 70/30 carbon-phenolic vapor in thermodynamicequilibrium were calculated by a free-energy minimization program, and are put ina table form for double interpolation with respect to pressure and enthalpy. Betweentwo consecutive table values, the temperature and the logarithm of specific volume
and of species concentrations are assumed to be linear in enthalpy and in the logar-
ithm of pressure.
50
APPENDIX B
RADIATIVE PROPERTIESOF AIR AND ABLATION SPECIES
The absorption coefficient of a given species is the product of its particle numberdensity and absorption cross sections. For a given mixture of species, the absorp-tion coefficient is the sum of theseproducts. The cross sections of air species andablation vapor species are discussedbelow.
B. 1 Absorption Cross Sections of Air Species
B. 1.1 Continuum absorption cross sections. - The effective cross sections ofneutral nitrogen (NI) and neutral oxygen (OI) given in Tables 4-3 and 4-5 of ref. 7
are used. These cross sections include the bound-free and free-free photon absorp-tion contributions and are per particle of NI or OI. For the conditions of interest,
the relative contribution due to NII is small. The incorporation of absorption contri-butions due to NII and N- continuum has not been made.
For the eight continuum band models described in Section 4. i, partial Planck mean[see Eq. (75)] cross sections are used for bands with frequency less than i0.95 eV.The NI cross sections increase in steps for frequency above i0.95 eV and average
values are used for each of the steps. The OI cross sections do not change apprecia-bly between 4.0 and 13.61 eV but increase in steps above 13.61 eV. For Of, partialPlanck means are used up to 13.61 eV and average values are used for each of thesteps above 13.61 eV. The selection of the band boundaries is based on the locations
of the step changes in cross sections. The band boundaries for the eight models aregiven in Table I. For frequencies above I0.95 eV, the cross sections for the highest-frequency band correspond to that of the lowest-frequency step within the band. For
example, in Model I, the average cross sections between i0.95 and 12.15 eV areused for the second band with boundaries at I0.95 and 30.0 eV.
For calculations with lines and ablation layer, the frequency regions are relatively
narrow. The value of the cross section at a specified point within the frequency re-gion is used. The logarithms of the cross sections are assumed to be linear in fre-quency and temperature between tabulated values.
B. 1.2 Line absorption cross sections. - Only the contribution of NI lines areincluded in the present STAGRADS code. The effective NI particle density for linecalculations is taken as the sum of the actual NI and OI particle densities. A total
of 75 NI lines are considered. Some of the lines are "effective" lines formed bytreating two or more closely situated lines as single lines. The f numbers and thehalf-width (at T = 10,000 ° K) of the NI lines given in Tables 4-9 and 4-15, respec-
tively, of ref. 7 are used. The line shifts are neglected and the half-widths are con-sidered independent' of temperature [see Eq. (99)]. See Appendix C. 3 for details ofline data used.
5l
B. 2 Absorption Cross Sections of Ablation Species
Only the continuum contributions of ablation species are included in the present
STAGRADS code. The radiative systems considered include the Swan, Fox-Herzberg,
Deslandres-D'Azambuja, Freymark, and Mulliken band systems of C 2 , the CO4+system, and the continuum processes due to the C + free-free contribution and the
photo-absorption of CO, H Balmer, H.Lyman, C ground and excited states, and C +excited states.
As discussed in Section 4.1, the division of the continuum bands for the ablation
species is made coincident with that of the line _,'_,,,_¢
The spectral ranges of the 21 bands and the average cross sections of the absorp-
tion processes are given in Table B-1. The cross section cro of the molecular bandsystems were obtained by wavelength averaging the results obtained by John Weisner(ref. 21) and by normalizing to unity f-number. The temperature dependence isdetermined from the cross-section values at two temperatures (3,000 ° K and 9,000 ° K
or 10,000 ° K) and is very approximate. The cross-sections of the continuum proces-ses are given by the same simple correlation, with f taken as a unity. No perturba-tion of the continuum cross-sections is considered in this study. The f-numbers ofthe band systems are given in Table B-2.
The C 2 Swan systems are the only band contributors considered within the first
five bands. Preliminary results of calculation indicated that the optical thickness forthese bands ( _ 0.62t_ ) was too low. As the absorption in the IR is generally strong due
to the presence of the polyatomic molecules adjacent to the wall and due to systems notincluded (e.g., C 2 Phillips and Ballik-Ramsey Band systems and H Paschen andBrackett continua), the C 2 Swan contribution within Band 1 is raised by a factor of 10 to
compensate for the neglected absorbing systems.
52
APPENDIX C
THE STAGRADS CODE
The listing of the STAGRADS code, for calculation in UNIVAC 1108 at LMSC, isgiven after the figures near the end of the report. The input data required for the
reference case (u_o = 15.24 km/sec , Poo/PSL = O. 166 x 10 -3 , R N = 256 cm)are then listed. Brief explanations are given for the code and the input. The programis very complex. The purpose of presenting the listing is for the benefit of those who,l_ve _l_v desire to modify the program and to make use of some of the subroutines forother related problems.
Representative output data, including details of the line data used, are also givenat the end of the report with brief explanations ....
}C. 1 STAGRADS Listing (See page 99. )
The listing of STAGRADS given is in FORTRAN IV. STAGRADS consists of a mainprogram and a number of subroutines and fun.ctions. In the order of appearance in the
listing, they will be explained briefly below.
C. 1.1 Main Program. The main program serves the following functions:
(1) Read all input data.
(2) Write a majority of the output results.
(3) Integrate the energy equation and iterate the enthalpy distribution for theair layer.
(4) Call the appropriate subprograms (i. e., Subroutine VEL for integration ofthe momentum equation to obtain the velocity distribution).
The built-in data TC (I, J) are used only in Subroutine ABSORP. They are repeatedhere for the purpose of checking (see listing near statement no. 72).
C. 1.2 Subroutine STAGAB, STAGAB (STAGrads ABlation) is for calculation ofstagnation ablation layer. It serves the following functions:
(1) Integrate the energy equation and iterate the enthalpy distribution, wall heat
flux and surface blowing rate for the ablation layer.
(2) Write output of ablation layer results.
(3) Call the appropriate subprograms (i. e., Subroutine VEL for calculation ofthe velocity distribution).
C. 1.3 Subroutine NGGBSP. NGGBSP (Non-Grey Gas) is used for radiation trans-port calculations involving finite bands of the air continuum contributions. As listed,
the exponential kernal approximation is used for E 3 (_). However, the actual
E 3 (_) may be used if only the air continua are considered.
53
C. 1.4 Subroutine CRLINE. CRLINE (CoRrection for LINE) is used for radiationtransport calculations when the contributions of the air lines and ablation speciescontinua are included. It is used for either of the two line-calculation options dis-cussed in Section 4.1.
C. 1.5 Subroutine ABSORC. ABSORC (ABSORption for Continua) calculates theair continuum absorption coefficients (NI and OI) for a specified frequency within theindividual line groups.
C. 1.6 Subroutine lINE. L_rNE calculates the equivalent widths or transmittance
for the line groups. It writes out the calculated results if requested by appropriatecontrols.
C. 1.7 Function FLR. FLR calculates the Ladenburg-Reiche function accordingto the approximations given by Eq. (113).
C. 1.8 Subroutine INTEG. INTEG (INTEGration) performs integration to obtain theequivalent width. It is based on a modified Simpson's rule. It writes out the smallest
integration step and other information if requested by the control.
C. 1.9 Subroutine INTRND. INTRND (INTegRaND) calculates the integrands re-quired for equivalent width calculations.
C. 1.10 Subroutine DOUBL. DOUBL (DOUBLe interpolation) is used to calculateparticle concentrations of NI, OI and e by double interpolation with respect to tem-perature and the logarithm of specific volume.
C. 1.11 Subroutine PROPT. PROPT (PROPerty) calculates air temperature, PVproduct, and compressibility factor for given pressure and enthalpy, using the corre-lation formulas given in Table A-1.
C. 1.12 Subroutine ABSORP. ABSORP (ABSORPtion) calculates the air continuumabsorption coefficients (NI and OI) for the individual bands for the eight band models.
C. I. 13 Subroutine SPABCO. SPABCO (SPectral ABsorption COefficient) calcu-
lates the ablation species continuum absorption coefficients for the 21 spectralregions.
C. 1.14 Subroutine PROT. PROT (PROperTy) calculates for the ablation vapor the
specific volume, temperature, and particle densities of C 2 , C, CO, H, and C+ bydouble interpolation with respect to enthalpy and the logarithm of pressure. Theproperty tables are input in the main program.
C. 1.15 Subroutine SESB. SESB (State Equation for Surface Blowing) calculatesthe surface enthalpy for 70/30 carbon phenolic in sublimation equilibrium at the sur-
face pressure. The surface specific volume and temperature calculated in SESB are
actually not used in STAGAB; they are recalculated by PROT for consistency with thevapor properties.
54
C. 1.16 Subroutine PLANCK. PLANCK calculates the fraction of black-bodypowers within two frequency limits and for a given temperature.
C. 1.17 Function EXPF. EXPF limits the absolute value of the argument for EXPto 85 to avoid numerical overflow.
C. 1.18 Subroutine VEL. VEL (VELocity) integrates the momentum equation forthe air layer and ablation layer.
C. 1.19 Subroutine TBLP4. TBLP4 is used for interpolation of variables such as
specific volume ratio, transformed distances and velocities in VEL.
C.1.20 Subroutine MODLI_ MODL2, ... MODL8. These subroutines specify thefrequency limits for the 8 air continuum band models. They also select the propercross section indices for calculations in ABSORP.
C. 1.21 Function E2F. (listed after input data) E2F calculates the second expo-nential integral by series and numerical correlations. It may be used in conjunction
with E3F in NGGBSP and CRLINE, when the actual exponential integral instead of theexponential kernal approximation is employed.
C. 2 Input Data Listing (See page 155. )
The input data for an example case (the reference case)are listed immediately
following the STAGRADS listing. All input data are read in within the main program.
The input line and line group data are first read. The 10 read statements after the
comment "Input Line and Line-Group Data" read in the values of the followingvariables:
NHV
NLST
GEE(I)
EPS(I)
FHVM(1)
FHVP
FHV(1)
ISOE(I)
Total number of line groups
Total number of lower states
Statistical weight of I-th lower state
Electronic energy of I-th lower state, eV
Lower frequency limit of I-th line group, eV
NUMINT(I)
NU(O
Upper frequency limit of I-th line group, eV
A selected (e. g., central) frequency within I-th line group, eV
> 0 if the approximation of effective isothermal region may be used forthe I-th line group; otherwise <_ 0.
> 0 if numerical integration is to be used (when required) to calculatethe equivalent width of the I-th line group; otherwise < 0.
Number of lines in I-th line group
55
NLINE
ND )
HVL(I)
FF(1)
GAMBA(I)
Total number of lines, calculated by summing all NU
Lower state index for I-th line
Center frequency of I-th line, eV
Effective f-number of I-th line (f-number of the particular transition
multiplied by a statistical-weight ratio, see Section 3.4.1)
Half-width (eV) of I-th line for 1 electron/em 3.
Next, some of the 4 input tables are read, between Statements 8 and 99 in the list-ing. Each of the 4 tables is headed by a leader card.
Table 01
L
CASE
PA
RHOA
UA
RN
NS
ES
HA
TW
MODEL
LINEOP
FCONTN
FDHAB
NRITE1
NRITE2
NRITE3
NRITE4
(Leader Card with value of NC = 0 1; near statement 10)
0 for two-dimensional and 1 for axisymmetric
Case identificationi
Ambient pressure, atm
P_/PsL
u , km/sec
_a N , em
•_ 10 number of air sublayers
/Ps
h/hsa t
T w , eV (ff < 0.02, reset to 0.02; recalculated as equilibrium surfacetemperature if ablation layer effects are considered)
Air continuum band model number, = 1, 2, ... , 8
{ •1 , only air continua with one of the eight band models2 , calculate air continua with one of the eight band models and then
correct for air lines
3 , 21 spectral regions with air continua and lines, with or withoutablation continua
Multiplying factor for air continuum cross sections (v < 10.95 eV ) forthe 8 band models
Multiplying factor for surface material latent heat of vaporization
> 0 print nondimensional entha]py distributions during iteration; < 0
no print
> 0 print equivalent-width information, final result only; _ 0 no print
> 0 print equivalent-width information during iteration; < 0 no print
> 0 print equivalent-width integration information during iteration; = 0
final result only; < 0 no print
56
NRITE5
NRITE6
NRITE7
CMAG
ERR
> 0 print line strength of sublayer, etc. during iteration; = 0 final re-sult only; < 0 no print
> 0 print line strength of air-layer regions, etc. during iteration; = 0final result only; < 0 no print
> 0 print nondimensional enthalpy distributions, QWT, etc,, when either
(but not both) the air or ablation layer iteration converges; -< 0 noprint
0. 0001 times absolute value of the average value of integrand
Integration accuracy control constant
With err = 0.1 and 0.01, the result is said to be accurate to 4-6 places and 5-7
places, respectively. Decreasing the value of err will increase the computing timerequired.
From the results of equivalent-width calculations for a variety of conditions, avalue of err = 0.1 is found adequate for most situations. Under certain situationswhen the lines are very narrow and with a small degree of overlapping, it is foundthat a value of err in the order of 0.01 is required for line groups 4 and 5 in order toobtain accurate results in equivalent widths by numerical integration. However, avalue of err in the order of 0.1 is adequate for flow field and total heat flux
calculations, ii
FRAC First integration step size divided by total integration interval
NSAB <_ 10, number of ablation sublayers
Table 01 must be input for each run of flow field calculations. For flow field cal-culations without the ablation layer, Table 01 ends with a card with -1 (value of NC)in the first two columns. For flow fields calculations with ablation layer, Tables 03
and 04 are also required; the "-1" card is then put at the end of Table 04.
For flow field calculations, each case must end with a "-1" card. For subsequent
cases of the same computer run, Tables 03 and 04 may be omitted if they are identi-cal with the corresponding ones in the preceding case.
Table 02 (Leader card with value of NC = 02; near Statement 80)
Table 02 is input for equivalent width calculations with given layer thickness, num-
ber of sublayers, and distributions of temperature and specific volume. No "-1" cardis required to end each case.
N
DE LTA
T(1)
v(1)
Total number of sublayers
Total thickness of layer, cm
Temperature of I-th sublayer, eV
PsL/P of I-th sublayer
• 57
CMAG, ERR, FRACNRITE1 --*NRITE6 ! see Table 01
Table 03 (Leader card with value of NC = 03; near Statement 7010)
IP
IT
Table 03 inputs the thermodynamic properties of the ablation layer vapor.
PRTAB(I)
TTAB(I, J)
ENTAB(I , J)
RHOTAB(I , J)
C2TAB(I , J)
C1TAB(I , J)
COTAB(I , J)
HTAB (I , J)
CPTAB(I , J)
Total number of pressure values in table (3 allowed in presentprogram)
Total number of temperature values in table (20 allowed in presentprogram)
The i-th pressure value, in lOgl0 (p , atm)
J-th temperature value for I-pressure value, °K
Enthalpy for I-th pressure and J-th temperature, cal/gm
Density for I-th pressure and J-th temperature, gm/cm 3
Particles/cm 3 for I-th pressure and J-th temperature, for C 2
For C 1
For CO
For H
For C+
EMTAB(I , J) For e
Table 04 (Leader card with value of NC = 04; near Statement 7020)
Table 04 inputs the f-numbers and continuum cross section multipliers for ablationlayer radiating systems.
ICOSR
COSR(I)
Total number of input COSR values
f-numbers for band systems. For continua, COSR = 1.0 if the standard
values of the cross sections are used. The order of COSR(I) is given inTable C-1.
HV
HV+
HV-
N
C: 3 Output Data Listing, Including Details of Line Data Used (See page 163.)
The data for the 21 line groups are first listed. The headings of the data are:
Frequency within a line group for calculating Planck intensity functionand continuum cross section, eV
Upper frequency limit of line group, eV
Lower frequency limit of line group, eV
Number of lines within line group
58
I
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
TAB LE C- 1
COSR(I) VALUES
Absorption Processes
C 2 Swan
C 2 F-H
C2 D-D
C 2 Frey
C 2 Mull
Unas signed
Unassigned
Unassigned
CO4 +
CO Photo
Unassigned
Unassigned
Unassigned
H Balmer
H Lyman
Unassignecl
Unassigned
C Photo
Unassigned
Unassigned
C+ Photo
Unassigned
C + + e____C + hv , (C + free-free)
Remarks
Use fvalue
Use f value
Use f value
Use f value
Use f value
Use 0 value
Use 0 value
Use 0 value
Use f value
Use 1 for standard value
Use 0 value
Use 0 value
Use 0 value
Use 1 for standard value
Use 1 for standard value
Use 0 value
Use 0 value
Use 1 for standard value
Use 0 value
Use 0 value
Use 1 for standard value
Use 0 value
Use 1 for standard value
LINE
NLST
HV(1)
FG)
GAM(1)
Line index
Lower state index
Line center frequency, eV
Effective f-number of line, same as FF(I) in Section C.2
Line half-width for 1 electron/cm 3 , eV/(e/cm 3 )
59
BFG(1)
ISOE
NUMINT
GEE(1)
EPS(I)
b'gf , Eq.(129) eV-cm2/particle
> 0, if effective isothermal region may be assumed; < 0 otherwise
> 0, if equivalent width is calculated by integration when its value
according to isolated lines exceeds 1 percent of line-group width; < 0if calculated according to isolated lines only
Statistical weight of I-th lower state
Electron energy of I-th lower state, eV
The next group of output data is the air continuum cross sections TC(I , J) usedin ABSORP. In ABSORP, TC(1 --* 14, 1) are temperatures in eV and TC(1 --* 14,2 -* 27) are values of 38 + logl0 (cross-section, cm2). These values have been
converted to °K and cm2, respectively, in the main program before being printed
out. Each index J (2 --* 27 ) is for a particular frequency interval (either a partialPlanck mean or a representative average).
J: i 2 3 4 5 6 7 8 9
Species: NI NI NI NI NI NI NI NI
vI , eV: 0.5 2.0 4.0 6.0 9.50 10.95 12.15 13.61
uu , eV: 2.0 4.0 6.0 9.5 10.95 12.15 13.61 14.50
10 _1 12 13 14 15 16 17 18
NI Ol Ol Ol Ol NI OI NI Ol
14.50 0.5 2.0 4.0 13.61 0.50 0.50 2.00 2.00
10.9530.00 2.0 4.0 (13.61) 30.00 10.95 10.95 10.95 10.95
19 20 21 22 23 24 25 26 27
NI NI NI NI NI OI OI OI OI
0.5 1.0 2.0 3.0 4.00 0.5 1.0 2.0 3.0
1.0 2.0 3.0 4.0 10.95 1.0 2.0 3.0 4.0
The next group of output consists of data of particle per air atom for equilibrium
air. Following a column of 14 temperatures (3,000 --* 24,000 °K), the particles per
air atom for NI, OI, and e (3 column as a group for one density) for lOgl0 (PsI_/P) =1.97634, 2.97634, and 3.97634 are given. The values of SPTABO (I, J, K) with the
main program are in - lOgl0 (particles/air atom). These values have been convertedto particles/air atom before being printed out.
The next group of output corresponds to Table 01 of input. The units of the vari-ables and the definitions of the symbols are given there.
6O
If Table 03 is input, the data will beprinted out. The data printed for the refer-ence case are for three pressures (lOgl0 p, atm = -1.0, 0.0, 1.0). Columns 1 to
9 are respectively temperature (°K), enthalpy (cal/gm), density (gm/em3), and
particles/cm3 for C2, C1, CO, H, C+, and e.
If Table 04 is input, the values of COSR(I) are printed out. See Table C:l foridentification of the index.
The next group of output repeats the information for the line groups including the
selected frequency and the lower and upper line indices.
The symbols appearing in the output are defined below:
NCOUNT
DELTA
PO
HO
TO
QWT
QSHOKT
I
YC
UC
VC
HC
T
V
Z
HV
HV+
HV-
NLINE
TAUC
TAUCT
LINEOP
EMD
Number of iterations for enthalpy distribution, one counter
for the air layer and another counter for the ablation layer
Layer thickness for air or ablation layer, cm
Air and ablation layer pressure, arm
hs/hsa t
T , eVS
Total heat flux to wall, watts/cm 2
Total heat flux to shock, watts/cm 2
Sublayer index from shock
z/A for air layer; y/z_b for ablation layer (measuredfrom center of sublayers)
Normalized tangential velocity
Normalized normal velocity
h/h s for air layer; h/h w for ablation layer
Temperature, eV
PSL/p
Compressibility
Selected (e.g., central) frequency of line group, eV
Upper limit frequency of line group, eV
Lower limit frequency of line group, eV
Number of lines in line group, eV
Continuum optical thickness for air layer
Continuum optical thickness from shock to wall
Line-calculation Option (See Section 4.1)
PwVw / p_ u
61
NQWT
NCONH
QWTAIQWTA2 !
QWTC1QWTC2 !
QWT1
HW
TW
VW
VCW
UR
RMOM
URMAS
IAB
NCI, NC2, NCONH, NC +
M
IPI, I
ISO. WIDTH
ACT. WIDTH
DNU* TRN
ISO-ACT WIDTH
WIDTH (IPl, I)
DELTAB
DELTAT
US
ZETA
NCNTRI
Number of iterations for heat flux to the wall for each
ablation-layer calculation
Counter for enthalpy distribution iteration for each as-sumed wall heat flux
Previously' assumed wall heat flux, watts/cm 2
Previously Calculated wall heat flux, watts/cm 2
New assumed wall heat flux, watts/era 2
hw/hsa t
Tw , eV
PsL/Pw
Vw/U _
Ur/U calculated based on interface streamline momen--
turn consideration
2 2PwV/p u
Ur/U calculated based on mass balance for a finitenum-ber o_ sublayers
Ablation sublayer index from wall
Particle/cm 3 for C 1 , C2 , CO, H, and C + , respectively
Line group index
Upper and lower index for region with indices IP1 and I
Equivalent width assuming isolated lines, eV
Equivalent width with overlap, eV
_v TRANS, eV
ISO - WIDTH-ACT • WIDTH, eV
Equivalent width _ eV
Aab , ablation layer thickness, cm
, thickness between shock and wall, em
US
Z
_p_dzPs A'
o
Counter for number of air layer enthalpy iterationconvergence
62
NCNTR2
NCNTRS
FNUL(KK)
FNUH(KK) (FNI/U)
FNU
AB
QW(KK)
QWTC
QDEL(I_)
QDELT
SQW(KK)
SQDEL(KK)
TAU(I, KK)
K,KM
JD
I
IPl , J
C-AM(I)
ss(I)
SST(I + 1)
SGAM(I, KM)
SGAMT ( I + 1)
TAUD(I)
TAUDT(I + i)
SS12
Counter for number of ablation layer calculationconvergence
Counter for number of either (but not both) air layer orablation layer convergence
Lower frequency limit of KK-th air continuum band, eV
Upper frequency limit of KK-th air continuum band, eV
Central frequency of air continuum band, eV
1.0, signifies with self-absorption
Continuum heat flux to wall for KK-th band, watts/cm 2
Total continuum heat flux to wall, watts/cm 2
Continuum heat flux to shock for KK-th band, watts/cm 2
Total continuum heat flux to shock, watts/cm 2
Spectral continuum flux to wall for KK-th band,watts/cm2 eV
Spectral continuum flux to shock for KK-th band,watts/cm2 eV
Continuum optical thickness for KK-th band from I-thboundary to shock (I = 1)
Line index and line index within line group, respectively
Lower-state index
Sublayers index from shock
Boundary upper and lower indices
Line half-width, eV
Line strength, Eq. (110), eV
I
SS(J), eVJ=l
SS(I) * GAM(I) , (eV) 2
I
__ SGAM (J KM ) (eV)21
Line-center optical thickness, SS(I )/7 GAM(I )
I
TAUD(J )J=l
Line strength between boundaries IPl and J, eV
63
SGAM12
GAME12
ZETA12
WID12K
WID12(IP1 , J)
TAU12
NCOUNT, NREJE
DNU
DXX(I)
Normalized Smallest Step
Value of SGAM between boundaries IP1 and J, (eV) 2
Effective half-width, SGAM12/SS12, eV
Argument of the Ladenburg-Reiche function, Eqs. (106) and
(107)
K-th line contribution to isolated line equivalent width forregion between boundaries IP1 and J, eV
Equivalent width according to isolated lines for region be-tween boundaries iPi and J, eV
Line-center optical thickness for region between boundariesIPI and J
Counter for advancing to next integration interval or for
rejecting existing integration step size, respectively. In
Subroutine EqTEG. Approximate number of evaluations of
integrand for each region between two boundaries =
(NCOUNT + NREJE) × 4
Line group spectral width, eV
Some integration step sizes, eV
= DXX/DNU
64
REFERENCES
lo
.
.
4.
,
.
.
8
.
Chin, Jin H. ; Inviscid Radiating Flow Around Bodies, Including the Effect of
Energy Loss and Non-Grey Self-Absorption, Lockheed Missiles & Space Company,Report LMSC-668005, October 1965.
Chin, Jin H. ; Effects of Non-Grey Self-Absorption and Energy Loss for Blunt BodyFlows, Presented at the Symposium on Interdisciplinary Aspects of RadiativeEnergy Transfer, February 24-26, 1966, Philadelphia, Pennsylvania, (to bepublished in J. Quant. Spectrosc. Radiat. Transfer, January, 1968).
Wilson, K.H. ; Private Communication, June, 1967.
Coleman, W.D., Lefferdo, J.M., Hearne, L.F., and Gallagher, L.W.; A Studyof the Effects of Uncertainties on the Performance of Thermal Protection Systemsfor Vehicles Entering the Earth's Atmosphere at Hyperbolic Speeds, Lockheed
Missiles & Space Company, Sunnyvale, California, (Monthly Progress Report forthe Period 1 June 1967-30 June 1967, Contract NAS 2-4219).
Sparrow, E.M. and Cess, R. D. ; Radiation Heat Transfer, Brooks/Cole Publish-
ing Company, Belmont, California, 1966, Chapter 7•
Gilmore, F.R.; Energy Levels, Partition Functions and Fractional Electronic
Populations for Nitrogen and Oxygen Atoms and Ions to 25,000°K, RANDRM-3748-PR, August 1963.
Wilson, K.H., and Nicolet, W. E. ; Spectral Absorption Coefficients of Carbon,Nitrogen, and Oxygen Atoms, LMSC 4-17-66-5, November, 1965.
Simmons', F.S. ; Radiances and Equivalent Widths of Lorentz Lines for Noniso-
thermal Paths, J. Quant. Spectrosc. Radiat. Transfer vol. 7, 111-121, 1967:
Penner, S.S. ; Quantitative Molecular Spectroscopy and Gas Emissivities,Addison-Wesley, 1959, pp. 42 -45.
10. Goody, R.M.; Atmospheric Radiation Vol. I Theoretical Basis, Oxford ClarendonPress 1964, pp. 237-239.
11. Hearne, L.F_ '• ,_Coleman, W.D., Lefferdo, J.M., Gallagher, L.W., and Chin, _J.H. i A Study of the Effects of Environmental and Ablator Performance Uncertain-
ties on Heat Shielding Requirements for Hyperbolic Entry Vehicles, NASA CR-Lockheed Missiles & Space Company, Sunnyvale, Calif., Dec 1967.
12. Hearne, _L.F. ,Coleman, W.D., Lefferdo, J.M., Gallagher, L.W., and Chin,J. H. ; A Study of the Effects of Enviromnental and Ablator Performance Uncertain-
ties on Heat Shielding Requirements for Hyperbolic Entry Vehicles, NASA CR-Lockheed Missiles & Space Company, Sunnyvale, Calif., Dec 1967 - Data Volume.
65
4
13.
14.
15.
16.
Yoshikawa, Kenneth K. ; Analysis of Radiative Heat Transfer for Large Objects atMeteoric Speeds, NASA TN D-4051, June, 1967.
Huffman, R.E., Tanaka, R.E. and Larrabee, J.C. ; Absorption Coefficients of
Nitrogen in the 1000-580A Wavelength Region, J. Chem. Phys. vol. 39, 4, 1963.
Viegas, J.R. and Howe, J.T.; Thermodynamic and Transport Property Correla-tion Formulas for Equilibrium Air From 1,000°K to 15,000°K, NASA TN D-1429,October, 1962.
Lockheed Missiles & Space Company, Final Report, Study of Heat Shielding Re-quirements for Manned Mars Landing and Return Missions, LMSC Report4-74-64-1, December 1964, Contract NAS 2-1798, p. 3-30.
17. Howe, J.T. and Sheaffer, Y.S. ; Effects of Uncertainties in the Thermal Conduc-
tivity of Air on Convective Heat Transfer for Stagnation Temperature up to30,000°Ki NASA TN D-2678, February, 1965.
18. Hansen, C. F. ; Approximations for the Thermodynamic and Transport Propertiesof High Temperature Air, NASA TR R-50, 1959.
19. Gilmore, _. R. ; Equilibrium Composition and Thermodynamic Properties of Airto 24,000_K, Rand Corporation Memorandum RM-1543, August, 1955.
T
20. Moeekel, W.E. and Weston, Kenneth C. ; Composition and Thermodynamic Prop-erties of Air in Chemical Equilibrium, NACA TN 4265, April, 1958.
Weisner, John D.; private Communication, April, 1966.
66
LIST OF TABLES
Table
i
2
A-1
B-1
Effect of Continuum Band Model on StagnationRadiative Flux
Effect of Environmental Variables
Correlation Formula Coefficients, Air
Spectral Range and Average Cross Sections for 70/30Carbon-Phenolic "21 Band" Model
Ablation Species Band System f-Numbers
COSR(1) Values
Page
68
69
70
72
73
59
67
_qm.<
X
0
Z .o ,r.D -_
°_ _ ,_0 I
0 o
._ x
l_ II
_o _
I11
°_.,i
O o
_ II
0
I O
I
i
I -To
I _i il i
o_O _
_m
o_
O
I OO
I OO
IOO
I OO
i
'r
I
' _-.I Ir " 1_ '
Ij_.£ £
I rm
'--]_
Ii
! i
i
I
,! :!
0,1 ¢¢
J
_'L .'L__' iI 0,1
!
'-I-, _
I
iI r
I n
i
m
O°r4
"O
o.
2_
0"0
"g
m
68
.<[-.,
"O
u
u'J
_ .-. ^
_ Cr _
_ v
_ ^
_ U • . o.
c_
<I
,.0
b
"_ c_
o)
C:L
g
ZoO
U C)
"0Q_
c_
0
:<Q_
.,.4
6_
I
<
<
Z
0
00
<;..-1
Z0N<
N
0
_A
...... _ ..... _ ._ .. ..... . ....... Z_ . _ .....
I|l
III I_
,,_ _
_'_d _ ..........
¢
+
........ _ .........._ _o
o
_d_ dddd d_ "_ Mdd_ _M_ M
IIII I I I II I
t_
..... , . , . , .
II IIII I I 141
III I
,,,, _ ,,, 1'
_._ _ .1_
_ ........... _ .......
°°_ _ __ •
;..1
7O
"13
::1
O
I=1
E_o
N
,,o
o
i o
????
**o.
_ ,-_ ..,.,.
o
o
qo
k
o
§
_q
IH.Q
m
N
+ =
m _1o e_ iN
+
N m
+ _
. no ,, ,_. _
Vl
1.,
VI
I,i
_v
7!
Band
No.
6
7
9
10
11
12
13
14
TABLE B-1
SPECTRAL RANGES AND AVERAGE CROSS SECTIONS FOR 70/30
_ARBON-PHENOLIC "21 BAND" MODEL
Frequency (eV)
Lower Upper
0.50 0.800.80 0. 965
0.965 1.20
1.20 1.3951. 395 1.60
I.60 2.35
2.35 3.25
}
3.2:5 4.10
i,
i
4.10 6.20
6.20 8.00
8.00 8.60
8.60 9.00
9.00 9.70
9.70 10.95
15 10.95 12.1516 12.15 12.7017 12.70 13.35
i8 13.35 13;85
19 i3.85 14.50
20 14.50 16.50
21 16.50 20.00
System
C2 SwanC+f-f
1.86(-161 1.2
C 2 Swan 1.14(-16) 0C 2 F-H 3.94(-18) 0.29C 2 D-D 3.26(-18) 0.65
C2 Swan 3.19(-18) 0.27
C2 F-H 2.44(-17) 0C 2 D-D 1.31(-17) 0.79C2 Frey 2.41(-16) 1.52H Balmer i 3.38(-17) 10.2
Band No.
8 plusC 2 D-D 4.5(-19) 1.17C 2 Frey 3.98(-16) 0.72
C 2 Mull I.5 (-17) 0
CO 4+ 3.52(-17) I.6
H Balmer 1.19(-17) 10.2
CO 4+ 6.23(-17) 0
H Balmer 5.55(-18) 10.2
C Photo 2.0(-19) 7.46
CO 4+ 6.75(-17) 0
H Balmer 3.09(-18) I0.2
C Photo I.3(-18) 2,67
CO 4+ 6.35(-18) 0.46
H Balmer 2.1(-18) 10.2
C Photo 2.05(-17) I.26
H Balmer 1.32(-18) 10.2C Photo 2.05(-17) 1.26
-- 1.8(-17) 0
CO Photo 1.{)(-17) 0H Lyman 4.85(-18) 0C Photo 2.05(-17) 1.26
- 1.8(-17) oC + Photo - -
NOTE: _, cm2 ;/_, cm-1 ;T, eV; (-16)=
Remarksi
= 2.43(-36)(C+) 2 T-l/2
3rd excited state
2nd excited state
1st excited state
1st excited state
ground state
1st excited state
ground state•excited state
= 3.4 × 10 -20 T-le -5" 4/T
10 -16 ; o" = fqO e-B/T
72
_J
TABLE B-2
ABLATION SPECIES BAND SYSTEM f NUMBERS
System
C 2 Swan
C 2 Fox-Herzberg
C 2 Deslandres-D' Azambuja
C 2 Freymark
C 2 Mulliken
CO 4+
f-Number
0. 0059
0.02
0. 006
0. 002
0.02
0. 017
• 73
_EDING PAGE BLANK NO'T.FI_.
LIST OF ILLUSTRATIONS
Figure
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
Geometry of Inviscid Flow Field
Geometry for a Finite Number of Sublayers
Calculation for Two Overlapping Lines
Qualitative Variation of R* and R
Effects of Exponential Kernal Approximation
Effects of Air Continuum Band Models
Effects of Line-Calculation Options
Effects of Ablation Layer
Temperature and Tangential Velocity Distribution, in PhysicalNormal Distance 84
Tangential Velocity Distribution in Transformed Mass NormalDistance 85
Particle Density Distributions 86
Continuum Optical Thickness 87
Spectral Heat Flux to Wall and Shock 88
Effects of 89
Effects of 90
Effects of 91
Effects of 92
Effects 93
Effects 94
Effects
by Perturbing Ablation Molecular Species f-Numbers
Equivalent Widths for Isothermal Layers of Different Thicknesses
Blunt Vehicle Cmffiguration for STRADS C alculations
Effect of Velocity on Radiation Heating Distribution, Blunt Vehicle
Page
76
77
78
9
80
81
82
83
Number of Sublayers
Nose Radius, Constant Velocity and Altitude
Velocity, Constant Nose Radius
Nose Radius., Constant Altitude
of Nose Radius, Constant Velocity
of Perturbation of Radiative Properties
on Ablation Layer Enthalpy Distribution and Heat Flux to Wall95
96
97
98
75
Shock Wave
fAZ 1
Iui
1
Zl.= 0
Z 2
AZ il I
!
Uo
I I
Z i '
Zi+l
Z n
/_Z °kI1 I : I_
Wall _ -Zn+ 1
Total Number of Sublayers = n
Fig. 2 Geometry for a Finite Number of Sublayers
77
o9I-
Air Continuum Only "_
0.8
0.7
0.6
0.5
Band Model 4I0 Sublayers
Uao
P_o/PSL
RN
P_/Ps
h/hsa t
Ps
TW
qw qs _a
Actual watts/cm 2 cm
E3( _ ) 2,087 4, 326 9.34
I I I I0.2 0.4 0.6 0.8
z/AO
= 15.24 knV'sec
= 1.66 x 10-4
= 256 cm
= O. 058
= 3.98
= 0.44 atm
= 0.331 eV
(3840 ° K
1.0
Fig. 5 Effects of Exponential Kernal Approximation
8O
\\
\\
Air ContinuumWith Line Correction
LINEOP = 2, 10 Sublayers
\\
Band qw qs
Model watts/cm 2
-- -- -- 1 3,056 5,546
4 2,903 6, 339
u = 15.24 km/secoO
poo/Ps L = 1.66 x 10-4
RN = 256 cm
P_o/#s = 0.058
h/hsa t = 3.98
Pa = 0.44 atm
TW
\
! I I0 0.2 0.4 0.6
z/_ a
= 0.331 eV
(3840°K)\
\
i
.0
Fig. 6 Effects of Air Continuum Band Models
81
1.0
e-
0.9
0.8
0.7
0.6
0.50
-- Air With Continuum_,qand Line _k
10 Sublayers
u = 15.24 km/secOO
poo/PSL = 1.66 x 10-4
RN = 256 cm
Poo/Ps = O. 058
h/hsa t = 3.98
Ps = 0.44 atm
TW = 0.331 eV
Band Model 4 for LINEOP = 2_"_ (3840OK)
qw qs _ _
LINEOP watts/cm 2
- 2 2,909 6, 339 8.59 _'_"x
3 2,969 6;581 8.53 "_,_
I I I I
0.2 0.4 0.6 0.8 1.0
z/A a
Fig. 7 Effects of Line-Calculation Options
82
1.3
b.
I--
Air With Continuumand Line
10 Sublayers
Ablation Layer With Continuum
I 0 Sublayers
Ablation
Layer
No
, Yes
u = 15.24 km/secoo
poo/PSL = 1.66 x 10-4
RN = 256 cm
P,,o/Ps = 0.058
h/hsa t = 3.98
Ps = 0.44 atm
TW = 0.331 eV (3840 °K)
Carbon Phenolic Surface
1 eV = 11605.7°K
!
qw ,qs zx,,a
watts,/cm 2 cm
2,969 6, 581 8.53
1,749 6, 798 8.55
I I I I0.2 0.4 0.6 0.8
z/AQ
1.0
Fig. 8 Effects of Ablation Layer
83
1.4u = 15.24 km,/sec
P_o/PsL = 1.66 x 10 -4
RN = 256 cm
! eV - 11605.7°K
AirLayer
AblationLayer
I-.
8i#1
_ Air With Continuumand Line
10 Sublayers
Ablation Layer With Continuum
_ 10 Sublayers
Carbon Phenolic Surface
= 0.44 atm
= 10.94 cm
0.097
0
I I I ....0.2 0.4 0.6
z/z_
0.8 1.0
Fig. 9 Temperature and Tangential Velocity Distribution inPhysical Normal Distance
84
1
1.0
0.8
0.6
0.4
0.2
0
Air
Layer
10 Air Sublayers
10 Ablation Sublayers
Carbon Phenolic Surface
Ablation
Layer
U
P_/PsL
RN
= 15.24 km/sec '
= 1.66 x 10-4
= 256 cm
I I I J0 0.4 0.8 1.2 1.6 2.0 2.4
Z
fp dz_ = ps A
0
Fig. 10 Tangential Velocity Distribution in TransformedMass Normal Distance
85
1018 t
Air
Layer
Ablation
LayerC2
C
H
1017N
O
CO
EU
¢)
U
,m
1016
1015
10 Air Sublayers
10 Ablation Sublayers
u : 15.24 km/sec
: 1.66 x 10-4P_/P SL
RN : 256 cm
Carbon Phenolic Surface
C ÷
1014
0
1 10.4 0.8 1.2
Z
f p dz: Ps AO
l1.6
Fig. 11 Particle Density Distributions
86
t
102
k
101
100
Be-
U
e-l'-
OU
"_- --1a. 10O
ED2_c-
e-Ou
10.2
10-3
10-4
u = 15.24 km/sec
p_o/PSL = 1.66 x I0 -4
RN - = 256 c-m_--_ ....
Shock to Wall
-.---.4"-
°
ocktoj Interfacef
10Ai_s_b!_yer_10 Ablation Sub!ayer_s
Carbon Phenolic Surface
! I I !0 4 8 12 16
/_i eV
20 ¸
Fig. 12 Continuum Optical Thicknesses
87
104
103
>
Eu
a 02
1I0
1000
U
P_/PsL
RN
: 15.24 km,/sec
= 1.66 × 10-4
= 256 cm
10 Air Sublayers
10 Ablation Sublayers
Carbon Phenolic Surface
I
I
II
I\
\\
--qwtv,
m" _ "" qw t l.J,
(s,.2.ght_yqAbove) s Iv i
1356 78
'111111 I I i
4il
II ./I
i
With Ablation Layer i,749
Without Ablation Layer 2,969
Without Ablation Layer 6,581
\\ \
o
Total
watts/cm 2
With Ablation Layer 6, 798
9 15 17 19
IIIIIRegion
I I4 12 16
I0 II 13
I II" 'IFrequency
I8
2O
\
2.1
v, eV
2O
88
Fig. 13 Spectral Heat Flux to Wall and Shock
>
I--
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
%o = 15.24 km//sec
Poo/PsL = 1.66 × 10-4
RN = 256 cm
uT
e-
--II
"_,, u/us
-- 1 eV = 11605.7 °K _
,
-- Carbon Phenolic Surface _.
-- C_bon Phenolic Su'r/ace _\ I
qw q_ A OwVw_J
I I I ! _0 0.2 0.4 0.6 0.8 1.0
Fig. 14 Effects of Number of Sublayers
89
olIOi.---
X
EL)
(3
x"
LI.
>
.¢,.
14
"(3
7
6
5
4
3
2
u_o = 15.24 km,/sec) (50 kft/sec)
poo/PSL = 1.66 × 10 -4 _212_ kft)
6 Air Sublayers
6 Ablation Sublayers
Carbon Phenolic Surfaceqs ab9
q$
qW
-g..e-----'----
0 I i I I0 2 4 6 8
RN , FT
0
9O
Fig. 15 Effects of Nose Radius, Constant Velocity and Altitude
105
104
E
a
o-
103 -
102
RN = 256 cm
• 0
• a
x A
P_/PSL
5.94 x 10-4
1.66 × 10-4
3.26 × 10-5
Altitude, kft
175 •
212
250
J
JJ
J
JJ
f
JJ
JJ
J mJ
JJ
J
6 Air Sublayers
• 6 Ablation Sublayers
Carbon Phenolic Surface
JJ
JJ
J
I xj I
JJ
JJ
12 13
J
Opened Symbols, Without Ablation Layer Effect
Filled Symbols, With Ablation Layer Effect
Tacked Symbols, Zero Ablation
I I I I14 15 16 17
u_o , km/sec
18
Fig. 16 Effects of Velocity, Constant Nose Radius
91
E
105
104
103
102
0
Poo/PsL = 1.66 × 10 -4 (212 kft)
• O
• 13
•( A
u km/sec KFT/secao •
17.38 57
15.24 50
12.20 40
Carbon Phenolic Surface
Opened Symbols, Without Ablation Layer Effect
Filled Symbols, With Ablation Layer EFfect
Tacked Symbols, Zero Ablatton
( i ) Interpolation
0 _
(i)
n_
_l_mm _ _ amJm _
a
• ...._....-iI
Jf
fI
(i)
_g°
I I I I I50 i 00 150 200 250
RN , cm
3OO
Fig. 17 Effects of Nose Radius, Constant Altitude
92
i0 4 --
E..u
.8--
0"
103 --
1020
P_ /Ps L u
• ® 5.94 × 10-4
• !_! 1.66 × 10-4
J( A 3.26 × 10-5
Opened Symbols, Without Ablation Layer Effect
Filled Symbols, With Ablation Layer Effect
Tacked Symbols, Zero Ablation
(i) Interpolation ._.
= 15.24 km/sec (50 kft/sec)
Carbon Phenolic Surface
(;)
©
----Ig-..----------Ig
I
I I I 1 I50 100 150 200 250
RN , cm
100
Fig. 18 Effects of Nose Radius, Constant Velocity
93
.j_
1.0
0.95
0.90'
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.500
*For band systems
I
u o = 15.24 km/sec
Poo/PSL = 1.66 x 10-4
RN "= 256 cm
10 Air Sublayers.
10 Ablation Sublayers
No Ablation Layer Effect
With Ablatlon Layer Effect
Carbon Phenolic Surface I..
o" * f_ "Y_ qw qsac abp < 10.95 watts,/cm2
n
3 N N N N 2,969 6,5811,749 6, 798
2 2N - - _ 2,573 4, 989
1 N - - _ 2,118 4,481
4 N - N 2N 3,108 6, 985
5 N - 2N N 3,208 7, 220m
6 N 2N 2N 2N 3,361 7, 7371,926 7, 988
"-" Not ConsideredI I
0.2 0.4 0.6 0.8 1.0
Fig. 19 Effects of Perturbation of Radiative Properties
94
4.5j
4.0
3.5
3.0
2.5
2.0
1.5
1.0
%o = 15.24 km/sec
#_o/PSL = 1.66 x 10-4
RN = 256 cm
10 Air Sublayers
10 Ablation Sublayers
Band Systemqw
watts/cm 2
Nominal Crab 1,749
2XNominal Crab 1,738
Carbon Phenolic Surface ?
'1IIIIiIIIIIIIIIIII.
II!
I!
1.0
Fig. 20 Effects on Ablation Layer Enthalpy Distribution and HeatFlux to Wall by Perturbing Ablation MolecularSpecies f-numbers
95
100
t = 1.24s eV (14450 oK)
P/PSL = 2.87 × 10-3
NI = 0.401 x 1017
01 = 0.148 x 1017
e = 0.806 x 1017
I0"I
4 4 0.195
10'3
lO-4 I 10.1 1.0 .... _0.0
_a ' cmI00
Fig. 21 Equivalent Widths for Isothermal Layers of Different Thickness
96
UoO
Effective Axls of
_" Symmetry
R0
\
J
Fig. 22 Blunt Vehicle Configuration for STRADS Calculations
97
I
\\
U
! I0
000
0
0
I[::0
°_mo
o
t-,.,4
0.,.-4
;>
,--4
.o
°,--1
C_
°_4
0
.o*,-4
"0
0
0
;>
0
ro
cq
98
C. 1 Listing of STAGRADS and Input Data
Z
!U
_T_
-}
iI
.J
u
t13..3
:>>-7_
<JF- u}¢<':7_r-_.J
F-
71_ O,
-"4:liJF_
u') X
C9 L_
l--
u) LI
tgV9=: 2C
b-- :,--c9 tD
.Jt
kJ tJ
_3G_r_l
__1J.<LI_
;C0
l--
_J:,9<
C_7...
u9L_7_
J
-r£F--
,<I
tD
_--4&K
:Z "4t--
5_ u)0LL
D,u
>2uJF-
+{D G", ox O_ Gx Ox G_ Ox O_D I I ! I I ! I IJ :M {'q (',l o,1 {XI {\J N c',J
:'_co
f',,] ,..0 _4L_ 0 C", oO o_,-_,--._ ,,-,.I N,-q r._ Cr-, D.--i"-.+ 0 q'-, I--- O,,-_m
99
o_ cO0 C)
C'_ O ",.'J
--}
I.I.,.i,.,9
.-}v
<•wB
<{..9
--.}v
..-}
._.1>-T-
.--}
..-}
{",.I
,--'1
G',
,3LLJF'-'-
_O:2"._E:
t_{",1
102
tL
7ZLL
--I
IL
I
Zh
_j 1._1 lm.I ,ILl IJJ IJa
II II II II II
>" >- _0 >-" C) Z
,.-._ _-.-4_---, _J I.--- "'..0 L) (_.h -< -<
L,J ._I .__._I.-- Z)C3,'J __-_">- 7__
_ L,,J ,LLJ I--- LLCO
•_ _ <r --I-LLI _ ILl_- _. C3 cD
cO, u) ,LLJtD
LLI LLI <'&' ._I,',_LI LLI 0"_
'LL ,LL :> LL
-3
II 3=
oooo oc-qC_ (,_ cq O] cq ,--I _) H
Zt-- D- F-- t_ I-- I--LL I---•:( ,_,-(.< _.. _ _< <<
O OOO OO DOLI_ I J_ _LL LL [1_ [L ;';][ LL
,-4 O4(',,I (q _ Lr_ '-O _--I O,JC) O O O <D _-4 ,-4
O O O O O CD. (__C', C_ G_, G_ (3", (h O',
IIIIII _-_I 0I ___I UI ,-4II
>LLI
0
UT
LLI l.IJ LLI IJ] uJ LL! [IJ
C)!--
0 •
i
U
>
Ja
Ld
U U
i--i
I1
r-.I _. I-- C'J (_J 0..I ,,'--I r-'.l "-0 _'_ ('_
tfl
g,--I
!I
z<0
U
cor_
,--4
-)
0
0 ,-_
k9 ,-_k9
}--
v
F-- I--< <:-£ Z-i
0_-0
,-4 C',I ,-4 ,-'.1
(D OL.", 0".
I---I-- F-- I-- b--I---I-- I--
<-< .:._.:< << < <."< <<
0000 000 0LL b_ II_ U- b_ ta LM LL
_", "49 i"-- 0.3 (7, 0 ,-4,-4 r--4 r-I r-I r'l (_.j .'.J0 0 0 0 c. 0 0 0C, (,', c[._ G, C', O', C:_ C_',
103
"2:£LIJ LLII-- I---
__ .c:. _>"LI ILl DJl-l--k'-
I--
U] tklI--- I---
-.T,il
",- II
F-P'-
£'<0_:_
IJJI--N
Of_7_
,4}LLI
I--
c_Z.
..1_Lkl
Z
c_kLII----
b--
c_22
r---t
LLI
i---i
".D,-.-.-I
c_
L,J
:£I
1,'-4
,1!.!
II
,,0i-.!
II
u}
I-..--i--11
Xtd
_!J
U
,'--I
L9
_J UJZ-L:
0 _I ,-,UJ _I _J
EC -_ C_
_ oGa t.__ ._l _1_0 hJ •
h]O _-£i- :TZ _:::
FY£u_ E_
104
oo 00o
r-i ,--t r+.i
r_ i1 O_
o3 C_ 0OO,-4,--I i---I ,r-I0.. EL Q_
t_,.+r_r-i
CO oC4
0_-_!-- II
0 _1LgkJ
C_C',
|--
i---i
XLIJ
J_J
U
c_
i--C.9
o
o_0ILl
do
r:_0 "
o,. • I o
+JJ _,_tJ u-) t2J
o _+i +__ul ¢',,J oOEL <C LL. _ I'-'- t"--0 t].. o LL.. ,--I L_t_ o, II © 0 0
i-,.--i
_._I
,tJ_.
04<C
,'_ 0u) .%
to tlJ r,tDJ r,,I I cq :_
0 :_ o :'._ c,,I
t_l L_
,-q ,-4 c,) 5Z LJ :_< £Z_ cq-cO c',.l U _ ;_ _ (,3 •
_ 4- _;.+ Dj;;< i_ >->- 9d :>- LU
f--
_2
Z
O
v
?200 t+,q cO0_, II ++J
0.. cOO0i-- QJ _J
0 ,.-, <EtOIL_;
OOC_,o o o
LLI 1-- 1---.--I CA) _
II -:] II o o _1 _4 o b0 r4 _ L_; 'J) >- >- -.1" O Lt] (_3 I1 tl _'_- _3 1-- O_ <r _ ._
[[ Z Z II II d II II 11 _-I tl H II II II II N Z_" r/ _ _- li ,"4 !+J d) -] -_ -_U LJ U _i- t_ II r-_ _ txl _1 N a- cO O_ O_ F I1 "_'t__ DJ "d LLJ >- _- II tl + +":Z 2-: 2:_: _- Lf_ r--I _J tf) _3 d} ---_ _.- F_. 3,- t-- ;;_ LLt L,; .%- )- >- >- )- "_ O Lt_ IL D_u3 cO u3 LJ U .._1 kJ ttJ LLt LiJ D _ ,-_ _ c,O LIJ < Ld _ E3 ('_ [3 ¢3 212 & _-_ _
_3
L)
d3
,'33<_
O4L_
a
Cd
O
0o4
Eb ,,';q
--3 1._<(EL.I--L.;
L!J ,E_
._2 <f.
b--2E
Cd K3
.... _..I .Q_Jl--
t.J t._,-4
107
"4}00,413..
0OCt.r-.j x..II 0
II_0t-q 2>
_v ,/_ _x/ :_z _',Z_,__< _x/ _ \/ x/
l-- F-- I-- _-- I--
23 _ D Z} 237.'£ 2--_" .--C_ 7-Z >__2[L U_ ,L 'J_ LL
-J -J -J -J --1_--) D _-3 _-D
LL L!_ 'LL _, LI_
L.q _0 r_ cOO_J 0 J 0 d OJ 0 ._!{'_ (_ {q C3 cq _ {q L_ o1 C"_co C} {o 0 o3 0 _0 co 0
:.C k£ : C .'_-_ ::f0 0 0 0 0l-- -J I-- -J l---J I--_I I-- J 0
,'I.2J
;& -oD !
'J] _-I'-D[./.
..,t- ,,p_-"
:k .-'E0 "--<_-- 0 L9
LO ::-- cO
;]'.;]--oqOo 4-
e O
II II ,--I
--J J -J J -J tl r_ .< 2- Z_C) ":_. () "--< 0 <[ (] ':f. '0 _ !:] II ':_. -:( D'__{.2)t.) tD U L9 U L9 U t9 U 2-f --} k9 LD k9
<_ Lq .4.9 1-_ co0 ,-4,--I ,-4 r4 ,--I r--I_'_ <-_
/:D cO CO 0 CD CO CM
108
cotO
r-I>- cotI >-O
lC. _ -<1"LgHC)
O CD O ,'M >- I--- 0",0 o li • |I _ P-i o _-" 1+"%
o H ".D ,--i _ -f- uO CJ Z C),---i II II II _'__/ II I t_9
,--4 _t ,-4 0 )- _ 'LL] _ 0+ + + N Ii + _ ,_ t--
_ t-J t-- u'} 0£ t-- _ _77- _C ,--+ +--+ <) l iJ ,--+ LL ]_ {.m_
(e'j
0I
00
r-'-I
!
030
oI
..._¢
w2>-I
o
r--4
LE_._-<
L9+<D
II
r-4
II
7E_<9
..j-c)c,,I
ia
.'-ZiD
Lt_
-.+-i"U
23 IL
LL "<
Li l LL.J_)Z r-_
Z)D C_
tL ;-2.-- D
_ C_3
_ L0D ZD <I2
tl_
--_ _j_ L)
_ UJ
• {".1 "3 >-
,_ -.t _ 7 00 -)_ LL •
0 ,--I Z 0 D __1 .o -} "-) +_ca + _ t-- 7-£ 3 0 I---I'-- _ _ 0 cO
I-- _ ,LL. PZ o ,0 ,O "3 _.] D __ bJ MO_ ,-_ ;,'30 + LL I".. 0 0 _ 23 _ <1. 7£ >t---- _ tO -C :D _ I ,.O cO c'3 13 _ 7_ .. ,--4 "
'._9 ;',_ -3 _ (7.) _:: -;g :;C L< tL -") / I-- >-
<--72"_LD _ o ,_ e ,-.--t,--I _',+_/. 2-) _ -",_ -3 -3 _-. :_% L'h rE) _ tO :2_" +
d We} _ II I1 h.l 0_ G'_ H b:f _-) --) --] -_ (f_ U-I {q O 7_) ,--t ,--i _-I _-_
tO }-4 o C} l:l +_ r--i '.2) "O vD "-3 + IL 2) 2-_" Z P4 o o + ;-:f bJ *', ._] _ )--II (-J + .'2) "I- -P hl + + II It 7_ LL t!_ o _1 +t ,£} IA_ 2_T '..0 I!J II
:.T'_ LLI II ->'Z t.'} b<3 7_'. r4 C;_ _ LL II II +-I 11 II + " .-2"__--" 5" _ D'-_'_ O o _ _-t ;--£ ;Z _ tiJ IAJ (L" ©--3 "-} II U {._3 II P2-_ 2.2] [IJ 2 2 _-4 EL! Exl {XI ,--t
{,.1 >-+ ,--_ I--- _ _ J t-- t-- H r-I_ _ tJ J 2_) 2£ J 73 I--- .J 1-- i-- tS} J L_ ,",1 -}-v} _ _ 2>_*I-- :J) _ _-_ _ II D 7-J 13 -) 2} 72) 77i Z3 _ --" D£ _-_ 7< __1 LLI ,-_
p.;> (] _ {.£) t..) t/} 2> _-t -_-: 2C (_22 "'{ it_ E'_ "12_ LL I.k [J_ II _ 2: Id- U -:; 7__-[L) £'1 (+_ _-l--'i r--t
trl ,.+9 r --+ O_ (.3 ,--t <1" O ,--t:2:) O O O O (_ C. {"4 ¢4ENI cXI C,J C',.I .--I ,--t ,--.1 E',J {x1
109
I---I--4
XL_J
--I..J
U 0
r-I 0 _" -"
!_I _ 221 I-- "t" "-"
<_ Z'-Z-|-F-- II 0 II
:-_ d ,--III _ {')_ o-_0 t'-Io_ _-- LL I--- O_
cq
,--Io
v
I--
v
>
0
F--
0E_-%
_J_!<.ik.)
c,"i
..T-,.-t
I--i
o,
o
I-.-
,t'_
x
o
<t-I.-
i--
o
b.->
H
>
or-tL9O__!
O<
,._, -)O
II
m---e _J
>L.J
-.1-O
ek
L_OC_
L_O
I_JLIJ
tl II • I
Ii II < "1[ 0 \ '--_
_ _ ,--! ,--t m--_ ,--I r-lL_
_!,--I o o _1 O_ -I- "_
Lg_ 0 _-E_LLIL9 _ o ,-_ 1-- LD L) ,-_
C3 ¢3 -'_" ;2#. (", G _C_-_. .. C"; _.( '-'-" _ I
<_._ • o G'_ _ o _ _ LL _ _ ,-..+
II o o I-" I--- ,-+-.Ir'2 o ,--4 d N H U II tg Zl_J _ I--' I-- L9 L90m Lg 0_-I 0 ,-_ _J ;;: _ _ L9
"Z_ P_ o • <C "_. ;" ;-- L_ I U '--_ + _ tO,_ 0 _*_ < _ 0 _ O 0 or) ,--., ,-< d:)
:;-_. J +--J II J _-i:u --(LL.O O ,L L,_ LL LL O <_. 'L.-'#. :-50 +( tD L.-
LJ ."_. "-" H ,_ _ L'.'.9L-" '_ L.) U tD t..J L] t9 Lg
C) r--I H _-I
0
:E
ooj II
I! v
•-, _-E_L9
L_ o
cxl eoI-.-
,-+ ._I
_"-0
o +'_
l--i ,--,.I
.,.T_+--.T._4tD LDII _
C'5 !-- IL
-.#O(q
(---_ otq •
0 _
,-_ U 7i=u,"_ Lg -_9O + +
I _ •:_. r'4 C) '_( f"- c,.l. c,,I
:-_:_) L9000 OC9 t9
0 II II II II _ (q li II
lJ_ < L) <_.k) _ 0 +-_. L)LD LD L.h LD Lg t_ t9 LL_
cq (D 0 0dl t_-I Crl
ii0
<9L9
<9 o 0
_ C',I r--t Z
| -'-
II II C"x,,I_ _ Z) ,--o _
-- '0 @ _ H :E II
"_ N II _ _ ::.___ ,_.-r U ',D LL "_- C) _'
oc_cq ,.-q
v
09
i>
I
r-I
+ e422 ,-4
o9,-, 0_> !--- _ o41--+ __ H II
,-4 ext 0 .'%1 ._ <C LD
_ 0 r--I t--I e
_> _> 0 _ :,D +D _ ,-4II II (:_ 0 o 0 _ ,-4 II
_) CM (q ,.O 0"_ 07,1C_I 0 DI I--;£] + -l-_ _ II 7[Z II 7Z _-_ .--I
.(> ,"4 col tM
v
',.9
U%0
v
o4,<tD
r.I
L)t9
,--I
<L9
(DG,
,O
L_Z;
OO bJ
U'I :.-.4
- J
04 4"
0 ":C 0", kJ
+ ocNI + _ 0__=
Lid -") ':D gr" o c3"_tO _I t'_ I-- I-- _ rL
0 .J.=< I _Z LD 0 uOkJ <". LI_ I-- kJ o (q tl3.:_U_ _, :-_ L.'J :._ (3 (.% LoII e II _-D D I1 1,"1 0 _ _,9I-- II -'3 0 2.f N 7-_. II Ff:!2 << <_ U H C,'_ _ ,_ 0
IIi
d2}
1--
g
+--- __-£09 ,---i b U
I-- ",,.-_o+"_ [ul (<-1,____u)}2 II Z} tq I--
II 0 I--- b-- ¢,+:-10 II *<
-7: _} O --I _ il o'_ i U:_-] ,'._] k_9 I--- tO 1--- U 0 U T-7.
_ _ 2?. ._ _ _-C 0 (_9 -_ u_
C_-_:J COLLJ --2:. I LtJ 'J')O o072:
__I U 2:2 o_ _J I-- "_-1:tL]-< o <9 .-rt.,.!'aJtd El'_I-- (_ Cq O_f -: C_] D LE _ r23 o 7_77£ :£ _ U _ II U (1] 11 2-[ tO <) ++( £] U
_ H (_>-:O . "--"_ 2:2 {0--<>:J J --I J // (_) F-- iI --.! 0 I_ b'} u) i< 7_ :,")_ _ --.! I--- LE _11--21/L-- 11. LL. <I[ 2: t;) O _ -: U00 IL I]_ < .',,J_Lt_
r-I r-I C)
(m ¢q O
112
\.__/
,--III
,_ O,d ,--.1":,/ ,-H _B
Z:) _, I"-_. 11I-- 0_ D_ t--"0
>_/ LdO
p£ tD
II _ _'- o 0"_
X,.-I
d
-r"L_
_1---I U
_._
:EI--
r-.l ,_
,-+>'4
_U_0_J T
__<J-
X-J:0 _.1
7-) C,_0CC ,__--<9',O
hT_><'-O L_+_ ,-_.+
_t..)i-- "_.-)
0 :ifLL :_
r'-I
0+]
--)v
0,,_j
0
,-'I
7
I02:)
I--
-..)
ID
++
-)
::-E>
i+1_
-)
:>
LL
-.-)
>--)
I1 _
Lfl ijJ -]
_-_ 0d) .C <0C_ .-; t_)
,4.,--.I
LLI
t_
oLLI0+I
C',J
',qo
"-.T,--I
L<
3]o
r--Ib_.
.a-
l--
.<:EF'-"0LL
(q
0".
.:-i'_i--i
I--
0"U
0
i----
i<_
0
I-- iUL_
C_
0a
W
]>I--i
I--
L9
113
F-'I.U
,"..1
A
I---i
V
F-
U
I,J
i-.,i
A
o
c_0
_'_ F-# <
I--•--I _Jl+ LLJ
C3"r-. 0
,-4 ¢q
_<Z <::<2.
<_. IJJ _ _
2:tz.tr.q 11----
4,,LJ
-¢"
,-_ 7_t-- U
U ,,,I--
>-.- tn ;7-,_ o U
,--, C'd .-__
t_ C'h
..._o
o
t_
t9
o
t_
o1-,I
-r"
,--I
j_ A
L9 Cq
v
r-_ +---n
++-F" iP+
_F'4O_
v
Kt_
A
o
u_A
'--'0> ,--+
,,'-cO
"-,_ L'-J
-'r-.'-Z(3 H
LL ¢--3O,,l
A
'--'i
>--
A
Lg>
A
(_) c,7r-i _-I
L3 ,_'D :__-I '-M
115
L_
-¢ i9_qO #t'_ }-- --<
0OOZ
r_ <D _-_
L) o O] oi_- t-- _O 0L_) -:; ¢ *,+'__ II ,-4 C)
';? _ <3 ;C II II k I1 II II_ Ii II o"] _ '.LI .--I ,._J .._1
i L t L cO :!3 D:Z _.1 _...I 0 _ :']
<_I--o,.!U
H
_J
U,-..
"3O'J7
t--C3_<_
_UIF-- _,_
02_tJ _{3.. IJ_ _ cq
U •
0 _ ILl _ o .T_.-Z-+-E_ :_ __--/_ ,-4 LLI
_ o iL _ 0
-p
_0 .+-% _ _'=- ,.0
<< 4" _< +-+ ::-" ,LLII--_ 0 o uq I--;;Z_-! T" _-+ + ,-_+'dJ_ C£_ Lq I bJ _d
0 _ < cq ,.0 -,t cq E;U I-- :Z) o ,cO :_ 0
2: "_ C#-__ # _ ,--_ [9 oZ ;_ v) 0 ;;_ r4 I-- I < (.7
I-- ;E CC 0 0 Z< :; ;'*.:q _.: _
I-- ,--_ Or'__ 0 o o ..#.. _-- r_ -I- _ C_5 LL I--O L) _.. o ,_-'_ 0 "-.. 0 LL <C.,_._ ,,.0 ,LL1 0 _ "_ ;:_
_ _ -- Lr_ O II 11 0 o k II <0 o (q _ C' H 0 ::; _ ZZ)L_- _ H ,--I L_I II _ _ O C+ 0 0 k _-- 0 () k 0 r-I II II (0 -_ 0
,-_ + + U +-'( .+.-I ,-.I Lq Ii 0,,I o Z _ II o Z_. II II ',L] +-E U _ U__1 _ ). F-- It II t--- _ _ II _ o .--I .__ I1 I_. +-I q-_ ZE Z]Z rD -_ ;:Z I1 II ;_._J H 11 II "iLl t_ --I U t-- "-4 • i r-I II 0 LLI "'I II II -] Z II <_ 0 _]L]<Z. U "_;: ]-3 hJ _ LLJ Z '--4_O _ II +-4U -E _" ]-< (f) L'J Z "] L) L] ?- "<_14_U "-- > H- ,-:.+.qu') {i] 0 tO _:3 >- tm LL ._ tL) _!Z ,_++ 0 [3 72:;-f :Z:: ;: bJ W _-_
,--I
u% ,--io +q,--i ,-4
t.+n_
Lt..Ie_
II
(h":E
0
I--<
I--
DJ
>-
<E_J
E_"
0
<_Js]
C)
tO_q
<(
p"
LL
C.j-0
'Z_
bJC_<
(%1i
[,JC),--I
o -:IEo ;_
_.I I-- :,>"o ;r-;: I+LI
.r_ _ II7;'-" II }:
l.,l,-I _-_
LL __: :<
,-_ c]LO
116
//
IP
,Dor_
O t,O
+
L.9
,-_c9:Jc LJ O II II
ItJ F- _ ,-4---_ :.-c O G-, + =:ZII Q II 0 C _-_ t--4
r4 Z-L" I-- I_ 0,,I _ I-- oO>- _ -:_ I-- :22 T-_[tJ lJ.. (] 0 0 _-_ 0 I1
,,o o_(_:, oo-4 ,'-4
o
o,j
u'] e,,l+
-I- z_,<-
v
,--41--+_-_.-Eo9
II II
!--<0Jii
t_
o 7_
o--_
0 ,-4 0-.J + ,,LL ";_ r-!
r_ o> 4- _J_q_
O I _ 3> + + I--._O I!J © _ _'--_'_:_ +
cO _'T" H _ _ _00-f, =:: LD -,-'"_ _: :-%:LD <_
O ,,_ _:_ _ H _ _O ,-_ + _ ::> :> > 032co II O _-_ II II I1 ©
+ + + H | C' II _ + g: + + • "[
_-_ F_ :L- II :E r :_ :E _L O :';: -- N O P_ _ El-- --JL0 _O :'E:-'Z ::C -_- PZ H _ L9 _ :> U _ :> _ '-q
0 0..,-4. _'_ 0DJ (.' (D 0,"4 , c0 _0 co co
<:
I]_
>h
_m
>-LLI
LU
LL
2,<I:
z.
i-4
Ja
ti,
o@
I-t
2:J I-.-
ILlN
_-_ f-_
,o o
[_ oI.-- ,,,
r,f
o I--
F-- uO
_ OJ
14 ,-_ 0 DJ_-_ + _2>od 2_oZ ::Z 11 J
II :L: J
od
117
A
','1,o
r-.ILLI
te,
II
F--
co
l--
eelr-t
i
co
°
LLI 2::)2:) CO
A 11 "_"
_J
;k ,<wzL_ 0 D cd
4" _T_ LL Z.)LLI 02 =g I--
o ,-_ cE. C,L'_
,-i > LU _: <-;.I :}3 :l: _ O
+ _:: _2_ ,E] D2 > o,,t _ i_1--- C_ LL :i: 00_ _ v gh
o Z::) OO _ .:'_'LLIOJ E2) :_ g_'_LLI I---D _CO .:C I1 :-C ,0 :Z..r{ :_ LLI O II N ;:,.--1 ,--, ZE I1U td II O LU II C0 L.') EC _-,_'L.LJLU I-- :.:_i-- II II N H II :5: I _< 2;.. H H
I1 O U ...x::" ::: '..tJ C.) U :_ Ed _ h_ C) 2i
L.) 7Z:ZC LiJ !]J <[ LD _PL'_ 2_1 D,--, U :_r-'l
,-_ o'_ -q'-I_ o'/ o3
cO<I:
:Z)
o
cc
ILl
I--
Z3
U
>
l--
-r- 11
• '. LLI0 .'_
< 0
_..! :>LLI _ _.
,-_ .b ,-4 =_:2_ D:.D c.'_ O _" >- I.L•_ 0 0", ,-i,;Z_ _ :X. R:
LIJ [tJ 0"1 5- ::Z::):_ "'--7 IIF-- t_ cq _ LL 'L L9 _
N 11 II II II i--i_A_':r2d C) U _J U UZ: F:: .C_ .'- T_) _I
120
F--
OU
H
</:Zt_
Z;..9
o.
O_ *-i 0 -I-
f'__i LtJ :_ LIJ I D!"_) I_ =,"I F-- II Z_)
EL 'D_d L-: ,'_ 1_: LLI D! LLIr-t
c-D _4
0u
cO23(90
L.9 }--3 z3
_DO.'9 t__
D".Z( t,O0
t4
I---4
r--t
13L
i--ill
-.¢
Or-I
x
or..-I
:>
O,.---I
I--
o,-i
>-r_
!-- "9:----"£-Z
"0F--- "--
:-2 ()CF tJ,--I
C,,l o"1 ,.D f"-r--I ,--I
L9 L90tDZ Z'_ Z__
<_
o ,_
I--- (NI _
t£_ I:L _-xlxcq
o') ,IJJ ,_ o F--:-r _l_ CNI t-- o _."
--r" ,ih C,-/ _ II H
N II _ .'../_ "_._/;'6 -",_
:ELL Z') -.1-0 '_ 0 ,':) <_ ::: 0u LLI _) _,_ F-- I---&_
cJ -.1-
,-_ L.'_ ,.0 "-1- -._ '.0
L9 (.9 (.9 '..9 LgL9
121
122
i"-- 09 0", C) od o_ ,,,1- 0"_
_9 _9 tD L9 tDtD _D_D::--_ _ 2Z Y_C 2L 2: ;_
OI"..-
O, _ {M o'_I"-- _ c00_L9 t9 f,DLD
O
o _,
m,
i--.!
+
v
F--
LL
!
---L _x_
J_ rq I--
_ LLI
d3 _2 LLI 0_ b... tL '-O •
LL -I- + _: o ;,,_ -I-1-- _ _ b._/ ¢-q _ u7 Lt.J
I-- _ _._/ x/_ _x/tO _ "_/_ uJ o
", O _ _'-/_ u') "O _ 0 J- t.") O c,,,I
II o C) H _. O (,_ (._,F-'- -_: (j Cq F--b,{ O 11 II (_ II t?J II 4- t..,O,'_-._ II LLI :-_
2./_ - ......... ,[i; _ "-/.- -Z ::: ,,._'__-- .-_
:T.
r--I
II
-.? II _-}+ :/_ ,-_. + < o--J
(q "3 ,--+ _ Z _ U t),.D,+ _1-- t."x F-- L_ 7Z :.:- C_l
:".1 h_ pz ,_ ._ _ _jI LL. (_) 0 H I-- IL lJJ "7)
U_ L) (_'3 7:,: tiJ ,--+ T-_ C_
OLr_,-4 C,,I0,--1
123
\ -
L0--I J
Or--i
rH
41,
u"t
tJ LLIF--t--
L2,_"-3-b O_ D_
F-- c',l CxI I
II -" D I-" --')
l!J _- I_ 0 Lt_
H l]J LLI LD H
t£'Xf.
c)
../
-}.--),--I
o_:-2:F-
.'Z
--)
v
U2}
I
_Z6",
r--I
EL---)
U:D<1:F-v0_OLt_ Ho,"i ,--I!-JII O
701---
.'--" OL:J L')
c,JO
l---I
.IL
r-I
-E14
l--i
#
\_Z
--)
--)
D<I:I---
I
3/_
r--I
o_
--)
U
D<I:I---
lz.
It]II
L_.-a::t--ILl
C)
x_Z
C_=i-"
2_/_'-/_
i-..i
+
-./_
0
||
",Z.
0UR:(-5
125
cO _9 cO._] _J __I
co
+
<90
_/
]>tlJ
_-_ ,-4 ,-_ _ _-! _-_ &J C,J ,-j i',i
I'_ -..T L_'h C) -._ cD ,.0 0 0 :',_
126
+
t9
0
_I
>Ltl
+
<.9
0
_I
>
bJ
cO
+
t9
0
--1
>
ILl
co'co cO _0 _0 ,--_ 0 0 0 0 0 0 0 0 0 0 0 C: 0 0 _
127
') II 0 [L II 0 0 LI 0 l!_ I!. <_ II (]) 11 LL tl D_ 0 I-- () LL I-- :::_ 0 N "_1 LLl ::. __ L1J :<::
LD DZ_LD ,---, I_ _ I_ F-- LD '--' '_ LD H (D _ _ ,-_ _ I._ L/ _.D _ 7_ bO C3 ,r) <17 .:(u') u') C.d l_J
N d", cO u", 0 0 L¢'_ ,,0 b- ,C) --_ H N c_ uO :D (DN _q "q c,_ .;q --!" _ -..1- -:1" :2 -:,1",._ ".D
129
0=I
,oILlI..--H
L."_LLI
E::::
LL!F---I-,-,I
Od
4_
colIJ_,,II--.-I,-.,!
;:-..--
LIJ
-_1:L
0::£:-L0U
130
_ j
oo,,--i
0k.-
0tD
o@
r_:_
L'Jo
t--
?CZ)
i--.i
132
7
I-- O_U
1 c,j0 ,--Iu) _ '_
.TZ 2=c,'_ I O_
X ,_ c',.I
k----_- " HO I
I"-- 0 _--_ 0 11 C___
oO _q-O :£ r.>:
_.-I.
t_
Z.
D
C_
11-,-I
i=
ck
O
-"1
c3._
2=_
c-,lr_
,o
DIl--
r<):
"]) 0 0 I-- _._ [_.J _ i'J Z _-) [L. -J hJ lJ_ _,I hJ _I LLI T_.kJ D_ kD u') }.': C[ }:: p- DJ ,'_ 1_ _ t_ EC _ lJ_. D[I._ _ LLI
00 _ 0 ,--4 _ cq LL. ,_ c,,l _q --1"_- r-I (.3 0 CD CD
I--I
L,J b9I-'-- _
'_ r-'l
<,-._LD ,--.-._
,_rn
.LD
__l _,.
q-- ,.._
I.
U
0
LJ ,._
t.'.J t2J.:_ :D:_ :>7.I---I t--I
I-- 1--7-_" .:2OOL) U
O ,:D
,D b"
tll
14CO t'9I.o "Y
1:!_ 1.1..
t_ 0"7"
I--0 cO--7 1"+'4
g uJ
i_ _J CK_
o _ ¢2L_
ILl :ELL J I'--
LL1>::L"C) '_'0 0 0 tO
cO _
h_O _ t_J _ +I-- r__ ,--I +J'3 I-"
-I-- -J T::zX 0 :."-- 'L1 <_ 'JJ 2:):C _D C) "-) LL _._ O
_ 1"-- £C td o ,---+ U
cO :]2 -,.3" l-- ;: II -,.+ O ,T'J IIO T/ -r" II LL X F-
:,z :7:0 D__ lJJ >¢ 0 z_U ',1) H D II --) ::E:H & II ::3)LLJ _ I--- IIJ E) ill 0
Q.. C) i,J :-< r_ }-= ':-D I- --1: ULD HtD C__ C._ _ tO "D _0 X Z
NC) O
U U <,I" _,J -,,+
134
o
r---I
Q.
_I _ _I
II II _ M I II II
0 _-4 0 Zl. 0 _-_ II M M
0 r_ 0 <Z O0 _ 0 0
co oo o,4. co
0(7,o_
o_
o,<.J;2)xII
1IL
I'--I
+
Z
_J
XI
I---
Q
"3
r,..-t
4-
cOZ
U_
L_
4
x
T
._J LD0
X U
r_" 0 "
LLI II II
Ld _L_J _
C_ ZD "-) _ XOLdO X
0 L.J CF__-_.
_-J O0 o00
l.zJL_J X _. ,_---'.__ -, ,,
_-_0_ ]Z LLJLkJ U ILl 'L_
OLt. X _z'___ LL rd,._._
C_
_q3
IJJ
II
QZr,,"LLI
-r--J_
I'--
"0
_4r-4
Ld
I!
13-LL..IF--U3.
I--U'}LLI
JJ
>2
q.-q_
t!
!'--.-7_
0k.)
t.._
0,2_0
00t---t
135
LLI LLIJ Jr_ ,DD
F-t--
bJiL_J:>>O0
<<
LO CO
O4DJ >- 0p<__ ,-a._J ..J -*
_u9 < 0cO t_ .-.E 0o tl] _ ,.--I
L,'J _0
:c_4 ,Xo_
F- i-- I_-,0 C3. _-_*-£ ":_, _ -J I L-) 0 I C',f o C) [:Z;:+ e,.! cq _ "-_
co
H
I
32
O,J "-_
G_
CO C,J _ <1+ +--.4 C_
0 0 ::'- c. H }-- l'J I"-" O__ _i"-- C_I nS _ 7IS oh II 0_ 23 ,--_ _
.-J _-_ 0 I
-) -) -)
r-_ -] L'_It,J I].'II II II
F--- ,-._ "-)-) -_o (<'; -_ "- ;£ >- ;C-:( S,_. o F- .._ (2;0 o 13 i£ iil ._d ill ,-'-', -_" El. i-- I-- _-_
C; I _ I"-- 0 0 0 -J EL 11 0 _- II II II tl] ;M (':'_ <k ,,r; IL 0 1.L 0 II 0 LI. II ';?]) L-) _-I
(_ G L O _ (D <-- 7 G _ 4
138
,L
---) --) --,) +,-) -..) --) -.-) ---j .-.) +.-) +_) .__ ,..-) _._
co, .zJ- <+2 H C_ (_-_ -_+- L.q C, ,-_ C,] 0"+ --t L."_ 0 _,
II II II II II II II II II II II II II II
0
+-0
139
,L;
--_ -__[ t.J _] LL. 11 _ 1--- ) I-- t'rlt,) .C-1 li II I--_L L'-" II It II 11 >( II tl ii Lid II II I1 II --) I_Y
DJ
c.Oq
Z
D_L1J>-
<C__JI1
_J
÷
0 I I I ! I I I I I I IJ o_1 CNI _ e',,I fN1 cxI t",J r',.! t-,,l CXI C',,I
cOLul (,'1
a
140
r.-|
U
U
I-.- ._
,_ |H _ j
U ,J_1-- t--Z
--_L9
'--', ,--.l ,--..40d _ _ --}- Z'Q
0 o _ _ _J II Z_ N ._
I--- _ t-- II 7.Z !fJ (q _ .:< | ,---I
0 LI_ II ::] i---C/- C) "_.,_ .'3 :]]
LO _ rL" u9 _ H C_ -) -) -'< :-<
e,,i
i--z0ULL
U
.a'-
91'
0
-1
<o
o
@
o
0
o
F--_J
o
"D
IL_
tO II11 _-b Z)
}C }-- E]:E CL LLI ;ELO tO CC
0(q
142
I •
I J ,-.-I
:::)::2 El.O. cO
II _ LJ cS+
:+-+ t-- :_: l--F)_ ILl D_ !_J
r+t
41
oI
o_X
,--I
I
o
(<'la+
I--
G_
gI
X
co
I+t_J
c_
c_U+
r--I
!,LJ
o
r-I
;_ U C,.I
":_ D_U _ u_II c,') Z_. II ."<-
U II E L) ,'L'.--_ _ c,.J _ _ --9
D. t-- O_ l_J p_ lJJcO _ t+.OC_ cO CC
,.--IC3
p,..O'. ,-4
o LtJC) I.EII
CI_ +X LP,tJ] LII+ 4-
I F-I.LI _.
(,+'I I'_
U X
p-. :_
| ,--.4 ,--4
.':'d _+,'._ +-',¢U..t+_-t U_L. + -I+i--+I+_ l-...
P-i--
I o ,-.I_ ,-.,I I
L,J EL-"_-:_ ltJcO '+'+J :+:_
I '_O ,--.ILLI _ I0", I LLI,--+ +_+J uq
O i--I O
cq .:++"U <J"G" o Z3 ;',_
U_ D_ tJII _- l.++O2_.__IIL3 U il _._-t_)-) -I- c++ :) ;:):-_ :C 1- -:D_ F- EL L,J :+-_
r+H
o c9
I-- !--t-- I--
p-.-.Q'-o '411
P"- _1I I
v
Pt cJ_X xhJ 'ILl
(3", CO+---I r--I
i I
r_ o _q
G', cO cOH _ H
+o U UH + +;g co o-_
I IU L_ L.U
F- _-I oq
•-_ U UI + +
x ::::_ I IIJJ O_ ,ILl LLtg;: _+r'j ¢T-j l._P-- + C'd
,-_U o _._! Z} '-0 U t:-iLLI 32 ;,'.: ZJ ::::: _;_
u+l _';, a i::1+ u) Uo 11 _ I1 {0 :_ 11 :T.2 II
G.I U C_ U II Cc L..J p_-_"(...3:'_+ -+J D +-O u-',+Z) _:) :D Z:):..', : :: +_-- ::_. "::_ t-- : +:+t-- :Z.U P, :,+LID_ D_ hi D_. L,J CL+ ,Z) _'d u+) CO D: t;) E.+L'_ cOr-q
O H C)LPI LPI ",0
',0
i-4
I
L"t..XbJ
1'-4
Il.tJ
clo
c',,J#o0
Lt+tT,o,r--II
LLI_4
g
U+
F-
,,0
OI
O_XL.kl
!DJ
(el
kJ ,D _
_--3 ;+-" :-1+.:.C P, O',0._ cO U',7) ::+L" II 2+-:+" 11
II fZ U C_- U+,o _C, :D _ _-_+:- F-- + - t- :::2
l_.'J Pu_ Lt.] D+_2 .r_'CL;) C*( LO
",D P-
t-i-'-
C)C'-JI
t-'.LI
il
I_ ,'M
•,0 +c'q =0
,-_ ii
D_. _0X iD_I
P'-- L,'hr--i ,-'-i
u
o I ,-_o l_J I
c',,I ILl
I _" O
cO U Uo+L3 a3 + U
,---i :+:+_+-...++ : 7.+ G: Z.J-"g 1'3.-:.:: O_ O_ 7"+-
143
Ch
13_69II _
0
r_
DfOLL
t--
e,,) c,,,,tOC) oC'J o40J•_S n-1 LJ
._I
8
_.Jo'I
_ 0
F- C') COLL CO '_
_1 c) co;Z co ,_
_J c_ o
"(7-0
LJJ C_. _ ,--4'_. r...J ?- ,_
:.- r_-- t--•_ .O N v
-JO I I__ _J N >,(
I--- II _ _
(_ h,4t_ LL
r_
co
o
0I--
00
o
?o
0
l--
X!X
0
o
o
I--._I
O
A
o t--p.J
0 It-.J
II r-4 NC'_ 11 _
>- __1 _.I_
O4
O0
O O O O C) C) O ,--1 .--I r--t ,-4 ,--I rlC%1C_J c',.l ck,t C.J cxl c'J c_t o4 ('xl ('.3 o4 ("4 c-,,l
C) () II J Z O It :-{ O II II ( __,._f-_.)" " J 1.1_.O II ::,] O II
o'1 o c-q --1- "_- t_,O (D CC) 'C-) C_ OcO cO Ch (,.'3 C)_ Oh
144
•.0 P"- cO 0",,,---t r4 ,--I
t'-,4 c,j r+,j c.4
u3 e-g, t+..g _-"+
0,,-I o,,,1 ,Col q,',',',T,t__0,,,I c'.J 04 £xl CxI cxI,"-,J C,,1 c,,,,l 04 CX1 ,'-x,l
-'C ".--£._t __I
-_1 .-J
_.,I .J
I-- F-.->( >-
! I
".Z.._.i _ _..l
11 __]
.'._ 1-->--II --'( II
: ,_i (._'- L..}
,'--4
I
--)+3 tlII i:_
:- J
..-I._.1
145
I i' |
• 0 J_
C4 cqC'4
;..- {'_ II II .11 I1 11 II U II II :'.- _ .-_" /N II II 11 II II tl 11 II II 11 II tl II II ,u..IL 0 II 2.]] 0 ,-4 (_ 0-! <~ :,_ "0 b 0 [[ J D_ 0 II ;_/ () ]_ .:r _-] '0 £3 {'J II_ ]- b_ ".3 _ U ; "J llJ
t__ --J J ',..9 U U LJ U _ U ',..9 _ -J _J _-_ L_ __I _] L9 >- ""_. ](3 <.3 ,___ LLJ LJ_ >- LL LL] .'2] U 'i] CC
[q oJ L£1 C2, {'XI
,-4 0 C", C)O_ Ch 0 C, 0
{xl {'q c'xl
A
:E
d -£ _ 7£ ::- "£JJJ dJd
•_jd jjj
-J _J __I _1 _J d_ _,.J ._t _J d __l1-
I .'>- <C. cm L) f'_I I I I I
::£
J ::- _ :< :.< ".<" -..J _.1J ,.__1 __J
--J _..I _,.I J _.1 d__J_, J d_jF
11 F 1-- I-- F I--(-))- << :D U Eb+--'+ tl tl II II It
t-- ,--4 {xI t,q ,..,1" _{h
:(UUUUU
C)
0
r ,,.,I
146
ddJJ
._J ._JJJ
i-- I--L_] LL! i
"<- -}T
._.ld
dd._J_J
lz/ ,kL
©0
t-- b.--•._ <=PC C<+ +
_J _J-_I--I
--I-_I_J._J
1-- t--LLI LLII II
_D i__t.':>P'--U L.)
,-_ eq '.G -:_ _ ,,Dr--
L2 L._ L._ _J U U UI i ! I I ! I
L3 U 'J L._ U kJ U
-k 4- 4- 4- 4- 4- 4- C. C, O O OOr--4 c_ CG .._ u% ,49 t'_ o o o ¢_ o e _-_•--i ,".4 C,"l .._ u"/ ,.0 i"_ C; O CD O C.)C_L2 L) L_ L) U LJ LJ _ r4 _-t _-_ ,-_H 29II I1 1! it II II II 11 ii II 11 il 11 I---
>-- '_ ::[3 L..) _ bJ U- ,_- LL ,ILl :::::b L2 :.3 "d
-,_. J ..J ...J J _1.._1 .J_J _.t -J _J J -J __1 __1
FjJJJ IJJ
_>-<<. :_ U L._ 'LLILL::£ I I I I I I I
_J _ J d j ._I _ _]
i-- __]_J _.I d _.I _J_ _J;<:
_ >-.._ r._ U C] LLILL
:7-[0O0.0O0o
,_J _.- F- P-I--- P-- i- 1---- .< <--< --< _ _< <I:
_++ 4-+ + ++tXI
.-<
IIOt--i
r--ir-iOx
_J _J _J j _J .J _] G__J .J _J_J __I _J _] 0
_J --J _J _J .--I_I J ['4
t-- I-- H--l- I-- I-- t- O>... ,< :;3 U _,C._LLI LL I--II 11 !1 II 11 II 11 _:3
U U U U L)U U cSl_
co
Q
o
o
o
r-.-i
o
o
I!
H
I---F-
o4cO
o
cq
p-..cq
o
L_"t
Cq
tl
{q
_J
O. EL0 I
£D ,--,, ..1- H H
--I I--,-_ 0 l--k-
-:-< D_ + I-- I bII rt _ _-_ II II
O_ ,-_ H-,-, <9 32
<I',-4 {'4
147
_-t (NI _q @ tf_ ",9 _ 600", O ,-Ir4 r-4
J --I .J d / --,1--! _J__l _.1J
rl O- Q. l:&. EL Q. 13_ CL£i ,q (3..
1--41
t-- C,"I
II l-
F-- C__
f_Ii
F-
:3
it
J
7_
LL
U
ILJ2:
<( O_] ,-_L,q_ :_3
:3nd c9O
[L.
b.
-.1-o9
tl
v
(',,1 _
><: 0>,+ !h-- Ld.0 l.r'l,0 C,,I,DO
•,0 Ch,D tc'l,-40O',.0
• O
C),-0
# _',1c_!X-"< l-
4- cOx I# ILltP_ [P_
o *D
O LC/I ,--4
c'q¢q •
0q P..-
cq_
cq --_(q _"4
o+O'-Q
I
;_ bJ,'4 NX
-I-. oX cq
LD [_
0_ q3
NO ;<
II .J-_3 --3:3 !
Ii QJr-I
0"11"-'4
0_ H
q::) _' O ,"4
0 0 00", 0 _q- o ,4- o -4" o
O _ ,r-i C) _-q tf'_ ",-4 0
OH O I 0 I 0 I 0 !
4"r-i
0
0 I1 -3 LL. i-- (__ _L I-- (_ ILi-- :_) LL I-- 0L_ pz It_ ,-_ :£ L0 _ ::- tO _ 7.-_ tD ,-_ :£ LD
_ '.,0 I"_ CO G'_ C_ r'-I ,'4
148
''-1
J.J /J3. __ G.
.._ _r_ _0 i"_ (O O, O H _._ (_{o, {r_ cq G_ _q cq -_- .._ ....I+ N.-+
J J J J J J J J JJD_ EL O_ O_ LL D_ EL CL l:)_ E__
JJJJ
I--
i--1
I!
0 ;-_ ><.
,-4 tl H :.:: UI1 ,'_1 I1 II
1-- ,&ZI 0 -'.:: cxJ
:U IL C_ kJ "<
G-_ -..+r-.I ,--t
--.1-
_p
U
Oo
+e',lX
O
+
><
O
C_
t",l
5<
!
Lx_
el..
+
223
El..
I|
]
[!_
O _
C'_, ,-_ ++-1- -3
O_N N O'h
H C'_l N [_3 :]:1 ._J
0 _LL ,L Il'J ;:_ ¢_1 pal t-- o_ I L_ I .'._ ,--t
r_ :L] r-I LL __ ,-_ _j o ::] ::ZN LL I 11 ,-I 2D O I LL O H () LL lL_.
H II _ H + Z F 2:: II H- 11 t-- II D2-2 L] _ r__ +-, LL
0 ] _J- .L] II II C) EL'.J LL H LL _-_ X 0 ,-._
I--I
H
I.L o
t'-J I ,<"2 X
>2><:;<12[.
IJ- _-, o") O ,I.LI_=:
X U .-'-C _P_ I__ :_)
,q] 0:3 0 2] lJ.J 21 :_) LL II :4. {]J 2--"
LL 0 _ (L) LL £Z lsJ _!2 U_ _ _-( .'LLI E_ L'JO
N {q .d- LP, EL. N,_
F--
i---,i.
:>-
O
N
>- _I:
<[ LL
I.L ,-_
21_ _q
-J I--
U _
,,_ or)UJ ,_
4"
N t.t..
O
D D
- "=_ 0.J <_-
.-J _ "-_ 0_UJNI-- e.'_
I,.JN OC, H
'tJ :_- H o II;_ E'__ ;-£ _ C:; C, H "3
H _ (] ::>- II O 2_) O
::) __ t') *', rd II il ,--I 0
--J 0 l£. _" II (_.1
:D .'> _ _ bJ :.::, -_?. (+:_ O
•'-2_" t/) [L {"+ -:( N IJ_ 'EL.h_ F+_0 ,_ ,--_,
LL t.r--x
I--
I--4
149
H
::)h
LL
+>Z)
::>
-)H
H
_9_r.33
_O-_ -._._._1
t_U O_,'-q oxl --), X
__1 ,--_ ,'N1Lq _::-)D3 _ry (',,I t-- _t-- bLI bU_
O)- O Nb_.J I--C_ k _ IIJ II II ._,_ O :--( 0 Xt,_
0 C__.__
150
i
-i
t14#
Jav
xC_
A
c_l I, LL.
INI LL 11.
IJJ C'.I I.[. LL
I cqu_ + +o,.1 ._ ,o r_ __',_ EL- ,r, "_ "-) "-)
4- 0-; .'D_
g '_ ,0 ) II II_LL. _
LL. LLI 11 + +II II I.---_--__ --)-_
_--_ D_ I4_tJ_
O- 0
_0 _0
oc',,I,o
Lq0C_tOx.O
•", cG 4:_ ,",,I<j-OaF-E_#.O",lAj 'lAJ
•.O "_ d::) t:::_
I---
uJ '_)O
H_.I .-.!+ _. or) _
t%1
i--i
i--1
+H
v
:>
:)LL
v
7DLL
I.L
i--i
LL
o t-
(x] ijj"_,. ix,1
c0 E::_,--_L3 II -I+
•r_. O \ c¢3 ,-,I (_3 II Or) _..,+
Oi--- U U u'_r-.,l LU 1-- {_ _ :>- 0 ,_ C_.:.1o,,I <1: tl II tl ',D IjJ _ :D LLI 0 -F IAJ t--| O<_N C) H o I'_0 LL -,,_,_(':"l N II
--) -J H D--._ t--<(--) II I-- II II ',3 _ _ --1
tk II N _'- 0 I tJ IA_ LO 0 -3 _4J 0 _-_ LL.-:_:
C_ C_ 0 Cc_ (%
,O ,3 ,O
• 0Ii
od
H
b9
D3 + L_.0 _ _ _
._.1+ +,--I
LO, td O-- II II
_ <_: ,..0 d::_ Z:) cxl o'_II II 2:: -t- +
(-:-,_ O_ _--_7_ Z
C) >- C))-- O__4) :_ L9 L__ U c.9 ',/)
O u'_ O-.t -4" C;
7_UJ_0__LLI
--) ii-oic_cJ_
U__o
Uc_
_U
cO_-_o
_-c_
13. o_jc_
LLI
ZZ_0
•-...10 Z
0
LL
I--
,---!
I
1---
I
-- I I I
c,,l I-- _ c_ I--U Ci_ _ +'-;_"I_LJ ' .i.
•+_ ,-i I ÷ ,-_ + -I-
F.+ -.? _L_J _ I-- i _ 'L_I_ __ o +--+,---.+ .,-__ C_)b-- <,I-+-I --
-_ -I+ I-- I-- "-) I--I-- Z) <--C,-.i_I-""-)-,-" _::
II {L II 0 II _L tl I1 I,I II C,I II LL tl II LIJ-_
r_
_J
0°7
Era_"
O[L
I--
i--i
151
E_
LLJ
152
fxj (xj
H
v';._ .7 o.I_I_,,LL LL '_,_
_-I II II ,--III _ _ II _. ILl
"-' ,--_ P-, -') "-) i)
-'_ O_J _ OI--- ,--_ D__II N d _) +-41--- F-- )
@ './_ 0 _ :.:_ O-.Z 0 LtJ:_
o4O O,-4 C'4
_Z
-M
I---I---F--
H
L3
-.)
JJ
.7tL
".I"J(:30
L'J
1--4
-.I- F-.._I__-2).(300 c_d
b30LL
t--
J
,, .j
-,1"-,I-
LP_
-"Z
-,1" b__ _-_. Z ,",J 11
.,_ ,---I 11 II r--I
."._/0 _J D 0 i--,--_ E_
" 2£ ._D D _ ;-" F- r_,0 t,./_ 0 _£ 2£ 0 "_ O DI 7£
oo
lPll_
2._/ 0
I-- ¢,_ l--I-- _ xZx/b-- ,-__-_ xZ
2.) _ r-i
IL. _ 2-.}
",D2)._ID_
_ CD r-I
-.1_ -Jd_ --J JODD
. ":2-- _LI-D-
'--1OO
{M
0"--1"
e_D,Jc_
;g
_"/_ _ {'XI L_ 0,,.I
0,,1{',.I
U_ II
4"
(M I|
_ 5Z .-! D _'-.,£
-_l 0 _--I II II _0 11 _ _ II *, L_J
-J Z} u'} ,.,') L_ O x/ .... _ ._0 2£ ;_ {q ."ZO _1 D 01--- .-._(
0 Pr£ bJ "_LJ_=_. _ II cXI _l 2-) _q _-" I-- __
C}
I-- :,'_ ".',/ D _) I-- 2£ t-- EB•-11 :---I _x_/ O _ ;_ O x_£ OLD' ;_£
r-4
o o,--i {x]
153
:_ -J TO --/_
_.-/ _ ;:-_ o,I II
H II II HII _ _ II _ IJJ
:,z .... ---2:7-b/- 0 --.] D 0 t--.- t--.-J D-"
II _,I --J :D _-I F-- l-- ::::)
_x< (_) _ :._. 0 -"d C) QJ ;3_'-x_ ._ 'LL LL C:) _ [,,_) D'-- LLJ
O O,---I ('_.
154
t-_ l"- t'- c_ GO o_
1"'4
-:_ o,1 I!
II e, DLI
2} I-- L-_" I--- E__ ?.-
LL £'_ _ U Df l lJ t-
O C3 >_r"l ,_,1
X
F--
f---Z)0_
I"4
Ii0
0
F--
Z
__!12__<11.1
0£0I.L
LIJ
72
,_ :-Z
<Z'}
.2"F£l_J 0:._ h_7£ t,J0 _.]U
":Z{._9 LfJ:£-:E
0 UJ
--JO
LI_ _J0 __
:c£
<d
212
UU
0
t_
0
t",,l
(D 0 0 _ L_ 0 c_] ff'_ _0 _
g e _ o _ O o cot"..- o('q o o _ o o _ o o 0"]
P_ _'_ 0 0 O O u_ 0 c,,I ..t_ G'_ 0 P_- 0 0 cq P_ 0 0
A d'l co _ ,..o ,o o ._+ col o
LOI r-4 r-_ 0:} _1 r-'_ C:O _l .-'-I O0 ,-'--I
0,/
•._ o c) ,-._ 0", o I'--. 0_ _---_,4-
co o _ • {Xl o e {XI o • c'_
@ C} O O OrB O O O,O
o {q O (xl l_ _,.£} _-i o o O1"- o o
H o C<"t rl o H r--t H ,--4
q:)0
,2} (2) _ O C) O _ C3 o
u k_) U :J k._l _XI _ :_ ,_ " 0 _ o (, &. ,--I o .. C", +H
O
Or-i
oo
ooc3¢2;ooo
.--4
o
oH
o,--4o_-4
oH,o,-4o
o1--I
:o2D(2;:.)O.'+2.
o.'D-+2
D
Oc_coi--4
o
o1-4
oI-H
c__--4
o_4
C_,--I
c)cTc)oC),oc-';o(2)HC)I--t
C2;HC)I--4O,-4
C2_H(2H
or-t
o
v
c_ ,'xlI i
DJ LLI
ooo p- p_-
oc0c0..J-o oOH_-I,,xJ++-i_--I
0 1 1
o
o
c3
,H 0 C}
,-40
0
0
C_D
0 u'_ I"--
o
oc_o
<i"c_]p-
c.)_9o
2.) ,0 ,o
155
I i I ! I I | " | I i I I I I I I I I I I I I I | I ! | | I I I I I I I I 1 | I i
[_ LLJ _ t!J !_J !LI IW LU Ld ,LLI UJ LLJ _ _ L;J L,'J LI.J I,J L_J L_J DJ U LLJ Ld 'D] L'J LLI ILl Lt! [tJ [L] LLI [d W DJ _ _ IJJ L]J
]'_ I"- I"- "-._) _ ",.D ,,_ _- [_ _ I"-- I""- C'G @ kO _") 0 "_+ 0 "_ _ 0 0"_ CC'p0"3 c"_ [q (q (q r-4 r-4 _ 0 _'q, (q Cq {*Nt ;."4 r-1%)
cO cO -d" 0_ -.4+ (q co :_3 c0 CD c0 ,-_ C_ cq -.1" -_ O 04 c.0."4 _ cq O', ah L'_ 0". _ G, Lf'_ L_ O C_ t"- I"- _- ,-_ ,--! _0
I i ! I I ! ! ! I I I i i ! I | I I ! I ! I I I I i I i i I I | I ! i I ! I I I'LLJ 'LLJ 'LLJ [_ L'J I__ W LLI _ LLI ILl _ LU L'_ L'-I LzJ LL/ W N N N ILl LtJ N LLI N _ LLI LLI bJ IJ "-tJ LLI _ bJ L_ ld It] [LIW
156
o
IKI
r4 ,--I C'40 ('4 O,--! C._ "'4 0" 0" ':.q ,--'10 ,'-40_ O OG" G"OO _-_ ,--I ,,"_ ,'-40 L,-"' OO O O'_ 0"_Dd _'_f_I _ :,4 ,,_ _ _ C_ • ; , : 2,: Pj ._.'p' _,, ,_,!r_1 _ ,-4 r\:P.I r'_t _ D_ :_.] P'.I ,-4 _ C_l _ r_l rlI I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I
L!J UJ i!d I.d ":J _ LLt LIJ ,ILl LU l±J LtJ l'.J [_!J hJ '.lJ l:J LLI ILl t].J ktJ [,J !d LLI EJ LLI _,t LLt t,I Ld IJJ t,J LIJ
Cq "0 _-4 (c/ ,.--I rq ,0 _) C.J _'_ _ ,---I _.O c_, [,"1 r--I C4 kO C_ l"- C, '-D 0,"- '_.) 4- O_ '0 C"I Cq _Q t.'3 t"-U', 0.3 "4:) C', -',1 _:_ @ C_ ,-4 _ c.-t ,4D -d" U', C'.J ,.{') col O", _O (q g_ 0", 6", (.'q @ 0", _, O"_ Cr_ Crt 0", ".-0 0.'1
I I I I I I I I I | I I I I I I I I I I I I I I I I I I I I I I IIU LLI ,LLJ_ iJJ LLI LtJ LLI LU _ DJ _ DJ 'LLI DJ ._,J l.!J LLI LU tlJ Ltj L,_I LU DJ D_I _J L;J PJ _ _ _ _ 'ILl
O.
o
L_
0
0
0
LLI
:Z.:::9 W_
Z-
l0L_
15Y
o
tot
o
Is)
.TZ LetIaJ
i---O
I---IS)
+_ trl .
< OD"
O0<
LCl
>::OH r--'l
_o:-Z I'-- o
o LLI U)
LLt U_ LLt._.l C_ tr%
<1: tr) 7_)I-- !--- 2_.-:
<a:w-'vd.+_ I-- OO C_d__. _IL O LLJ LLi
LL .b: __3cO UJ : Z
t--t-- -7) u_o_" Z] ,-4:-< (_. I---L) ;_ t/) ;.'-:
•< r< :DOc'< o_" ..-< _1<.¢ 0 LLI 0d::_ ,LL >-:: kJ
t_
k.3 L) tO t_) L)
158
+ +++;++++++++++r--'l H m"f r-4; _I r-4 _--'1 ,-"4 ,-H ,'-I '-4 _ r--I e--I
+ ++
(D OO
C} OC)
O _h Cx,
+ ++t 3 QO OD
+ + +
O C3 v,D
OO..&..'C)C) (q
cq .._ ;'---+ ++
G"_ Cr', cO
°r++
o(2,,--4(-D C) ,--_(-.)00
_-.i (.,.j
+++
C7, 0 0! ! I
5-) 0 _{
23 :D
c,j -._3 C);-4 o3:q+ ++,T -4- :_
O (7.)0
+ + +C) '__'7)[q() ,--t f_.h,-9 ,--I O'_
-.1" vO _-I
-F- -t- +4- .4-
C) CO O-F ++c) C) O
:-) .;D C)
+{+ + + + + + + + + +I'_: (q _O ,-q cq I"---,'.0 ,----I --.t" @ -.&
'.:n{ ,-4 _ "-D r'..i o'] ,"q -T cO Cq ,...,3,-9: O <% ,---i O cO _ I'-- v0 v0 :__
-._i(q P" ,-4 P4 (q o3 ,-4 cq r-- ,-4+ +++ +++ ++++o0 :O ,:z) cO cO (O cO 03 i"-- i'- r_
+++++++ ++++_'_ ':q ,D cO o:3 ["- '.£', ,._ O ',-,% N..Cq (q O,I ,-.-4 Lr/ -<i" 'O u-_ I"- u% O
,D ,.D _ <3 aq _ ,,', .._ ,_ Cq c"l:0 _ ,.3 '.i'_ _ vq q'd _4 &3 -T ("4
+ +++ +++ +++ +
r'-- "Q) '-') L;."_ Ur'l_i- "b= tq (X_ _
++++++4-++++
('q i"-- ,:3 I"-- O_. -._ '2,'. t,:'_ G", O'. -..1".'4 (q -a _ C", 04 vO r--- :O _ @ L_
0.2, <20,---_ ,-4 C) ,".'I ,--._ r-_ O .._:'4 -q_ ,--4 (q ,-4 4- _ ,4_ H ,'q ,:0
++++ + ++ ++++-.'2" -:_ ..-k <.t -.1" t_ {:'_ _ u% '._h _:'t
O C} O (D O O C) O O O O
i I i I I I I ! I I I.:_, (._-) (-'t ,.-.-I _ O :'q O O ',"d ::"1C) L% 03 cO (D LG '_D cq I"- 0 ,0
++++ +++ ++++
L_% '.71 [('_ iF', L,'It2"_ '-,.'-_ :.'_ t('}L"I '-"_
(D 0 0 C} 0 0 CD (D 0 0 0
++++ +++ ++++
cq t_ ,.00m 0"_ co O :" O t_ rq?q (,'_ :_ a r't _,% 0", ,O O ,7.m j.)<) P'-- (00 C_ tS_ C', Ln (M I"- C',,I,4 ,-_ ,-.4 o,d cq ('4 c'.J cq -t- -.t- ._
+ + + + q + + + + + +
O O ,'2, C) C_ (_" C), (7:(_30 O
+ + + + + + -F + + + +O C) O Cb .0 0 (-3-0 0 0 (.3
,.h C) O (D O C O O O ,_'7)C;
,:+) C.) O O O ,--4 ,%1 ¢."1 -@" :C_ ,O
cO cO cO CD O3 _ o"I
r-_ r-'l r--,l r-I r--I r--_++ ++ + + +
(W vO co O b- N O
.J- _ (h _,.! LEO &o O
O (7, (:3 :O v0 u% O
c,.lr-d ,-4 ,--I,--I ,--I (°i
++9+ + + +co o:) CO uD c,3 I"-- (q
r4 r-'_ r-4 m'l ,--'1 r'-I
++4-'+++ +Iz1 G_ H _% --_ O
++++++ +
•.0 ".9 ".0 ".C u% ta_, C',,--.4 r--t _J. r--4 rH r'-I++ ++ + + +
,--_ l'---'..a_ co col oh O
<k <,;- -i" cq C, cq C):0-<_- c'J ,-4 ...1- _ ,.z-,,
++2+++ +
++ ++ + + +
,.3 _ .fF <]- 00 C.) O
c..I CM P-- ,.0 C'.l cA cO
('G i"--- ,--4 .._ ,'%1 _ cO
++ ++ + + +
I"_ -O ,.0 v9 "-0 uh '&,
+ + ++ + +('4 I"_ oq (q _ _ C2,
d'_ ,D ..0 u'_ ta% ,-4 C.
."40 ,-4 _i_ O O ('J
+ + +-F + + +
r_ 0 C_ 0 G', CO (D,--_r-_ r4,--40 C ."4
++ ++ + + +
O _.3 -_ _O G', ',,"_ O
L:'_ cO -4" -4- u'-'_ -:2"FNI .:.O (,] H C% _:l
+ + ++ + +
•.% IL; .5"t :_ L"_ u% C',JO C) O (.30 0 0
I I I I I I iP- ,D Ln c3 cq _ OCO C) ."3 (.3",_ C_•.O _ ,---I _ ,O ,-_ -.t-
oG (q ._'q .'%1 _ C.J _q
++ ++ + + +
C) O ,'D C', d"D O O
+ + ++ + + +P".- O', D-,.D I'-- [_ O
:,.'_ u"_ v0 ,-.'3 '4:) cO t,q+4-++4-+ +
L'h Lf'l :.% L_ ;Dl ::% ,-4 @O ,"-;. LD ,:} ':--:-_ O O ()-I- + + + + + + +
C) ".:3 (D O C] C) O C)::3 (m. :D 0 (-) :.'2. 0 C>t-..- ,O O'_ ,LC)C'J [r_ O Lq
r"_ _--I r-I ,--'I r-4 r-I r'-_ _-"I r'_ rq _--; r_l r--_ r--{ m-I
+ + + +.+ + + + + + + + + + +
O O I" OJ t£i _ O', C_ t_ Ox G_ _ C',,I cO coC) O 0". o-_ (q ,4") L_ _-I. _. <I- 0", CM r-I <110 O-.i- b-- cq .d- D- ,'-,.I ,-q ._ c9 c,.t q-_ u_ ,O
C,.,I 04,-4 ON GI cO ,--I(,_ch _ 0-, r.4 ,--_. ,---1 r.-..(
++++++++++++++4-
-d- t_ _ p-.- i"-- r-,- cO ,_ o.'} oJ oD C:', O', _ C',r.-i r--I H r_. _--I _--1 r-t _---I r--i r4 rt t--1 r-1 r-,4 _i- I
+ + + ++ + + + + + ++ ++ +C) OCO :_ ,_ t"- eq OO, i'--- -,.1- C', c,,I o3 tr_
O O ,O ,"4 ('O H vO cO -._ C', f"'- _ ,_ cOO O H O (q t_ 1"_ O cC. CO C', O e,,,I ,_ (q
+ + + + + + + + -I- + + + + + 4-
G, O' O"- 0", Ox O_ C_ _ O, C> 03 cO cO cO cO
+ + + ++ + + + + +++ ++ +C'_, O0 0",..I" o% F--- ,--4,,'4 _% .Q ,-q I'_ tr'_ ,.0
C) 01"-- _q \D CO ,.O f_ &, H C_ CO P--- '-D
G", ,"--- O _ O 1"'- t_% cG ,-4,' O _..q , O 0 _O ti_-.1- (q {q ."4 ,",d,--_ .-4 _ ,-4,' H _ _ t:% o'_ {NI
+ + + + -t- + + + + + + + + + +
""_ _0 "? "C "? t"- _- "C' 'C2' ,m. ,_ .? .+ r-_ _.
+ + + + + + + + + -F + + + + +O (D I"-- ,O O _;3 C3 L_'_ C,J ,.O c:,.) cG cO -..t" O,J
C- 0 cA I"'- _ '7,_ Cq "-O ,-4 C) _5 o,d 0"] .,._ cO,'%10 :C'_ _x _C_ O 0-J _ tG [_ t_ _ C00q ,--4
03 I_- _rl cq C,J _ _ :n ,--I _ H U", ,--I ",0 C,J+++ +++ +++++++++G', '_ (% _ U-', 0", O', 0", C', C', G', G', O_ tO :O
+ +++ ++ + +++ +++ +++
C: 0 _ _ O cO .,,T (3", C', ,'q t'q O _- Q3 C,d00_I- C", Ch ,-D _" ,--I cO vO r'*-- ,.o t._ I'-,- c',+.t
0 ,O u_ ,---I ;:] co ,--I .-I" ,.D C', ('q ,.,9 r--,I 1"" ,--I
+ + + + + + + + + + + + -I- + +
O', ._ L:', :3:0 _ r'-- '.O v9 "42, _C_ :.% i_ ,@ <;-
-I- + + + + + + + + + + + + + +(-_ O C'4 -_ v_ CO _O C,", (c_ (,q Od D"- _0 ,a% ,O
o o c4 4- ::.:_oD _ r_ uq cfl c-, N c,,J,.n ...,*O O O q3 :,5", 0 @ ,-.,._ ,.rD (',I P"- CX) -.._ r-"l C.r', ,.OH I"_ 4" _ '.q _ (<1 ,--4_O _q _q cO ..& ("J O', -:a_"
+ + + + 4- + + + + + + + + + + +
0d ,'q :','_d] .c-, ._h '.'q_ -'k 4- -_" -T -._ -J" <-
000 0(2, 0 0 LO C3000 C) (3 0
! I I I I ! I I i I I I I I I
0 .'D ,.O :.J ,-4 C_ D- -4 vO 'v3 c,', o,J i_ <OO', ,.aq r'_ _: -._ o-4 vO c_ cO --.1- C> k_ _ :,O ,-0
C} L_ <.I- (;D L_ o'_ ,-_ 0 (D G', cO O'_ ,-4-..,+ O\
r-'_ LPI N :-4 ,H ,--,! H _ _ ,"- '-D tf _, _% '--_ ("_+ + + + + + + + + + + + 4- + +
O .:) 0 0 0 00 C:. O 0 0 C) 0 00
++ + 4-+ + + ++ + ++ ++ +C30 v...r.o,'O-.I- "-O -.T u'] -_ ,-4 c0 ,-4 (q c-O vO
i"-- _r'-_ C) :f'_ :O ,--t -_ ,"_ 0x '0 0 (q :-4 .-++1 (]
,--40 _ ,.D r---- o', OH o] ,.0 O.-t- _ q'- C',o3 .4 _-q ,--I ,--4 _'-,.I :NI c,d {',.1 Cq (_ (q _'- -q_"
++ + ++ + + + + + +-k ++ +@ _ @ -d- 4" <,_ IG ,_ :fx t,'-'+ t£_ _ t;3 iG l_
Ck v-) :L) C) O ,:Z+C_ C; CZ: <--)C) (D (-3 _D O
++ + + + + + ++ + ++ ++ +
(] O O O <D (D O O (b C) (7) O (D <D O
". O C> (D O (-) C) L-_ (-) C: _.%- (.) (7) CD (Z_(D 0 t-) tD CD O O ,--4.c,j (,3 -_ '_,', ,O P~ o3
r_ ,:2, P-- -') _ r--I _4 ,-4 r-'l ,--I ,+4 r--I _ ,-4
-159
\
,--i c_lc_
IL LL 11_
bJ ILl I_J
Z
2--"O
__I_I
OLL
LLJ
ZI--
>-EQ
t'JZl--a
._Ir_
U
E2
<( o_Z
EJ F%L9
LgO
_X_
I-- 'LL C_J
_) C_JIIJ(19 t'_l ;;_
IDO_
F-- !
,--, Z LL
_ 'LL. XX t_
bJ
ILl Z II
IJ_O4L_
F--
Z
DL9
r_<(
<(
-_< LLI
,',__
O4 O IilJ _', 0_ 7_
;;_ ("4_C) N
L_ U L,J Sf [L _ 7-: LL0Q_ ,--I 0
(7",t--
U L) "1_
LL It. U_ It_ ',0 _ LL LLLL LL LL LL('_I 04 ("4 C_IIL N 0_I NN (_J ['_INLI] DJ I1! LLI C'4 'tJ 'LLt ILl LLI LLJ ,I.%1LId
UJ
161
_leoe oeel •.......... _....._ .... .................
1 :
i * i
J i
!_o _ tft
¢o o •
eo _-,
,f) ,:h
o.-4
f_
tM
c_tM
uft
..4
tfl i(D
I
._ . .._$
I
r'-I _:o O_ ct .-4 t_l _")
, 1
$
g_
c_
o
.4
n
164
I qI
IIIIIIIIIIIIII1'1111
IIIIIIIIIIIIIIIIIII
_oeeee=ee.oe_oe •
_RNNNN_NNd_NNd_d_d
,-4 ,-4 .-4 .,.4
f'- _.3 ,_ UPI
I
.,; ,-; _; ;I
;
I
U3 _.'1
..1" ,,,0-..4..4
I.,-ZI--I
I
,_,.4
I.II II
eeeeeooeoeo
_2__!
eeeoee_eeoo
! !
e=oeeeooeee
I I
co+÷
eeeeeeoo=eo
_09÷
eoeeeoeeooe
#÷
eoeoeoeoooo
e.eeoeee.oo
oeeoeeeoee.elOll
165
I 1 I I ! I I'll It
It t! Ol II il t| 11 II
_ ;,", ._, _ _ _J
',_ '_J ,_J
c_ c_
II
ij..
--4
J
.'1
;I
0t.d
J
I
c
It
q._d
N
_J
ll"
4.1:2
g
II
T
,-j
fL...;
uJ
7t.-4
.D4[
166
!
LL. ......... IJ. tL _,-L',t..... I' f_. ,0 P'_- _C_ _; :_I""._,_Ilg_
J J .J.J J.J-,,I .J_ _ !
Z¢J_ 4J'_
I I I I I
0 .:." _ ,,_ 0÷
I I I I
=r
i I
• • 2 I,,- ._
eeelloeell
_ ,,,. ,,i- ÷ .,_- .t- -p _ 4..t..,-t PC) i,_) .-_ rc. (_i ..-T ..,* _ (._
• • • • • • • • • e
_''',---_--_
• • • • • • • • • •
! I I I I I I I I I
_:___ _ -IIIIIIIIII
l_l_llllll
_'_ _'_-_L_-
_..r_.f...p.p,.P..l_.p..p.. p_
J_ _._i _', _'_ .'_ ..., _ C_ i__. .'%
. • • * * • e • • •
_-" c,J ,._-.oc: --g ./_• _:
168
oee._oeoe_e.l.eeeoeee
e..o_e.OeOle_.QO.l.ee •
11C.C',42
i;
bJ
Z.,
0
/_
e, _j
. _1 ...e -._ ..4 -e :_1 ¢_1
.J
tJJ
_=.
.j
I
r_
3
I
g.
..+***+._ +
eeeeeeeeee ee
_. :_ _ _ ._ ._ _ • ,C ,_
II
,j
...*
I.-
.it i!
O2:
•_- I I
169
,..4
i
II
,.4
c
.?CC_
_- c(%
I
a_
3e__-_,=_
p_
IIt!!Ii111
+++_++++++
2 ._ . ._ • •
•e•l••eel•
_C _!
170
III _III
Z0
I--
..I
u_1
u
Z0
.J
I
.o.°..o.._ I|
I ___
_ ÷+÷+÷+++÷+
o_.o.ooeoo
IIIIIIIIII
÷ IIIIIIIIII
• o.e..o.o..
+.,.., #..
oo ÷÷,Fit I I I I +
I
0._1"
8--o..J ¢,_
"', .'z'4
I I I I 4 I I I I I
I I I I I I I I I I
• • • . • • o • _ •
÷ Mo
_JN
I!
,JbJ
I--
,,F).f)
II
l--ZU
:::::::::::::::::::::=
||
+*÷÷÷÷+*'41111111111
eoooeetlloeoloooooee
llllllllllllllltllll
.o.olelleoooeeooooo_
II
¢_g@gg@gg@@@gggg@xg@I_IIIIIIIIIIIIIIIIII _
!!i!!i1111111tlti!it
1--
J
II
JoA
171
172
II
>-
_J
+
II
J1,1
II II
0
,i .%1J
II Q,. k/1
Z Z'" "2:)l n,-
c_ ,,,,=t +,,,I C4'1 • ÷'4. •t,-+ ,=t. [,,-,e .0,
I-.C" I-_
CL i _.. I
C_r _-, ,"- .=,-.-¢ _._
_ | _._
C:_ @ I • + I •
-'., C, I/I
'4- Ld _D Id <_I
;._ t,l I
o <o- <=_
i-- @ I • 'el •
_F I _" I,m_p (lip
C_ l.,O .,-+ t,.O +W_
*"+ _ IU") ,,,,+L/1 re)
II •
,.4 ,,.,,
°i:-'• w | •
n _
N.
,d
u'lr_
I
I!
I I.'I
W.J._I
e...
N
1-/
!
i!
o
¢0 I_ .,-*
+
c¢_
I!
r,-
W
r_
O"
II
Jo_ III
II ) III"4 p,. "-'r"