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e>|V[EMORANDUM RM-2826-PR NOVEMBER 1962
I THE THERMODYNAMIC PROPERTIES E rj AND SHOCK-WAVE CHARACTERISTICS OF CD r- A MODEL VENUS ATMOSPHERE
^W. C. Strahle
A S T 5 A ■ .■■--'■
• ä i i'.
a
PREPARED FOR:
UNITED STATES AIR FORCE PROJECT RAND
mwd Ifa rv"H I I VJ (fyvifi&tatccut SANTA MONICA • CALIFORNIA-
1
MEMORANDUM RM-2826-PR NOVEMBER 1962
THE THERMODYNAMIC PROPERTIES AND SHOCK-WAVE CHARACTERISTICS OF
A MODEL VENUS ATMOSPHERE W. C. Strahle
This research is sponsored by the United States Air Force under Project RAND — Con- tract No. AF 49(638)-700 —monitored by the Directorate of Development Planning, Deputy Chief of Staff, Research and Technology, Hq ÜSAF. Views or conclusions con- tained in this Memorandum should not he interpreted as representing the official opinion or policy of the United States Air Force. Permission to quote from or reproduce portions of this Memorandum must be obtained from The RAND Corporation.
m n D* 74£ HC-H I I \j ßonfeanatunt
ill
PREFACE
This Project RAND Memorandum extends the study reported In RM-2292,
Thermodynamlc Properties of Carbon Dioxide to 2^O00oK - With Possible Appli.
cation to the Atmosphere of Venus. Whereas that earlier study assumed an
atmosphere of pure carbon dioxide, the present study assumes an atmospheric
composition of 85 per cent carbon dioxide and 15 per cent nitrogen by volume.
The calculated thermodynamlc properties of this mixture, as well as the
normal-shock-wave properties at selected altitudes, are presented as an aid
to studies of the aerodynamic phenomena of high-speed vehicle entry into
the atmosphere of Venus,
ACKNQWIiSDGMENTS
The author is very grateful for the many instructive discussions with
J. L. Raymond, S. H. Dole, and F. R. Gilmore. The assistance of Jeannine
McGann and Cartslyn Huher with the calculations and graphs is also greatly
appreciated.
vil
SUMMARY
In this Memorandum a model for the atmosphere of Venus is developed
which may be useful for design studies of early atmospheric-entry vehicles
and for wind-tunnel simulation application. This atmosphere, derived from
many questionable assumptions and scant experimental data, consists of 85
per cent carbon dioxide and 15 per cent nitrogen by volume.
The thermodynamic properties of the derived atmospheric composition
are presented over the temperature-pressure range of l50oK to 24,000oK. and -h 2
10 atm to 10 atm. To further assist in aerodynamic entry calculations,
normal-shock-wave characteristics of such an atmosphere are also presented.
The fact that nitrogen is present has Important consequences in the thermo-
dynamic properties and electron concentrations over the full temperature and
pressure range. The effects are of the order of the percentage of nitrogen
addition to pure carbon dioxide.
A graphical method of obtaining obtaining oblique-shock-wave data and
a method of obtaining electron concentrations at low temperatures are in-
cluded as appendixes.
ix
CONTEMTS
PREFACE , üi
ACKNOWLEIX34EMTS v
siM>iArar vii
LIST OF SYMBOLS xi
Section I. IMTRODÜCTION 1
II. A TENTATIVE ATMOSPHERE FOR VENUS 3 Conditions and Assumptions 3 Basic Data 2; Theoretical Development g Model of the Venus Atmosphere 12 Discussion , j7
III. THE THESMODYNAMIC PROPERTIES OF 15 PER CENT N„ AND 85 PER CEMT COg BY VOLUME 7 21
Conditions and Assumptions 21 Method of Calculation 22 Discussion i^c
IV. NORMAL-SHOCK-WAVE CHARACTERISTICS OF A TENTATIVE VENUS AOMOSPHERE 1^
Conditions and Assumptions W Method of Conrputation l^Y Discussion ^ _ # l^o
Appendix A. TABLES 57
B. THE COMPUTATION OF TWO-DIMENSIONAL OBLIQUE-SHOCK-WAVE CHARACTERISTICS FROM NORMAL-SHOCK-WAVE DATA 90
G. CALCULATION OF ELECTRON CONCENTRATIONS AT LOW TEMPERATURES IN THE C-N-0 GASEOUS SYSTEM 119
REFERENCES jp^
H
o
in
n
xi
gYMBOLS
2
specific heat, - Bt^ 55-, ~^r ' lb-mole ^^^ g-mole OR
A area, cm^
C
E specific internal energy, cal/g-mole
e specific internal energy, cal/g
F Gibbs free energy, cal/g-mole
g acceleration of gravity, cm/sec2
specific enthalpy, cal/g-mole
h specific enthalpy, cal/g
E specific enthalpy in Ref. 10, Btu/lb-mole
:Kp equilibrium constant for partial pressures
L Avogadro's number, particles/g-mole
M molecular weight («f mixture if without a subscript), g/g-mole
molecular weight of "cold" undissociated mixture, g/g-mole
mass, g
number of g-moles = £ n. 1 i
p pressure, atm
p standard pressure, 1 atm
0 polytropic exponent
R universal gas constant, r §|.u - . ca:L 7 lb-mole VR' g-mole ^K
r radius from center of planet. Km
S specific entropy, cal/g-mole ^
s specific entropy, cal/g ^K
T temperature, 0K
To standard temperatxire, 273.l60K
xil
velocity component normal to a two-dimensional oblique shock wave, ft/sec
velocity component parallel to a two-dimensional oblique shock wave^ ft/sec
w free-stream velocity in two-dimensional flow = -vAi + v , ft/ sec
ß two-dimensional oblique-shock-wave inclination angle to free stream (< 90 )
J ratio of specific heats = C /c p' v
6 two-dimensional oblique-shock-wave flow deflection angle
p density, g/cm
Po density at p , T , and M , g/cm^
T entropy function tabulated in Ref. 10, Btu/lb-mole 0R
SUBSCRIPTS
E earth
. . th 1 1 species
0 standard temperature, 0 0K unless as specified by T
p constant pressure
s surface of planet
V Venus
v constant volume
1 state upstream of shock wave
2 state aft of shock wave
SUPERSCRIPT
o standard pressure, 1 atm
I. IMTRODUCTIOH
In oräer that design studies may be carried out concerning space-vehicle
entiy into planetaiy atmospheres, it is necesseoy that the atmospheric compo-
sition-pressure-density relation with altitude for a particular atmosphere in
question be known to a fair degree of precision. Since shock-wave, deceler-
ation, and aerodynainic-heating calculations are all dependent upon this re-
lation, it must at least be bracketed within reasonable limits. (l) This study
is concerned with estimating this relation for the atmosphere of Venus.
Unfortunately, very little is known concerning this atmosphere. Through
theoretical argunents, spectrescopic, phoWtric, and radiometric measure-
ments, several theories have been advanced, no two of which, however, appear
to be the same.(l-8) An attempt is made here to bring about reasonable agree-
ment between two theories that are approached from different viewpoints. The
resulting atmospheric model could then be used as a tentative irean which could
be compared to other theoretical and experiiiental models until more experi-
mental data are gathered and a better understanding of the Venus atmosphere
develops. The establishment of this mean model atmosphere and its therao-
dyna^nlc and aerodynamic characteristics is the objective of this study.
The^ first theoiy on the Venus atmosphere which is considered is that
The partial pressures given in Ref. 3 correspond to a per cent
by volume equal to the reported per cent by weight. Since the molecular
veight of N2 and C02 are markedOy different, this cannot be the case. Two
methods for calculating atmospheric composition have been employed. Using
the basic data on pp. h-6 and the theo^ of Dole, the probable range of
atmospheric compositions has been recalculated; a probable average coBiposition
of Dole '3^
according to this theoiy has also been calculated. The second method em-
ployed makes use of some of the available experimental data, one of Dole's
basic assumptions, and one assumption concerning the pressure-density re-
lation of the Venus atmosphere belov the cloud layer. The assumptions in-
volved are open to debate and the experimental data are highly uncertain,
but the remarkable result is that the two methods yield quantitative esti-
mates of the atmospheric composition in close agreement with each other.
It is taken as the basic statement of this Memorandum that these methods
do yield a reasonable quantitative estimate of the atmospheric composition.
The composition arrived at is 85 per cent CO^ and 15 per cent N2 by volume,
and the atmosphere is then modeled; i.e., the pressure-temperature-altitude
relation is derived. The thermodynamic properties of the chosen composi-
tion are calculated over the temperature-pressure ranges of IJO K to 2^,000 K
and 10 atm to 102 atm, respectively. Finally, normal shock-wave properties
are presented for selected altitudes in the chosen atmospheric model.
The appendices include a method of obtaining oblique shock-vave properties
from normal shock-wave properties and a method of obtaining electron concen-
trations at low temperatures. The first is useful when it is impractical to
compute and plot oblique shock charts because of uncertainty in the charac-
teristics of a particular atmosphere. The second is useful for determining
the temperature and pressure influence on the electromagnetic environment,
for example, of radio signals to or from an entry vehicle.
II. A TEMTAJIVE A3M0SEHBRE FOR VENUS
COMDITIONS AND ASSUMPTIONS
Two methods will be used to deduce the corapoaition of the Venus atmos-
phere; one is that of Dole, -while the other is based on a few selected
experimental measurements. The first approach •will merely require that
slight numerical refineiaents be made in Dole's paper; the trace of argon
mentioned in the paper will be neglected, but the theory will be assumed
correct. Ihe second approach will require assumptions concerning both the
validity of some highly speculative experimental data and the pressure-temper-
ature-altitude relation below the Venus cloud layer. More will be said later
concerning the acceptance of these data and the assumptions involved in
choosing the final model. However, before continuing with the derivation
it is necessary to state that this is not intended to be a rigorous, absolute
derivation of the composition of the Venus atmosphere. It is solely intended
to provide some justification for assuming a particular model atmosphere.
The discussion at the end of this section will shed some light on the diffi-
culties associated with the acceptance of this model; as for acceptance of
soiie of the experimental data, the reader may consult the referenced literature.
First, some general assumptions pertinent to both methods are neces-
sary. The constituents of the atmosphere will be assumed to obey the per-
fect gas law, p/p = Kr/M. The gas mixture will be assumed to be of con-
stant composition throughout the atmosphere; this neglects "settling" of
heavier constituents, diffusion, photodlssociation, and ionization. The
complete gas mixture is assumed to be in hydrostatic equilibrium in a
constant gravitational field, with centrifugal force due to planetary
rotation being neglected. Now, as pointed out in Ref. 1, all of these as-
sumptions except the perfect gas law may be far from realizable, especial-
ly at high altitudes. However, they introduce far less uncertainty than
do the assumptions to follow, and they are fairly good assumptions for the
altitude range of aerodynamic interest.
Following Dole, the ameunt of N in the Venus atmosphere will be as-
sumed equal to that in Earth's, but scaled down to make allowance for the
difference in the sizes of the planets. Since the atmospheric gases are
mainly generated at or near the surface of a planet, relative masses of N
in the two atmospheres will be scaled in proportion to the surface areas of
the two planets. Moreover, the Venus atmosphere is assumed to consist of
only CO and N , the trace constituents being neglected. These two assumptions
are basically deduced by Dole, and they are basic to both the numerical re-
finements in Dole's paper and to the second method.
The second method will require, in addition to the above, that the
atmosphere of Venus exist in adiabatic equilihrium from the surface to "the
top of the cloud layer." This is another veiy basic and important assumption.
The numerical data consistent both with vhat is known of Earth and Venus
and with the above assumptions are presented below. For detailed information
on these data, the reader is directed to the references. The composition of
the Venus atmosphere will then be calculated from considerations of these
two methods or approaches and the data below.
BASIC DATA
(a) A. Coniposition of air by volume for Earth
0 = 20.9^6 per cemt
K2 = 78,084 per cent
+ trace elements (a)
B. Molecular -weights (chemical scale)V7/
Mj. =28.016 g/g-mole 2
MC0 = !4i!-'011 g/g-mole
MAir = 28-967 g/g-mole
C. Specific heats (k^0oK)^10^
(C )N = 7.OI7 cal/g-mole 0K P 2
(C L = 5.031 cal/g-mole 0K
(C )co = 10.280 cal/g-mole ^ P 2
(Cv)co =8.294 cal/g-mole 0K
D. Mass of CO per unit area of Earth's surface (in the atmosphere, sea
vater, carbonates, coal, etc.)^'
mco -j~] = 7.23 - 9-66 kg/cm2, estimated limits
S IE
,mco01 21 A s
=> 8.144-5 kg/cm (computed from Ref. 3)
Average
E. Plane taiy data
m^/mg = O.8I9 (averaged from Ref. 3)
ra /ra = O.950 (estimated from Ref. 3, including an V E
estimate for the height of the cloud layer)
r = 6371 kn/11^
g /s = 0.9071+8 (coniputed from above)
2(n) g = 98O.665 cm/sec
SE
F. ExperlmeirfcaL (and highly uncertain) data concerning the atmosphere of
Venus
Surface temperature
T = 600oK(l2)
s
Temperature of reflecting layer above -which there are 1000
m-atm of C0o
^ = 3000K(2'13)
Temperature at or near the "cloud top," but assumed to be
above the 300^ reflecting layer (data compiled from original
references in Ref. 6)
T2 = 235^
THEORETICAL DBVELOFMENT
Dole's Approach
Dole's theory is based on the assumption that since the Earth and Venus
are similar in mass and size the only differences between them must be funda-
mentally due to the amount of solar radiation received by each. Hence, an
evolutionary theory of the origin of the Earth's atmosphere expressed by
conservation equations of atmospheric constituents may be looked at by de-
scribing the effects of placing Earth in Venus' orbit SOHK 3 or i+ billion
years ago. Briefly, the results are that due to the increased surface temper-
ature no surface water could exist. The water vapor would be subject to a
One m-atm of a gas is the mass contained in a tube 1 m long and with a 1-cm cross-sectional area, when the gas is at standard pressure and temperature.
higher rate of dissociation due to increased radiation, and the result vould
be a dry planet. In the absence of liquid vater, CO« produced by volcanic
action would not go Into solid carbonates or sea vater. Atmospheric turbu-
lence would be very high, wind erosion would be Increased, and, coupled with
a higher surface temperature and normal isostasy, surface oxidation would be
very great. Together with the oxidation of CO produced by volcanic action,
surface oxidation would remove most of the Op from the atmosphere. The bal-
ance of N2 would be only slightly affected. Therefore, the net result is a
dry atmosphere of COg and nQ. This enables a quantitative estimate of these
constituents.
Under the ass-«jirption of static equlllbrium with no centrifugal force,
and considering a spherical coordinate system fixed to a planet, the hydro-
static differential equation may be shown to be of complete generality as
dp = - gpdx (i)
Now consider an atmosphere of constant composition consisting of "i"
perfect gases in over-all hydrostatic equilibrium; also, consider the ac-
celeration of gravity to be constant through the depth of the atmosphere,
^y integrating the mass of the atmosphere through its depth above a unit
surface area using Eq, (l) and perfect gas relations, it may be shown that
paass of species i in the atmosjiiere |_ 1_ _ji 1 per unit surface area / g M ^i
' s s
or
pi = hr M: e8 (2)
■
Alternatively, since
m1 = niMi
'is "11- I MSe (3)
Therefore, if the mass of species 1 in the total atmosphere is known, and
if the compoBition percentages are known along with the physical size of
the planet, the partial pressure of species i at the surface may be found.
Note that this equation requires that each species in a constant-composition
atmosphere be not In hydrostatic equilibrium with itself, and that the surface
partial pressure of a particular species be not generally equal to the weight
per unit area of the gas above the surface as was reported by Dole. This is
therefore a refinement in Dole's findings. It is known that for perfect
gases partial-pressure ratios represent ratios of percentage by volume, not
by weight. Iherefore, it would be Impossible to have the surface partial
pressure of a particular species generally equal to the weight per unit area
of this species above the surface of the planet. This refinement will de-
crease the fraction of CO^ reported by Dole by approximately k per cent.
The method of computatian then becoaoss
1. Consistent with the original assumption, the acceleration of gravity
on Earth is assuiaed to be a constant with altitude. From the air composition
data on p. k and from Eq. (2), the mass of N2 In Earth's atmosphere per unit
surface area is calculated.
P"" ^ N2E .,.
M ; g-- = 780-11 em/cm2
'Air &sE
2. From the data on p. 5 concerning the COg and from step (l) above,
the coiaposition hy weight of COg and N2 is found. Under Dole's theoiy, and
neglecting tha trace of argon, this comprises the atmosphere of Venus.
(% CO by weight) = 100 x
($ N2 by weight) = 100 x
3- From step (2) above, and the molecular-wel^it data previously-
mentioned, the composition by VOIUBK is found
/^ (% C02 by weight) jj-
(> CO 'by volume) = — ... (£ CO by weight) f\
1 + 100 M - 1 vo.
(i N2 by vol-we) =
'MC0 (% N by weight)!—-21
1 \
1 + (% N2 by weight)
IÖÖ
'M CO.
^
4. From Eq. (2), the molecular weight and planetary data, step (3), and the
assumption that the mass of N2 in the atmosphere is proportional to surface area,
the surface partial pressure of N2 is found for Venus. Essentially the mass
of N2 per unit surface area is assumed to be the same for Earth and Venus.
M- ^ C02 by volume^
IÖÖ 'f N2 by voluBe
N
löö I %
M
10
5. From steps (3) and (4) the surface partial preesure of cCu is found.
f CO "by volume PC02 ^ ^2 I ^ W2 ^ volume
SV SV
6, From steps (k) and (5) the total surface pressure is found.
p8
= % + PC0 SV SV
The results of this computation are presented in Table 1. It should
not be construed, however, that the significant figures shown are indicative
of the precision of the theory.
Second Approach
For an atmosphere in adiabatic equilibrium the following relation must
hold
Sa - % (ka)
where a and b denote any two positions in the atmosphere. Under our original
assumptions and with the above-mentisned data, this relation must hold below
the layer at 300oK, above which there are 1000 m-atm of CO . In particular,
it must hold between the points where T = 300oK and T = 600oK. Using the J_ s
data and methods of Ref. 10, the temperatures T and T , and Eq. (ka), a
relation among ps, p^., and the composition is easily obtained.
n^ R In -i + (3 - &) n+ npQ R In ^ + (? - ^L) n = 0 2 Ps si N^ TJ2 ^2 ps si c0 C02
r c where j5 = / ^01 + constant
11
Since 1000 m-atm of CO^ = k,k6l g-raole/em of CO^ the partial pressure
of the C02 at point 1 may be found Tjy Sq. (3) .
> r^)« pco0 " -IT- M es
ISxen from the law of partial presBxires, the total preaBure at point 1 may-
be found as a function of the compositian.
Hie mass of N in the atmosphere of Venus obtained in step (If) of Dole's
approach is also assumed to hold here:
f-j >= S7.8I*-5 g-mole/cm2
Now, from the law of partial pressures, Eq. (3), and the amount of IT, the
total surface pressure p may be found as a function of composition.
Biese three relations yield a relation for determining the composition.
12
P_ (V^) H/f%+ V)a ^
/v s A.
CO,
R IQ|-—*!+ m . w. cor
R 1 + ITB . ^ cor
yihidh must be solved for n^ by trial. Carrying out the computation gives
the following composition
CO =1 81.6 per cent
NQ = 18.k per cent
longpaxed to the average values obtained by Dole's approach
C02 =. 87.3 per cent
N2 = 12.7 per cent
©IUS it is apparent that both theories give quantitative results in close
agreement -with each other.
MODEL OF THE VENUS AIMOSEHERE
Assuming therefore that these two theories do give good quantitative
estimates of the atmosphere of Venus, the remaining problem is that of
adopting a tentative composition for our model Venus atmosphere. Rather than
investigate the probable effects of experimental uncertainty in the data,
nonuniform gravitation, trace elements, imperfect gases, etc., *hich would
13
in fact invalidate all of the original assumptions and require more accuracy
than is obviously contained in this model, we make a fundamental assumption:
that the temperature lapse rate with altitude is slightly less than dry adi-
abatic. The experimental data and all of the original assumptions are re-
tained except that of a dry adiabatic lapse rate below ^ As an alterna-
tive to Eq. (ka.), the well-known relation
^ " ITJ (4b)
could have been written for an adiabatic atmosphere of perfect gases. It
was not used because 7 has a strong variation with temperature for this
mixture, and a "proper" average 7 vould. have had to be chosen as In the "C"
data on p. 5. If a polytropic atmosphere is assumed where an adiabatic atmos-
phere was previous^ assumed, the following equation holds by definition
p /T \q/q.-i a fa
Pb'lV (5) /
For an adiabatic atmosphere, q = 7 ; for an isothemal atmosphere, q - 1;
therefore, ve will choose 1< q < 7. Using this criterion and Eq. (5) in
place of Eq. (ka), and repeating the computations of the second approach,
it is found that the composition must contain more than 81.6 per cent CO
by volume. This tends to reduce the discrepancy between the t^o approaches
The atmosphere of Venus is now arbitrarily chosen to consist of 85 per cent
C02 and 15 per cent N by volume.
With a composition assumed, the atmosphere is modeled as follows:
Ik
1. Fram the known eompositlQn, aioaunt of N , and Eq.. (3), the
total surface preseure is found.
2. From the knowa eorapositian, amotmt of CO ahove point 1, Ta,
T., and Eqa. (3) and (5), the exponent q is found.
3. This polytropic atmosphere is aseumed to hold true until point
2 is reached, »here T = 235^^ the assumed "cloud top."
k. The atmosphere will be considered isothermal ahove point 2,
■where T = T = 2350K and g = 1. Now it is highly improbable
that this situation precisely prevails, hut It should represent a
reasonable assumption over altitudes of aerodynamic interest. From
the standpoint of radiative equilibrium, the temperature probably
drops to about 190oK at-55 km. There may also be a slight increase
in temperature above this level due to 0 heating. This will
be much more suppressed, however, than the case on Earth because
of the absence of much 0 at these altitudes (provided, of course,
that there is no appreciable amount of 02 in the atmosphere of
Venus). Effects of this kind are neglected in this model, how-
ever, because they are inconsistent with the accuracy possible
through use of the other assumptions of the model.
5. With the above information, Eq. (5), and the perfect gas law,
Eq. (l) can be integrated. This yields a complete pressure-density-
tensperature-altitude relation consistent with the assumptions and
data. This relation is presented l>elow, and it is represented in
Table 2 and in Fig. 1.
15
i.O
IGT
iO -2
10 -3
a1" IO'4
IO-
10"
10 r7
'S \
^
\
VTs
\
\
\P/f > s
P> /P\
\ v Assu ned < gas p roper ties ^ V
at the surface \\ Ts = 600° K v v ^s = 5.174 xlO"3 g/cm3 \ \
Ps = 6.12 3 atr n
^
20 40 60 Altitude (km)
80 100
Fig. I—Tentative atmosphere of Venus
16
For 0 s: (r - r ) s lj.2.16 tan, the cloud "top" altitude.
i , I? (<I - 1? fg (P - r ) i R q T vr rB;
8
q/qi-l
(6)
^xMig^f» (>„£.) i-i/q
R q T (7)
£_ X R q fj; \T **! ^ s
-, 1/q-l (1/q)
(8)
For 42.16 < (r - r )
T = T2 = 235^
p8 vp£
exp Mg8
W2 (r " r2) (9)
iL.= 12 a V s
exp Mga
KP; (r " r2) (10)
vhere consistent with the assumptions and derivation of the model
T = 60CrK s
p = 6.78^ atm
P= 5.734 x 10"3 g/em3
r = 6052.45 "Km s
p a 5.464 x 10"2 atm 2
17
P2 = 1.179 x 10'h g/cm3
g = 889.9 cm/sec2
■^1 " 5-1^3
M = hi.612 g/g-mole
r = r + h2.l6 km 2 s
R = 8.31^33 x 107 erg/g-mole 0K
DISCUSSION
It is necessary at this point to review exactly vhat has been done in
order to see *hat assusrptions are basic to this atmospheric model. In analogy
to the Earth, it is immediately evident that the assumptions of unifom
gravitation, constant composition, perfect gases, and hydrostatic equilibrium
with no centrifugal force are not the major sources of uncertainty, it is
In the statements that followed that great uncertainty was introduced. In
essence. Dole's theory was aasumsd correct at the outset, and was investi-
gated as to compatibillly with SOEE experÜKntal observations. It was neces-
sary at the start to assume that C02 and N,, are the only major constituents
in the atmosphere and additionally to assune the total aaxxmt (not proportion)
of N2 in the atmosphere. Then it was necessary to assume accuracy in soae
experiinental data that are known to contain a high degree of uncertainty.
Further, it was necessary to speculate on lie nature of the cloud layer sur-
rounding the planet so that the nature of the temperature lapse rate with
altitude could be specified. At this point it would appear that the entire
problem had been assumed away. However, there are indications that this may
not be the case.
18
Consider flrmt that Dole's approach is accepted and that the question
is one of cwnpatlbility with some experimental measurements. In dealing
with the second approach, it is found first of all that any uncertainty in
the anwunt of N chosen greatly affects the derived compoBition. Uncertain-
ties of 1 per eent in this amount affect the resulting fraction of N by
approximately 1-2 per cent. It is also found that uncertainties in the
data concerning the amount of CO above the "reflecting surface" greatly
affect the derived composition. For instance, if there were only 500 m-atm
sf G02 above this surface instead of the chosen figure of 1000 m-atm, the
derived camposltion would contain approximately 30 per cent N by volume.
TSis results are also highly dependent upon the temperatures chosen and upon
the nature of temperature lapse rate below the "reflecting surface." In
other words, there are no chosen parameters that tend to suppress uncertainty
effects or that cause less uncertainty in the results for N composition
than is contained in the chosen parameter. It is then apparent that the
close agreement of the two approaches must either be a remarkable coincidence
or have soms basis in fact. However, this model is inconsistent with the
measurements of Ref. 7, and consequently with the results of Ref. 8.
It must also be noted that the amount of CO reported by Dole is approxi-
mately one-half that which would result if the figures for the amount of
Earth C02 from Ref. Ik were used. Attention rnust also be called to Ref. 15
in \Aiich a method similar to the second approach here is used to derive a
coiqpositlon of approximately 75 per cent CO and 25 per cent N .
All that is claimed here is that under the assumptiens a tentative
atmosphere of Venus has been derived. Kow if the assumptions involved ulti-
mately prove reasonable and the experimental data, and consequently the
19
conclusions derived therefrom, are reasonably correct, then the close agree-
neat of the two affproaehes IMleates tfeat the «&o«en caopoaltlao a^ *»
renreaentatlve of the Venus atmosphere. Further, the pressure-taaperature
distrihution below the cloud layer may be raaaomMy correct, übe «iffl-
culties In accepting an isothexmal atmsphere above Hu* doud l^er ham
been discussed prevlouaJy. Above 100 tan, photodissociatien and Ionisation
will surely beeeme appreciable, and In going to these higher altltuÄes the
assiss^tion of a uaiform gravitational field beeones poorer, n» eut«ff at
100 km vas chosen because the density at this point roughly corresponds
to that density in the Earth's atmosphere (at approximately 300,000 ft)
at vhich incipient aerodynamic effects associated with re-entry nose cones
are no longer negligible.
One immediate objeetion to the aceeptanee cf this model concerns the
arbitrary choice of the temperature lapse rate belov the cloud layer. In
reality, the real Justification for choosing a lapse rate slightly less than
an adiabatic lapse rate is that this choice reduces the discrepancy m atmos-
pheric eompositian between the two approaches. However, it may be partially
Justified by other arguments. For instance, if the clouds are composed of
water and ice ciystals, the lapse rate in the cloud layer must be less than
the dry adiabatic lapse rate because of the release of the heats of vapori-
zation and fusion as altitude increases. The chosen lapse rate may then be
considered an average between the surface and the top of the cloud layer.
Yet another possibility may be considered. If the surface temperature is
slightly greater than. 600% as may well be possible considering the un-
certainty in the data, Eq. (1+a) requires that the surface pressure be higher
than that originally derived. Assuming that the amount of N is fixed, Eq_. (2)
20
requires that there be meye CO . übe ohoscm lapee rate nagr üben he c«n-
sidered as a eeo-rectleai device to fix the «urfaee tesqerature at 600°!:, bufc
still he censlsteitt vlth the amount of CO in the atnosjhere. Of oeurse
this argumeat wwitB in reverse if the surface teawerature is less than 600°!:,
It does not seem profitable to investigate all of these olBor effects in view
of the uncertainties already introduced, nils lapse rate was chosen simply
because it serves to reduce the discrepancy hetwaen the two approaches and
appears to be as good as aay other assunption at this tlae.
It is interesting to note that if the elouda eurrouading VenuB are as-
sumed to consist of ice eiystala, ^ ^ accerding to the present model the
vapor pressure above ice at T2 ■ 235^ indicates that water vapor eoi^rlses
0,29 per cent ty volume of the Venus atmosphere above the cloud layer (as-
suming a constant mixing ratio above this point). Kils ia larger by a factor
of approximately kO than that which would exist In this model aeeording to
the recent measurements of the Moore-Ross balloon flight. ' It must be cau-
tioned, however, that these measurements contained a high degree of uncertainty.
In view of the above-mentioned difficulties, this model Is proposed only
as a first approximation with which aerodynamic entry problems concerning
Venus may be studied. It may be used as a mean to which perturbations
may he applied; this mean may be bracketed by other existing theories untH
more experimental data are aeeumulated. The significant figures preseated
for this model are obviously not indicative of the precision of the knowledge
of the Venus atmoephere, but are consistent with the proposed model if the
experimental data and assumptions Involved are taken as "exact."
21
III. THE OHERMDDYHAMIC PROPERTIES OF 15 PER CENT N AND 83 PER CENT CO^ BY VOLUME "~2
CONDITIONS AMD ASSUMPTIONS
The thermodynamlc properties of a mixture of 85 per cent CO and 15 per
cent N2 by volume will be presented on the premise that they may be useful
regardless of specific application to this tentative Venus atmosphere. In
this light, the calculations of this section win be carried out for exactly
the stated composition and will contain far less computational uncertainty than
does Section II concerning the constituency of the atmosphere of Venus.
The gas mixture constituents are assumed to obey the perfect gas law
pM/RpT = 1. Investigation of the critical constants of the constituents
shows that this is a good assumption over the selected temperature range of
from 10000K to 2k,O00OK. COg deviates most from the perfect gas law (the ef-
fects of compressibility for this constituent have been discussed by Ray-
(17) mond)0 since N2 is better-behaved in this respect than CO^, the com-
pressibility at 1000OK and 102 atm for this mixture is slightly depressed
from the value of 1.026 quoted by Raymond for pure COg. However, this as-
sumption becomes poorer at temperatures below 1000^ and at the higher pres-
sures. The Justification for neglecting the effects of intermolecular forces
at these lower temperatures is that the thermodynamlc states of interest with
respect to the atmosphere of Venus do not occur where large effects of inter-
molecular forces are apparent. When the pressure is nearest the critical
pressure (for C02) the temperature is high; where the temperature approaches
and falls below the critical temperature, the pressure is low. Again, the
presence of Ng will generally tend to influence the mixture to behave more
like a perfect gas than will pure CO .
22
The mixture vill be assuraeä t© be in thermoÄynÄmlc equilibrium. From
the standpoint of the use of these properties, this assumes that for any-
particular state to which these properties are to be applied, e.g., aft of
a shock wave, the time required for relaxation phenomena to take place is
much shorter than that in which the gas changes appreciably in state. The
equilibrium composition of the gas mixture is then computed by the method
of Ref. 18 under the criteria that the Gibbs Free Energy is a minimum for a
selected particular temperature and pressure.
Below 1000 K the mixture is assumed to consist solely of molecular CO
and N2. This assumption is justified by noting the calculated equilibrium
composition at T = 10009K and p = 10" atm. Using perfect gas relations,
the thermodynamic properties of the mixture can be calculated from the basic
data in Ref. 10. However, a slight correction must be applied to these data
in order that they may become consistent with the data above 1000oK; the
method of correction will be described later.
METHOD OF CALCULATION
T > 1000OK
Under the criteria of significantly contributing to the thermodynamic
properties of the mixture it was first assumed that the mixture could con-
sist of the following possible species: COg, Ng, Og, NOg, CO, NO, 0 , C, N,
0, C+, N+, 0+, Og, 0', C0+, N0+, N*, Og, C++, N++, 0++, and e". Using the
basic referenced data in Tables 3-6, and the method of Ref. 18, the equi-
librium composition was calculated over the temperature range 1000^ < T <
2k,0O0rK. Constituents present in amounts smaller than 10 g-mole per origi-
nal "cold" g-mde of mixture were then considered negligible and not listed.
23
Thus, thermodyBandc properties are preBented for the foUo-win« significant
canstituents in the temperature ranges specified below:
1000^ < T < kOOO^K
C02, N2, 0^ CO, NO, C, N, and 0
^OOO^ < T < 2k, 000^
a02' N2' 02' C0' m' C' N' 0' C+' N+' 0+'
0", C0+, N0+, Og, C"^, N-^, O"^, and e"
■Die equilibrlxm composition is presented on the basis of one g-mole of the
"cold" mixture in Figs. 2 and 3 and Tables 7 and 8 over a temperature range
of 1000 K to 2l4-,000OK and a pressure range of 102 atm to 10"^ atm.
Since the thermodynamic data are presented on the basis of one g-mole
of the "cold" gas mixture, i.e., kl.6l2 g, the following relation applies
n M
o 2 n n vJ-W i
since no = 1 and 2 n = n. Thus, n./n represents the mole fraction of i
species i. The datum density, p is chosen to correspond to p =1 atm (on
the surface of the earth) and To = 2T3.l60K for the original "cold" mixture.
Thus
p M P„ = •^-2 = 1.856^7 x 10"3 gm/em3 o ~ RT
o
Then
£_ (p/po)(M/Mo)
^ = fTT (12)
24
9.0
5 10 15 Temperature, T (thousands of 0K)
Fig. 2 —- Total number of moles as a function of temperature
25
o
o
en O CL
E o o
o
o ro
en
in o
u 'sajouj ^o jaqainN
in ö
26
C\J
o
^ C o o
in ^_ +_ o • —
O) in o Tj Q.
o E in o 3 o o
-C t_
o 1- o (1) ^
O => o h. 0}
E ro QJ
CP
in
ID O in o in CM <NJ
u 'saiooi ^o jaquunfsj o
27
IT) CM
O CM
^ c o o
m ° to X3 o c- Q.
E 3 o O u
-1=
*•—' k^.
t- o - o
OJ ^ rD 1
o 2 o ro
E CTi CD
LD
if) o
u "saiouj p jaqtunu IDIOI
o o
in — o E
tu
E
5 10 15 20
Temperature, T (thousands of 0K)
25
Fig. 3d — Molar composition
29
10 15
Temperature. T (thousands of 0K)
Fig. 3e — Molar composition
20 25
30
5.0 - /
4.5 - /
4.0 - ey
3.5 — c
in 0) o 3.0 E
p=I0"3 atm _X o s^' 1 2.5 E 3 2
— j 2.0 - /
0 / 0+
0X 1.5
1.0 V c + \ / c++
"~~^\ A
0.5 - f ]Px/ \ N + X
_J^\ Ng A "27 /A / >Cx (^^\~ S 0 yA/^o^\x ^x 4^-^ \. J^^ ^=r^_
10 15 Temperature, T (thousands of 0K)
20 25
Fig.3f—Molar composition
31
iO 15
Temperature, T (thousands of 0K)
20 25
Fig. 3g— Molar composition
32
Hhe molecular wsights and densities are shown in Tables 9 and 10 and Figs.
h and 5.
The datxon state chosen for the internal energy equal to 0 is ■where
carbon (graphite), gaseous molecular oxygen, and gaseous molecular nitrogen
exist at a temperature of 0 K and zero pressure. Then the internal energy
per "cold" g-mole of the mixture may be calculated as follows:
E = £ n, E. i 1 (13)
where
(E0 - E0) + E0 M
and E is the heat of formation of species i from the ground state mentioned
above. In dimensionless form, the computing equation becones
E__ = T_ ET T 7 nl o o 1
/E0 - E0 o RT h
The basic data are taken from the previously mentioned tables.
Since the enthalpy is given by
(15)
Hl = El + PlV = E1 + ^ RT (16)
the computing equation Is
H !_ RT - T 7 1 o o 1
E - E c RT
_o RT + h^ + 1 (17)
33
10 15
Temperature, T (thousands of 0K) 20 25
Fig.4—Molecular weight versus temperature
3^
10s
10'
e o \ E en
c a>
10 -2
10'
— —
-
-,
\
— \
\
\ p{atm)
-^ \io2
\ \ s.
_ \ \ \
\ ^^\
-^ v Vo ^^
\
— \ V \
\ \. \^
-. v \I.O ^ -^
\ \ — \ v ^
\ \ \ .
- ^ N^O"1 ^^ ___
1 1 1 1 1 i >o 1 1 1 1 1 1 1 1 i i i i 5 10 15 20 Temperature,! (thousands of 0K)
25
Fig. 5a — Density versus temperature
35
icr
10"
10"
E o
E w
c
10'
10"
10"
- \ - \ - V p(atnn)
\ sio-1
\
- \ N \ \
\ \
\ SIO"2
\
^
\
\ \ \ \
\ - \
\
-, \ v
\ Xio-3 ^
- \ \ \ \^
- \
- \
\io-4 ^^
— \ — \ - \ \^
- \^
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ! 1
5 10 15 20
Temperature,T (thousands of 0K)
25
Fig. 5b — Density versus temperature
36
The Gibbs Free Energy Is given by
^i = Hl " TSi (18)
Also
P = Fi + RT t*1 Pi (19)
Then
S E - F" (20)
or
Since
'E0 - E° /F0 - E0\
RT
y1 - £n pi + 1
p. = — p ^i n
(21)
(22)
and
1 = ^(1) i :
(23)
the computing equation is
^ni
E0 - E0 v \ F0 - E° + 1 (24)
Again, the basic energy data are taken from the previously mentioned tables.
The values for internal energy, enthalpy, and entropy are shown in
Tables 11-13 and Figs. 6-8, respectively. High-temperature Mollier
charts are presented in Figs. 9 and 10.
37
8000
7000 -
5 10 15 20 Temperature, T (thousands of 0K)
25
Fig.6 Internal energy, E/RTQ, versus temperature
38
8500
8000
7500
7000
6500
6000
5500
5000
£ 4500
x ^ 4000 ^. D
JC
fi 3500
3000
2500 h
2000
1500
1000 h
500
0 -
-500
p(atm) /I0-;4 A
/3x,10
10" 3
_L 5 10 15
Temperature, T (thousands of 0K) 20
Fig. 7— Enthalpy, H/RT0, versus temperature
-3
25
39
260
20
0 _L 5 10 15 20
Temperature. T (thousands of "K)
Fig. 8— Entropy, S/R,versus temperature
25
■ ■■■■ •■ ■ ,
o o m m ID
ID ID
o 8
0ia/H 'Ad|D4tu
o o ID
(D in
O
8 in
o 2 to
R 8 g m (3 IT) CM <M — 8
0ia/H 'AdiDM+ug
kl
o o ro
_L S2 Q o o o o in o L?) T ^- ro
0±a/H ' Ad|DqnJ3
R g O o
IT) (M <M
in 2 o
o 00
er CO
o CO
O
o c\J
CD
X
E
CD
O > o
o
O O
LO
c o
o o o C\J
O ^J- c:
Lü ^f- o r\J
^ ü CM o LD
00
o o
o
CL>
ö
CD
3000
^3
pCatm)
?500
2000 -
_p 1500
uj 1000
500
3x10""
7,000
6,000
5,000
4,000 N3,000
2,000 ,ooo-''ÄJ22S^~~! • 500
T(0K)
-500 50
J_ 100
Entropy , S/R 150 200
Fig.lO —Mollier chart to 12,000 0K for a volumetri mixture of 85% CO2 and 15% N,
To convert the thermodynamlc properties to the mass basis, it ie iserely
necessary to divide the properties by the "cold" molecular weight. Thus
-S- = -L J^L (25) RT 14 RT o o o
*_ = i. iL_ (26) o 00
and
Rf-^irl (27) o o
T < 1000OK
The thermodynamic properties in this range were computed from Tables
12 and 17 of Ref. 10 for a mixture of O.85 g-mole of CO and 0.15 g-mole of
N . In this reference the zero for internal energy and enthalpy is taken
at 0oK. It vas necessary, therefore, to add the heat of formation of C02
to the enthalpy -of CO tabulated in Ref. 10. In order for these data also
to match the computed properties at 10000K but still retain the zero at 0 K,
it is necessary to multiply the enthalpy data in Ref. 10 by a factor only very
slightly different from unity. In effect, this means that the specific heat
at constant pressure is multiplied by this factor. The computing equation
then becomes
RT RT o o
Kco hco 0.85 1 j2^ 2 - 173.10W 0.3 5 \ 2 y (28)
■where the K's are the multiplying factors to match the data at 1000 K and
K^ » I.OOO32U
KJJ = I.OOOI4.675
45
Now the entroEjr function * , tabulated In Kef. 10, represents the
following:
F0 + R (29)
Also
/?<*T + constant (30)
Bius, the save multiplying factor mst be applied to this function as is
applied to the enthalpy above. The final matching of these data with the
computed data at 1000^ is then taien care of by the additive constant in E<1« (30). 131113, the computing equation is
R = O.85
where
Kco/co2 + B^ - in (0.85p) + 0.15 w 2 2
R + B^ - in (0.14p) 2
(31)
Bco2 = 0-011k
^ = -0.0008 2
The values for the enthalpy, internal energy, and entropy over the range
of 150oK to 800^ and 102 to 10"4 atm are shown in Figs. 6 - 8 and Tables 11 - 13.
In Table 2, values are also Included for the entropy at the particular pressure-
temperature combinations that occur in the model of the Venus atmosphere se-
lected In Section II.
DISCUSSION
Since much of the basic data were taken from Gilmore's paper,(9) the
reader is referred to Ref. 9 for an adequate discussion of the accuracy of
the basic data. Constituents having a large error in the basic data (due
k6
to a stiramatlon over too fev energy levels in the internal partition fltnction)
are present in such small amounts at the temperatures and pressures ■when the
error is quite significant that the error introduced in the thermodynamic
functions may be considered very slight. It should also be mentioned that
the data are significantly in error near the liquefaction points of the CO .
used.
For more accurate results in this region, better data for CO should be
(19)
Selected cases were calculated over the complete pressure range at temper-
atures of 5000oK and 12,000oI<: for a mixture of 8^.7 per cent CO , 15 per cent
N , and 0.3 per cent HgO vapor by volume. This was done to investigate the
effect of water vapor on these properties in the event that a substantial
amount of water vapor actually exists in the atmosphere of Venus. The ad-
ditional constituents considered were HO, H , OH, H, H+, H^, 0H+, H", and
OH". Table 8H shows the effect of this addition on the composition and thermo-
dynamic properties of the mixture. It may be stated that the thermodynamic
properties are not affected by more than the order of percentage addition of
H20.
The three most important differences between this case and that of pure
CO^, are the reduction in internal degrees of freedom at low temperatures,
since N^ is present; the appearance of NO at moderate temperaturesj and the
reduction in electron concentration at high temperatures due to the high ion-
ization potential of N„. Since NO forms quite readily around 5000 K, more
energy goes toward dissociation at this level than in the pure-COp case.
Since NO ionizes quite readily, the electron concentration at this tengperatur«
level is higher than that for pure CQ-,. All effects mentioned are generally
of the order of the percentage of Ng addition to pure COg.
^7
IV. NORMAL SHOCK-WAVE CHARACTERISTICS OF A TENTATIVE VEMJS ATMOSPHERE
CONDITIONS AffTD ASSUMPTIONS
The gasdyriamic properties of the atmosphere of Venus depend upon the
gas mixture assumed and the thermodynajaic properties at the high temperature
and pressure ratios usually expected through a nomal shock at high entiy
speeds. Therefore, it should be recalled that the gas mixture assumed is
85 per cent C02 and 15 per cent N2 by volume (Section II), and that the
calculations of the themodynamic properties are based upon the assumption
of thermodynamic equilibrium (Section III).
In addition, it must be assumed that aerodynamics of continuum flow
apply> that is, that the shock thickness must be veiy much smaller than the
characteristic length of the object causing the shock. This is necessaiy
in order that the conservation equations may be applied across a veiy thin
flow discontinuity. Since the highest altitude (and lovest density) chosen
corresponds to the altitude in Earth's atmosphere *here this assumption still
generally applies, the assumption does not impose an appreciable limitation.
However, it must be cautioned that the mean free path at 100 km is of the
order of 10 cm.
METHOD OF COMPUTATION
Several methods of calculating shock-wave characteristics were invests
gated, however, because the upstream state must be specified and be-
cause of the desirability of a rapid convergence of the iteration procedure,
the method of Ref. 2o was selected, since it appeared to be well-suited to the
solution of the problem. Applying conservation of mass, momentum, and energy
across a flow discontinuity
Continuity: p^ » p^ ^
^6
2 2 Mtmemtum: p^ + P^ " P2
+ P^-Q (33)
Energy: 1 ..2 1 ..2 ^ + I ^ " h2 + 2 ^ w
SubstitutiDg Eg. (32) in Eq.. (33) and manipulating, the preasxure ratio
becomes
^2 *X + Pl Mo 2 f . " Po po po ^o ^ \ ul/
Bearranging and manipulating Eq. (3*0, the enthalpy hecomes
RT " RT KT^ 2 000
/ \2
"l
(35)
(36)
Fr«m Sectien III the equation of state is preBented in Tables 10 and 12 in
the form
H-, T£- =• f (T0, p0/prt) RT 2' jr2/irO'
(37a)
^ = f (V P2/Po) o
(37b)
Finally, from Eq. (32)
pl/po ^2
"l P27P0
(38)
1+9
With tli© equations above, the machine Iteration procedure follows this
pattern:
1. Pick an upetream thermodynamlc state and veloclly tu, and enter
all known quantities,,
2. Aasune Ug/iL^ ■ 0. Calculate PO/PO fwa- Eq.. (35) and H^/RT tram
Sq. (36).
3. Enter these value« In the tabular relations (Eq. (37)) and find
p2/po-
k. Calculate Ug/t^ frem Eq. (38) and enter this into Eqs. (35) and (36).
5. Repeat this procedure until u^/u, obtained in step (k) above differs
fram that used in the prevleus iteration by less than 0.0001.
6. Print out p^, ^2/\, P^V V11!*
The results of these comgarutatisns are presented as a function of altitude
in the adopted model of the Venus atmosphere in Figs. 11a - lid and Tables
Ika - Ikd over a flight-speed range of 2,000 to 14-0,000 ft/sec. A typical
sound speed is 910 ft/sec (at an altitude of 314-.65 km in this model) so
that the free-stream Mach number range is roughly 2.2 to 14-14-.
DISCUSSION
It is interesting to note that to a good approximation at low Mach
numbers, air may be treated as a perfect gas with constant specific heats
as far as shock-wave relations are concerned; however, this atmosphere
cannot be so treated because of the marked variation with temperature of
the specific heats of CO in the normal temperature range of Interest. It
may be shown through an error analysis that the iteration procedure can be
cut off when two successive values of u /u. differ by less than 0.0001 and
yet generally yield errors in the thermodynamlc-state variables of the order
of 0.01 per cent.
50
3000
2750
2500
2250
2000
a. cT
1750
1500 -
a> 1250 ol
1000
750
500
250
All altitudes above 138, 327 ft (42.16 km)
(approximately)
75,791 ft (23.10 km) • Surface
J 1 I L 10 15 20 25 30
Velocity, u,(thousands of ft/sec)
Fig. Ila-I—Normal-shock-wave characteristics of a tentative Venus atmosphere
(pressure ratio, P2/P1)
35 40
51
200
180
160 -
140
120 c^
o 100
a> ■—
3 (/> ü> 0)
Q. bU
60 -
40
20
All altitudes above
,791 ft 3.10km)
2 4 6 8 10
Velocity, ^(thousands of ft/sec) 12
Fig.!! a-2—Normal-shock-wave characteristics of a tentative Venus atmosphere
(pressure ratio, p2/p|)
52
45
40
35
30
.2 25 o k_
0)
I 20 O) o. E n - 15
1 1 262,480 ft (80 km)
10 15 20 25 30 Velocity, u, (thousands of ft/sec)
35 40
Fig. Mb Normal-shock-wave characteristics of a tentative Venus atmosphere
{temperature ratio, Tg /T|)
53
25.0
22.5
20.0
17.5 -
"^ '50 a. o
i 12.5
10.0
7.5 -
5.0
2.5
i r Altitude
328, 100 ft y
262,480 ft (80 km)-
96,860 ft (60 km)
138,327 ft (42.16 km)
_L 5 10 15 20 25 30
Velocity, u, (thousands of ft/sec)
Fig. lie — Normal-shock-wave characteristics of a tentative Venus atmosphere
(density ratio, P2/P{)
35 40
0.50
0.4t
0.40
■?h
0.35
0.30
0.25
0.20
0.15 ~
0.10
0.05
38,327 ft (42.16 km)
196,860 ff (60 km)
262,480 ft (80 km)
328,l00ff (100km) 1 L
5 10 15 20 25 30 Velocity, u, (thousands of ft/sec)
Fig.lld-I — Normal-shock-wave characteristics of a
tentative Venus atmosphere
(velocity ratio, u2/U| ; U|,from 0 to 40,000fps)
55
0.50
0.4S
040 -
0.35
^ 0.30
=3
o*
S 0.25
« 0.20
0.15
0.10
0.05 -
5.0 7.5 10.0 12.5 15.0 Velocity, u, (thousands of ft/sec)
Fig.lld-2—Normal-shock-wave characteristics of a tentative Venus atmosphere
(velocity ratio, u2/U| ; U|,from 0 to 20,000fps)
APPENDIX A
TABLES
59
OJ H fO CO
EH
Per Cent Composition
(at
all altitudes)
o >
t- OJ t— O -=t CVJ
"0 ^ 'O lÄ CO t- co CO CO
K~\ CO tf\ cr\ ii\ t-
100.000
100.000
100.000
4^
■a ■rH 91 >
^H 00 -d- VO OJ J- oj ir\ i^
O QJ H ON OS ON
CT\ OJ VO KN t~- ir\ t-- J* ^t
ON t— CO
100.000
100.000
100.000
to 4)
m (U to o (U aj U <H
AH f-l
rH CO a)
•H 4-> +J a)
to n O vo ^f^ H t- -=t ■H K-N t--
*"N ir\ t— t— co r- ITN ^1 KN J" VD ifN
ir\ -^ o t~- 00 CO t- t- t—
CO O f- 3 t— ITN K> cr\ H ITN ^O NO
m <u
Pt m o a -p <»;
^H O ^t t~- ON lA H »A t— O ^H O
ON ITN CO ON oj vß H K^ QJ o o o
O lA OJ t— ^ QJ KN t— O O ^H rH
VO CO f- rH rH rH t- ON CO
Case
L=low C02
H>=h
igh
CO
cd
1)
CO
>
4 ( c
4
4 t s (
c
u
J
3
OJ
8 OJ
CO rH OJ
60
£
H K"\ 0.1 iCN VO VO t— r<-\ cn ro CO K ON r~- ai (JN KN h- -d" -* KN KN ai
OT r-H H OJ M-N VO ro OJ rH o ON CO
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Table 3
BASIC PHYSICAL CONSTANTS AND CONVERSION FACTORS
Universal Gas Constant (chemical scale)
R = 8.31^33 x 107 erg/g-mole ^
m 1.98717 cal/g-mole 0K
= 82.O56I atm cm3/g-mole 0K
Avogadro's Number (chemical scale)
L = 6.02306 x 1023/g-iiK3le
Energy Conversion Factors (chemical scale)
1 cal/g-mole » 4.33605 x 10"5 ev
1 thermochemlcal calorie - 4.18400 x 10' erg
Pressture Conversion Factor
6 p 1 standard atmosphere ■ 1.01325 x 10 dyne/cm
Ratio of Fbysical to Chemical Scales
«I^B^chem " Wachem " ^<xx^^
62
Table k
ENERGIES OF FORMATION BC
Constituent E0
o kcal/g-mole
All obtained from Ref. 9 by method indicated below
co2 -93.9639 Direct
N2
0 Definition
02
0 Definition
CO -27.1992 Direct
NO 21.479 Direct
c(g) I69.99 Direct
N 112.535 Computed
0 58.985 Computed
c+ ^29.832 Computed
N+ 448.051 Computed
0+ 373.033 Computed
V 277.9 Computed
co+ 295.981 Computed
NO+
++ c 234.879
992.125
Computed
Computed
N^ II3O.969 Computed
0^ 1183.77 Computed
0" 25.485 Computed
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Table 8l
THERMODYEAMIC PROPERTIES FOR ATMOSPHERE WITH 0.3 PER CEHT WATER VAPOR
Property-
Pressure (atm) M PA*. E/RT H/RT S/R
T = 5000OK
10-* 16.211 2.1323"6 459.25 506.15 96.563
lO-3 19.k9k 2.56^2"5
2.8198"4 257.77 296.76 79.747
io-2 21.438 170.54 206.00 70.147
io-1 22.135 2.9115'3 145.09 179.44 64.317
10° 22.529 2.9634"2 134.78 168.52 59.437
io1 23.594 3.U035"1 117.16 149.39 54.222
io2 26.752 3.5189 74,32 102.74 47.842
T = 12000^
IO"* 7.2981 3.9999"7 2J459.4 2739.4 172.87 io"3
7.4171 4.0651" 2429.4 2675.4 158.38 10-2 8.2261 4.5084-5
2072.7 2294.5 137.31 10-1
10.319 5.6554-4 1420=0 1596.8 110.99 100 12.442 6.8188-3 1006.8 1153.5 92.503 io1 13-784 7.5544-2 822.47 954.65 80.722 IO2 14.642 8.0246"1
731.09 855.70 71.731
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75
Table 8l
THERMODYNAMIC PROPERTIES FOR ATMOSPHERE WITH 0.3 PER CENT WATER VAPOR
Property Pressure
(atm) M p/po E/RT 0
H/RTo S/R
T = 5000OK
10-3 16,211 2.1323"6
459.25 506.15 96.563 19.494 2.5642"5
2.8198"4 257.77 296.76 79.747
10 d 21.438 170.54 206.00 70.147
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1 22.529 2.9634^ 134.78 168.52 59.437
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io2 26.752 3.5189 74.32 102.74 47.842
T = ; I^OOO0^
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4.0651-6
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5.6554-U
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ID2
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81
Table l^A
NORMAL SHOCK WAVE CHARACTERISTICS OP A TENTATIVE VENUS ATMOSPHERE
PRESSURE RATIO, Vj^
Velocity Altitude (Kin) (ft/sec) 0 23.10 1*2.16 60 80 100 20 x 10^ 2.692 h.Olß 6.773 6.775 6.773 25 x 10p 1U275 6.37^ 10. T^ 10.749 10.745 30 x 10% 6.217 9.269 15.638 15.644 15.637 35 x 10d 8.50^ 12.709 21.^57 21.1*66 21.456 ho x KC II.I32 16.669 28.208 28.219 28.206 ^5 x 10^ 1^.135 21.136 35.838 35.853 35.81*0 50 x 10^ 26.178 ^.3^3 44.365 44.359 55 x 10p 31.8140 53.921 53.951 53.952 6o x 10^ 37.932 64.381 64.471 64.526 65 x io; 1^.687 75.769 75.943 76.074 TO x io; 51.9^1 88.288 88.607 88.799 75 x io; 59.702 101.501 101.893 102.744 80 x 10^ 68.066 115.679 115-9^5 117.192 116.666 85 x io; 77.092 130.909 130.893 132.296 133.069 90 x 10^ 86.829 1^7.284 ll*6.8oo 11*8.152 150.660 95 x 10p 97.252 164.881 163.811 164.694 168.701
100 x 10^ 107.835 I83.7I1O 182.021 181.81*0 187.056 105 x IDp 203.744 201.498 199.605 206.266 no x io^ 223.790 222.31? 217.973 226.280 115 x 10^ 244.778 244.674 237.744 247.149 120 x 10^ 266.723 268.396 259.574 268.807 125 x io; 289.673 292.695 282.934 291.282 130 x io; 313.686 316.602 307.874 314.606 135 x io; 338.867 341.410 334.455 339.194 lljO x 10^ 365.243 367.136 362.782 364.812 1^5 x io; 392.321 393.809 392.937 391.695 150 x io; 419.986 421.1*69 1*22.555 420.034 155 x io; 1*48.384 450.164 452.956 449.700 160 x 10^ 1*77.191 479.947 483.937 1*80.775 165 x io; 506.834 510.883 513.4D8 513.345 170 x lOZ 537.374 5U2.993 543.843 547.574 175 x io; 568.907 575.236 575.691 580.072 180 x 10^ 600.325 606.519 609.391 607.563 185 x 10^ 632.794 639.119 643.543 642.936 190 x io; 667.007 674.160 676.304 678.634 195 x 10^ 702.689 709.215 711.085 712.1*60 200 x 10p 738.494 744.626 7^7.457 747.360 205 x 10^ 775.064 781.77b 783.186 783.837 210 x 10^ 812.435 819.664 819.833 822.144 215 x io; 850.412 858.164 857.787 862.758 220 x 10^ 889.810 897.848 897.297 906.208 225 x io; 930.670 939.077 939.31*8 952.368 230 x lO^ 973.108 981.827 983.982 998.529
82
Table ih A (cont'd)
Velocity 1 Altitude (Km) (tt/sec) 0 23.10 k2.l6 60 80 100
235 x lo! 1017.023 1025.1H7 1029.857 101*2.823 2^0 x 10p IO62.885 1070.229 1077-1*03 1087.627 2l»-5 x 10^ IIO7.59I 1116.307 1126.683 1133.103 250 x 10p 1152.523 1163.631 117^.318 1179.510 255 x io; 1197.995 1211.820 1221,9^3 12P6.841*. 260 x 1C£ 12M4-.175 1259.769 1270.1*47 1275.156 265 x Ißt, 1292.193 1308.61J-9 1319.892 1324.633 270 x 1C£ 13^2.5^5 1358.935 1370.325 1375.053 275 x 10p 139^.287 ikio.k96 11*21.859 1426.863 280 x 10^ 11*47.309 1^63.583 lk.7Kk.9Q ll*8o. 148 285 x lo; 1501.529 1518.38^ 1528,308 1535.206 290 x ICC 1555.679 157^.651 1583.374 1591.342 295 x 10; 1610.311 1629.161 1639.790 1647.116 300 x 10^ 1666.009 1684.821 1697.658 1704.202 305 x lo; 175^.163 1762.225 310 x 10^ 1810.509 1820.444 315 x 10; 1867.597 1877.703 320 x 10^ 1926.238 1936.733 325 x 10p 1987.145 1998.332 330 x 10^ 201*8.959 2061.871 335 x 10; 2109.716 2123.668 Sto x 10p 2171.228 2186.632 3^5 x 10p 2233.716 2250.555 350 x 10^ 2297.411 2315.431 355 x lor 2362.016 2381.326 360 x 10; 21*28.009 2448.545 365 x 10% 2516.619 370 x 10^ 2585.719 375 x 10% 2655.571 380 x 10% 2726.395 385 x 10; 2798.288 390 x 10% 2871.417 395 x 10^ 2945.489 1400 x icr 3020.505
85
Table IhB
NORMAL SHOCK WAVE CHARACTERISTICS OF A TENTATIVE VENUS ATMOSPHESE
TEMPERATURE RATIO Tg/l^
Velocity Altitude (Km) (ft/sec) 0 23.10 U2.16 60 80 100
20 x 10p 1.222 l.lt07 1.824 1.824 1.824 25 x 10p 1.383 1.655 2.270 2.270 2.270 30 x 10p 1.566 i.gho 2.776 2.776 2.776 35 x 10p 1.776 2.258 3.340 3.340 3.340 kO x 10p 2.009 2.618 3.958 3.958 3.958 1*5 x 10p 2.261+ 3.012 4.646 4.645 4.645 50 x KC 5.^38 5.385 5.382 5.570 55 x 10p 5.893 6.174 6.164 6.136 60 x 10p 4.378 6.997 6.942 6.862 65 x io; 4.892 7.847 7.700 7.503 70 x 10p 5.1*00 8.711 8.438 8.048 75 x 10p 5.893 9.554 9.208 8.474 80 x 10p 6.367 10.515 9.977 9.034 8.747 85 x 10p 6.810 11.054 10.716 9.655 8.796 90 x 10p 7.208 11.666 11.1*02 10.280 8.753 95 x lOp 7.551 12.187 12.006 10.901 8.931
100 x lOg 105 x lOp
7.989 12.571 12.491 11.507 9.368 12.866 12.853 12.087 9.819
110 x lOp 13.508 13.066 12.628 10.277 115 x lOp 120 x lOp
14.137 13.090 13.122 10.737 14.742 12.985 15.516 11.191
125 x lOp 15.311 13.015 15.814 11.629 130 x lOp 135 x 10
15.850 15.497 15.996 12.043 16.284 15.980 14.054 12.559
11*0 x lOp 1^5 x lOp
16.651 14.457 13.895 12.628 17.092 14.925 13.552 12.851
150 x lOp 155 x lOp 160 x io2
165 x lOp 170 x lOp 175 x lOp 180 x lOp
17.641 15.573 15.524 15.019 18.251 15.799 15.552 13.122 18.985 16.195 15.617 13.152 19.740 16.549 14.517 15.091 20.519 16.867 15.044 12.908 21.517 17.441 15.792 15.215 22.355 18.492 16.555 14.749
185 x lOp 25.522 19.656 17.516 16.545 190 x lOp 24.747 20.954 18.856 18.088 195 x lOp 26.021 22.294 20.4UO 19.541 200 x lOp 27.244 23.672 21.936 20.394 205 x lOp 28.651 25.095 23.090 21.224 210 x lOp 50.229 26.364 24.089 21.804 215 x 10 51.556 27.404 24.915 22.104
8k
Table IkB (cont'd)
NORMAL SHOCK WAVE CHARACTERISTICS OF A TENTATIVE VENUS ATMOSPHERE
TEMPERATURE RATIO Tp/T
Velocity (ft/»«c)
220 225 230 235 21*0 2^5 250 255 260
10' 10: 10^ 10^ 10^ lot 102 102 iof
265 x 10^ 270 275 280 285 290 295 300 305 310 315 320 325 330 335
x 10' 10; 10^ 10: 10^
10 102 102 10p 10p 102 102 10„
3kO x 10' 3^5 350 355 360 365 370 375 380 385 390 395 UOO
10, 10: 10^ 10^ 10^ 10^ 10' 102 10^ 102 102 10
23.10 Altitude (Km) U2.16
32.660 33.57U 34.313 34.863 55.216 35.967 36.811 37.654 38.495 39.252 39.774 40.239 to. 656 41.036 41.585 42.186 42.770
60
28.290 29.002 29.523 30.085 30.574 30.990 31.344 31.707 32.245 32.727 33.146 33.493 33.758 33.928 34.045 54.638 35.247
80
25.543 25.951 26.108 26.208 26.187 26.029 26.319 26.782 27.250 27.658 28.062 28.425 28.755 29.040 29.281 29.470 29.602 50.060 50.771 51.599 52.505 55.481 54.528 35.638 56.821 58.099 59.151 40.126 40.995
100
22.094 21.754 21.565 21.942 22.411 22.885 25.555 25.755 24.145 24.491 24.791 25.058 25.220 25.518 25.407 25.661 25.926 26.206 26.615 27.401 28.241 29.117 29.995 50.806 51.557 52.255 52.965 55.659 54.267 54.844 55.591 55.949 36.486 36.992 57.465 57.897 58.525
85
Table IhC
NORMAL SHOCK WAVE CHARACTERISTICS OF A TENTATIVE VENUS ATMOSPHERE
DENSITY' RATIO, Pp/p,
Velocity (ft/see)
20x10^ 25x10^ 30x10^ 35x10^ UOxlO^ ^5x10; 50x10^ 55xl0; 60x10^ 65xlo| 70xio; 75x10^ 80x10^ 85x10^ 90xio; 95x10^
100x10^ i05>ao^ 110x1 o;; 115x10^ 120x10^ 125x10; 130x10^ 135x10^ 1^0x10^ 1^5x1o; 150x10; 155x10^ 160x10^ I65xl0; 170x10" 175x10^ 180x10^ 185x102 190x10^ 195x10; 200x10^ 205x10; 2ioxio; 215x10; 220x10^ 225xl02
0
2.201 3.< 3- U.772 5.^65 6.131+
Altitude 23-10 1+2.16
2.8V7 3.697 3.81)4 4.724 1+.775 5.625 5.629 6.1+21 6.353 7.125 6.922 7.675 7.^90 8.097 8.139 8.656 8. 52I+ 9.O95 9.064 9.491 9.476 10.056 9-792 10.332
10,163 10.619 IO.65I+ 10.984 II.323 11.491 12.137 12.191 12.370 13-125
14.212 14.461 14.681 14.901 15.154 15.478 15.914 16.512 16.935 17.057 17.029 16.742■ 16.443 16.172 15.970 15.403 14.943 14.834 14.893 14.703 14.505 14.304 14-053 13.943 13.961
(Km) 60
3.697 4.724 5.625 6.421 7.125 7.676 8.IO3 8.666 9.171 9.649
10.362 10.680 IO.827 10.932 II.O67 11.311 11.728 12.340 13.202 14.509 . 16.220 17.651 17.750 17.804 17.838 17.879 17.955 18.093 18.318 18.662 19.131 19.037 17-971 17.280 17.319 16.972 16.477 16.331 16.130 15.860 15.696 15.682
80
3-697 4.724 5.625 6.1+21 7-125 7.681 8.122 8.697 9.272 9.837
IO.626 II.692 12.169 12.308 12.253 12.025 11.655 11.194 10.695 10.522 10.846 11.393 12.198 13.337 14.970 17.377 18.933 20.1+18 21.632 20.577 19.800 19.463 19.726 19.644 18.408 17.846 17.660 16.927 16.302 15.854 15.656 15.895
100
II.568 I3.I83 15.171 16.495 16.840 17.015 17.032 16.918 16.703 16.428 16.139 16.129 16.223 16.513 17.089 17.903 19.019 20.556 22.773 22.658 I8.651 19.301 19.590 18.466 17.622 17.122 17.025 17.394 18,341 20,010
86
Table Ite (cont'd)
NORMAL SHOCK WAVE CHARACTERISTICS OF A TENTATIVE VEMJS ATMOSPHERE
DENSITY RATIO, p /p
Velocity Altitude (Km) (ft/sec) 0 23-10 k2.l6 60 80 100
230x10^ 235xl0; 2ij.0xl0^ 2^5x10^ 250x10^ 255xl0| 260x10^ 265x10;; 270x10^ 275x10^ 280x10^ 285xl02
Ik.083 15.857 16.501 21.400 1^.308 15.966 17.194 21.596 14.695 16.140 18.096 21.567 11+. 695 16.38O 19.258 21.476 Ik. 566 16.677 19.645 21.375 Ik. 391 16.940 19.697 21.277 14.203 16.926 19.741 21.200 14.175 16.946 19.783 21.167 14.354 17.027 19.837 21.202 14.581 17.187 19.931 21.344 14.848 17.456 20.062 21.638 15.143 17.871 20.245 22.174
290x10^ 295x10^
15.276 18.377 20.499 22.727 15.348 18.308 20.846 22.849
300x10^ 15.440 18.283 21.312 23.093 305x10; 21.156 23-319 310x10^ 20.774 23.314 315x10^ 20.394 22.776 320x10^ 20.180 22.526 325x10; 20.246 22.698 330x10^ 20.289 23.097 335x10^ 19.950 22.805 3^0x10^ 19.623 22.611 3^5xl0; 19.332 22.432 350xl0; 19.108 22.267 355x10; 360xl0;
18.894 22.129 I8.761 22.074
365x10; 370xl0;
22.002 21.951
375xl0; 380x10; 385x10^
21.867 21.795 21.747
390x10; 21.731 395xlo; lj.00xl0d 21.7^9
21.761
87
Table IkV
NORMAL SHOCK WAVE CHARACTERISTICS OF A TENTATIVE VENUS ATMOSHIERE
VELOCITY RATIO, Ug/i^
Velocity Altitude (Km)
(ft/sec) 0 23.10 14-2.16 60 80 100
20x10^ .k5h • 351 .270 .270 .270
25xlO„ .32h .260 .212 .212 .212 y O
30x10^ .252 .210 .178 .178 .178 p
35x10^ .210 .178 .156 .156 .156
IvOxlO^ .183 .157 .IkO .11+0 .11+0
^5x10; .163 .Ikk .130 .130 .130
50x10^ .13h .121*- •123 .123 P
55xl0; .123 .116 .115 .115 60x1 o;; .117 .110 .109 .108
65xl0p .110 .105 .104 .102
70x10^ .106 .099 .097 .09I+
75x1°^ .102 .097 .091+ .086 P
80x10^ .098 .09I+ .092 .082 .086
85x10^ .09I+ .091 .092 .081 .076
90x10^ .088 .087 .090 .082 .066
95xl0„ .082 .082 .088 .O83 .061
lOOxlOp .081 .076 .085 .086 .059 105x10^ .070 .081 .089 .059 110x10^ .O69 .076 .093 .059 ii5xio| .068 .069 • 095 .059 120xl0p .O67 .062 .092 .060
125x1Op .066 .057 .088 .061
130xl0p .065 .056 .082 .062
135xl0p .O63 .056 .075 .062
lit-OxlOp . O61 .056 .067 .062
1^5x10^ • 059 .056 .058 .061
150xl0p • 059 .056 .053 .059 155xl0; • 059 .055 .01+9 .056
l60xl0p .060 .055 .01+6 .053 165x10p .06l .05I+ .01+9 .01+9
170xl0p .062 .052 .050 .01+1+
175xl0p .063 • 053 .051 .01+1+
iBOxlOp .O65 .056 .051 .051+
185x10p .067 .058 .051 .052
190xl0p .O67 .058 .051+ .051
195x10^ .O67 .059 .056 .051+
200x10 .068 .061 • 057 .057
205xl0p .069 .061 .059 .058
210xl0p .070 .062 .061 .059 215xl0p .071 .063 .063 .058
220xl0p .072 .064 .061+ .055 225x1Op .072 .061+ .063 .050
2^0xl0„ .071 .063 .061 .01+7
235x10^ .070 .063 .058 .01+6
88
Table IkD (cont'd)
NORMAL SHOCK WAVE CHARACTERISTICS OF A TENTATIVE VENUS ATMOSPHERE
VELOCITY RATIO, Ug/u.
Velocity Altitude (Kin) (ft/sec) 0 23.10 42.16 60 80 100
240x10^ .068 .062 .055 .046 245x10^ .068 .061 .052 .047 250x10; .069 .060 .051 .047 255xl0; .069 .059 .051 .047 260x10; .070 .059 .051 .047 265xl0; .071 .059 .051 .047 270x10^ .070 .059 .050 .047 275x10; .069 .O58 .050 .047 280x10^ .067 .057 .050 .046 285xl0; .066 .056 .049 .045 290x10^ .066 .054 .049 .044 295x10; .065 .055 .048 .044 300x10^ .065 .055 .047 .043 305x10; .047 .043 310x10; 315x10; 320x10^ 325xl0; 330xl0; 335xlo; 340x10^ 345x10^ 350xl0; 355xl0; 360x10;
.048 .043
.049 .044
.050 .044
.049 .044
.049 .043
.050 .044
.051 .044
.052 .045
.052 .045
.053 .045
.053 .045 365x10; .045 370xlo; 375x10; 380x10; 385xlo; 390x10; 395xlo; 400x10^
.046
.046
.046
.046
.046
.046
.046
8.9
Table 15
EQUILIBRIUM CONSTANTS, K (atm)
T C^) Reaction
1 2 3 k 5
1000 3.022-68 2.525-69 1.4604"59 2.397-60 2.216"56
1500 1.215-^ 1.375^5 3,492-39 5.054-^ 5.438-37
2000 8.970"33 1.219"33 6.332-29 2.664"33 3.135-27
3000 9.25i+"21 I.793-21 I.855-18 I.979-21 2.519"17
4000 2.828"15 4.191"13 2.IO8"15 2.765-12
5000 1.686-:L1 7.60I-10 9.868~12 3.290~9
6000 6,088"9 1.209"7 2.995"9 3.984"7
Table 16
EUSCmON CONCENTRATIONS TOR PURE C0„. n - 2' e
(g-moles electrons/44.011 g mixture)
p (atm) " ' '
T (OR) 102 10 1 10-1 10-2 IQ"3 10-*
2000 3.32-17 I.52-16 6,9Q-16 3.21-^ 1.43-^ 6.79-1* 3.O6-13
3000 3.56-11 1.58-10 6.84-10 2.61-9 7.27"9 1.15-
8 1.4Ö-8
4000 3.59"8 1.29"7 3.20'7 5.19'7 LOS"6 3.40-6 1.63-5
5000 1.77-6 3.93-6 8.26-6 2.61-5
1.4-* 1.1-3 7=4"3
6000 I.33-5 4.41"5 1.8"* 1.2-3 7.9'3 3.3-2 l.l-1
90
Appendix B
^ffl . COMPUTATIOH OF TWO-BIMEUSIQNAL OBLIQUE-SHOCK-WAVE CHARACTERISTICS FROM NORMAL-SHOCK-WAVE DATA ~
Hie practical problems usually associated with aerodynamics generally
involve heat transfer and aerodynamic forces which arise due to the motion
©f a body through a fluid. In the present application, both of these problems
require a knowledge of the so-called inviscid parameters associated with
both shock waves and expansive-type flows such as the Prandtl-Meyer expansion
around a comer. The purpose of this appendix is to furnish a practical way
to calculate the oblictue-shock parameters. Since aerodynamic parameters
«hange only in the direction noimal to shock, only the oblique shock, angle
is required to solve the problem, when the normal-shock solution (Section IV)
is known.
a» present method may also prove useful for confuting oblique-shock
characteristics f»r aiay selected gas eompositian once its equilibrium thermo-
dynamic properties are known. While the generally uaed iteration scheme for
obtaining noimal-shock data(20) is highly convergent, the iteratien scheme
for oblique-ahoek solutions presented in Ref. '20 is not, especially near the
detachment point. It is presumed that if it is practical to obtain normal-
shock data for a particular composition, the oblique-shock characteristics
can be obtained by this method for a particular case.
As previously mentioned, all of the physics of the problem are contained
in the noimal-shock-wavB data and the fact that the velocity component paral-
lel to the shock wave remains constant across the shock (this follow^ directly
fyran a conservatlon-of-mass-and-monentum analysis across a flov discontinuity).
Therefore, no physical assumptions about the flow are necessaiy.
91
What is uBually apeeifled in an obli<|ue-shock problem ie the flow de-
fleeti«i»n angle, the upstream thersacxfynamie state, and the upetream velocity.
If the -wave inclination angle may be found, all downstream parameters may
be calculated. Unfortunately, no explicit expression for the wave inclination
angle exists, and a practical iteration scheme uhieh always converges must
be found.
Referring to Fig. 12, the following trigonomstric relations across an
obllqrje shock becoios obvious:
"! = ^ sin ß (39)
U2 .. tan (g - 6) V^ tan ß (W)
The same conservation equations apply to the normal stream cociponent across
a flow discontinuity as apply to the stream across a nomal shock wave. Since
for a given upstream state u^ vs u1 Is usually plotted for a particular
gas if nomal shock wave data are given, this relation must also hold for
the noiTnal stream components across an oblique shock if the upstream themo-
dynamlc state ±s ^ same- ^e general form of the relation Is shown In
quadrant 2 of Fig. 13. This curve always assumes this form for a perfect
gas. Real-gas effects will usually destroy the monotonlc nature of this
curve and will, m exceptional cases, cause a positive slope in a small u, (20) ^L
range. However, this presents no problem, as will be cone evident later.
Equation (39) is very simple and presents no computational problem.
Considering Eq. (1,0), the following equation for ß may be developed after
some manipulation:
92
P2,T2,/02
I ¥ 'III!!! 1111111111 Jt i I i i i i i i i i i i 1 l f i
Body
/////////////
Fig. 12 — Oblique - shock diagram
93
'-1 4 U|
UJ/UI vs U| from
normal shock wave data (general form)-
^z.. tan(/3-g) ui ' tan /3
/3*
u2
(D 1 ©
, © /3:90o ^ Quadrant
Legend
Fig. 13—Graphical solution for U2/U. and ß
(weak-shock solution)
ß = arctan
2 I -— I tan 6
91+
■ - ~ ^1 + 1/ 1 - 2 f^ Vl + 2 taxFe] +{U2
\ A A/ (hi)
Differeatiating Eq.. (^0)_, the maxiinum of u /xu vs ß with 0 a fixed parameter
occurs at
and
ß = ^5° + | (42)
1 - sin 0 1 + sin 0 (^3)
/raax
Also, referring to Pig. 13, it is seen that the negative sign in Eq. (kl)
corresponds to 0 < ß < (45 + 6/2), and the positive sign correBponde to
(14,5° + 0/2) < ß < 90°. Note that 0° < ß < 9O0 is the only range of interest.
For reasons that will become apparent later, it is usually ß which must he
calculated from a known u /u,, rather than vice versa. Since Eq. (4l) re-
quires a rather tedlsrus computatlen If done by hand in an iteration scheme.
Fig. 14 is included for 0 < 0 < 3O0.
It may be shown that ß < (45 + 0/2), i.e., ß < ß correBponding to
(u /TL. ) for a given 0, must always be in the weak-shock-wave region. max
Stating this another way: if, for a given upstream state, ß were plotted
vs 0 with upstream velocity as a parameter, ß < (45 + 0/2) would always
be less than the ß dictated by the line connecting the ma-rirmm Q solutions
for a given upstream velocity» This may be seen for a perfect gas Ttgr plotting
a line ß « (1*5° + 6/2) on pp. 42 and 1+3 of Ref. 22. The Intersection of this
95
-ht^ :^ s ■ f
0
10
20
3
0
40
5
0
60
7
0
80
9
0
Wav
e a
ng
le,
/3
(deg)
,^ .-* -.'• ■n^P* ■^^ W
/ ^ ,S ', -n^P^ ̂ <: ̂ ^^
/ ' ^ M ̂ ^^ ̂ ^^ / / / ///
| ̂ //////
/ ( / / / / // //MA / O)
it / / / // T/ 7 ^///// / en
i 1 / / //// / //// / / ]
o I >
r ^ n //
r > // V
» /
f >< o E
CD l I / / / 7/ V / 7 y r > o 5 / / / fill / • S
3
! fill M ■
' 1 | x'
■ '
3 o o
i_ X .[ ' (fl
^ ^ 1 I o
*> ^ ,-"
\ ^-' \ \ \l\ vW o
I \ \ W \\ K
—
en i \ \ \ \ ^w k \
\ !
H \ \ \j\ \\ \ i \ i \ V Xi- I l\l\l\ \ \ \ v ^ o \- x\ — H- \ i \ ' V A \\ \l NjS\ N
sSs^V o o
I \ \ \ V \ \ \^o\\ NyX^V sz c — l \ ^ \ \ \ \ sV v ^S X^S^s Q-
a
CD *o \ \ V \l ' vN \^
\J sNJ SsVsv^s
T) — I \ \ \ \ -N 'V
Nj^t vN ̂ x^SsT" 0) .^
\ i \ N \ .°s s. \ vs >vj ̂ O^I^
— Ll_ l\ \ \ -\ \ \ H. TN^ ^^^^ ""S^, »J-
<u a> "a ^
\ N
s. "\ ss v
\ > ^ •^s r*' en
<u « > v .s \ Ns v. ■s.
"% ^
LL.
0 £ j 1 a \
n o>
M
nCD
v N "--v ^ i
E => o ( c
_ c c \ V
^^ -
o <u *- in
v ^
" -- H —- ». c: —. z ̂ H — —
o 1- i ■■■ —
C ) a c
) ) c
> >
c M — J 3
c > c c
96
0.35
0.30
0.25
0.20
0.15
0.10
0.05
I
1
/ 1 /
/ /
1 1 /
/ 1 f /
/ ' j j -J.
j /
/ / 1 / /
j / / 1 / 1, /
e = 2' ->
A f0
50l 8°/
/ / , / / /
10°/ 12°/
14°/
16°/
8°/ / /
1 1 / /
1
/ 2 0°/
/ 2: -"
0 2 4 6 8 10 12 14 16 18 20 22 24
ß Fig. 14a
97
0.35
yü
0.35
0.30
0.25
0.20
0.15
0.10
o.oe
e=280.
/
^
30° \ ——.
*=:
^ \
y'
\
\
44 46 48 50 52 54 56 58 60 62 64 66 68
ß Fig. 14c
9,9
0.35
Fig i4d
100
0.65
0.60
0.55
U2 rr^ 0.50 u
0.45
0.40
0.35
0.30
/ / z /
1 /
/ 7
i / / / 7
|
1
I / / 7 /
/
/ / / / ' /
9 = 2°
40
/ 6°/
/ / / /
1 1
/
8°
/
'
10°/
12°
/
1 40/
16°/
0 2 4 6 8 10 12 14 16 18 20 22 24 ß
Fig. I4e
101
v.ou / / ^S 1 1 1 \ \ ^ :=>—
/80
/
0.60 / / ,'
e = ic y /
0,55 / \zy /
/ /
/ /
/ ^
iiy / u2
— 0.50 / / / ̂ ^ ul
/ / 16^
0.45 / / 1
18V / ^
/
/
/ / / 2C )y /
0.40 / / / / / y
/ / / / /
2.Z°/
/ / /
' / / /
/' / ZAy ^
0.35 / / / / / /
/ ' 7
/ / ' /
/ / -
26>' y
0.30 - / i / /
/ / / / zsy
22 24 26 28 30 32 34 36 38 40 42 44 46
Fig. I4f
102
0.65
0.60
0.55
0.50
0.45
0.40
0.35
0.30
\
h-^ ^
10^
e «12°
\ "^
^^ -—
\
-^
\14 o
^ — ^ ̂
^16°
18° \
\
,-
\
^ -^20
-22°
24°
O
^ \
\
/
\
~- ^
^ \
' / - 1
\ 2^
28°
)°
- \ \
\
/ /
^
-
\ 30' >
- ̂ \
44 46 48 50 52 54 56 58 60 62 64 66 68 ß
Fig. !4g
103
0.65
66 68 70 72 74 76 78 80 82 84 86 88 90
Fig. 14h
icA
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
/ -^
/ /
s
/
1 / /
/
e = 2°/ /
/
/ /
/
1
4 O /
/
/
/ /
/
6°/
%"/ /
10 12 14 16 18 20 22 24
Fig. !4i,
105
0.95
0.90
0.85
0.80
0.75
0.70 —
0.65
0.60 22 24 26 28 30 32 34 36 38 40 42 44 46
Fig. I4j
106
0.9
0.90
0.85
u2 0.80
0.75
0.70
065
0.60
107
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
^^
\
se = 2 o
i 1
^ x
\ \4° \
\
\ \6"
\ \
\
X
\ .8°
\
\ \ \
\
\J2 o
10°
\ \ \
\
\
66 68 70 72 74 76 78 80 82 84 86 88 90
ß Fig. 141, /
108
line with 0=0 „at Mach = <» is not a coincidence, hut, as may be shown,
a consistent result. The existence of real-gas effects does not alter the
original statement, and as the upstream velocity "becomes large ß = (k'y + 0/2)
approaches the limit of the -weak-shock solutions. At low upstream Mach
numbers a large range of weak-shock solutions lies where ß > (45° + 9/2), as
may be seen on p. k2 of Ref. 22 (at low Mach numbers a gas usually behaves
as a perfect gas). At any rate, the majority of physically possible solutions
■where the upstream Mach number Is greater than 2 will occur where ß < (k-5
+ B/S) J this will turn out to be fortunate, considering convergence of the
method.
Now a solution satisfying Eqs. (39) and (kO) and the normal shock
relation of vip/u, vs u.. for a given upstream state, w., and 0, is desired.
A qualitative plot is now constructed in Fig. 13 as follows:
Quadrant 1: A functional relationship between u /u, and ß for
the given 6 (Eq. (J+O))
Quadrant 2: The normal-shock relation u0/u_ vs u, for the given
upstream state (contains the physics of the problem)
'Quadrant 3: A functional relation between u, and ß for the given
w, (u, = w. sin ß)
Quadrant k: A 45 line to graphically convert the ß axis in
quadrant 1 to the ß axis in quadrant 3
Line ABCDA can be seen to represent the weak-shock solution. The procedure
to arrive at this solution is as follows:
1. Assume ß = 90 in quadrant 3 such that u, = w.. (point E).
2. From vu = w find u /^ in quadrant 2 (point F).
3. From u /u-, find ß in quadrant 1 (point G).
109
h. Carry over this ß to quadrant 3 through quadrant k (point H). It
is now apparent that through the monotonic nature of all the functions the
solution has laeen bracketed between ß = 90° and ß„ = ß^. G H
5- Obtain li. in quadrant 3 (point J).
6. Obtain u /u- in quadrant 2 (point K),
7. Obtain ß in quadrant 1 (point L). Again it is obvious that the
solution is now bracketed between ßT and ßn = ß-, ij GH
8. Continuing in this clockwise fashion, it is seen that the process
is rapidly convergent.
It is Important to note that any attempt to construct this solution
in a counterclockwise manner would have failed, since the process vould
diverge. It is easily seen that this process converges in the sauie manner
■when the normal-shock relation in quadrant 2 contains inflection points
but retains negative slope. It will be stated without demonstration that
this process also converges when the nertnal-shock relation contains a small
region of positive slope as previously mentioned. In this case, however,
the convergence to the solution Is from one side only; the solution is not
continually being bracketed as above.
One other difficulty may be encountered. On the second iteration
point K may correspond to a Ug/i^ > (Ug/i^) . In this caB« the starting max
point for the third trial is the ß corresponding t© (tu/u, ) , I.e.,
. o / ~ max ß a (lt.5 + 9/2). If ©n the third trial the ratio VL/VL. obtained, ia qijairant
2 iß greater than (u /y.) , the solution must be where ß > (1*5° + ©/2). max
Figure 15 illuatrates the case mentiened imn«dlately above. Following
the lines JKLMN and OPQR, it is apparent that the solution must not be
where ß < (^5 + 9/2). Now starting again at point J and proceeding to K,
110
Quadrant Legend
® f o
(D i ©
Fig. 15 — Graphical solution for U2/U| and ß
(strong-and weak-shock solution)
Ill
continue the KL line to S where ß > (45° + 0/2). Continue with a clockwise
iteration starting from JKSTV, always selecting ß > (45° + e/a) in quadrant
1; "this is seen to converge to the solution ABCDA. Alternatively, starting
from (ug/Uj^) at point 0 and proceeding counterclockwise. the iteration max
begun by OWXYZ is seen to be converging to the solution EFGHE. It may also
be seen that if ß is originally selected such that ß < ß < ß the clock-
wise iteration will converge to ABCDA and counterclockwise iteration will
converge to EFGHE, It will also be stated without demonstration that these
concLusions are not altered if small regions of positive slope occur in the
normal-shock relation (quadrant 2). It then becomes apparent that EFGHE is
the weak-shock solution and ABCDA is the strong-shock solution.
With all cases now considered, the general method to be followed in
obtaining a solution is outlined below.
Given: upstream state, B, v±> and the normal-shock relation between
^A^ arid ^ for the given upstream thermodynamic state.
A. Vfeak-shock solution
1. Assume ß = 90°, i.e., ^ = w
2. From the normal shock relation obtain u /u.
3. From Eq. (1+1) or the included graphs, obtain a new ß, choosing
ß < (1+5° + 0/2)
k. From this ß compute a new tu from Eq. (39)
5. Prom the new u^ and the normal-shock relation obtain a
new Ug/i^
6. If this UgA^ < (ug/t^) for the given 0 from Eq. (1+0), max
repeat this procedure to the desired degree of convergence
7- If this Ug/^ > (Ug/uj^) assume ß = I4.50 + 9/2 and compute a
112
new iL. from Eq.. (39)
8, Using this v^ and the normal-shoek relation obtain a new
U2/Ul
9. If this Ug/x^ < C^/^) proceed as in step 6 max
10. If this Ug/^ > (Ug/ujJ , the solution will not lie where 0 max
ß < (^5 + 0/2); asaume Ug/i^ = (u^/u^) ; obtain u^ from ▼no ^r max
the nonnal shoek 3relati©n
11. Cojspute ß from Eq, (39) and this u.
12. Using this ß compute u /^ from Eq. (40) or obtain ß .from
the incluied charts
13. Using this Ug/t^ obtain u_ from the normal-shock charts
Ik. Repeat steps 11 - 13 until the desired degree of convergence
is reached
B. Strong-shock solution
1. Assume ß = 90°, i.e.., IL = w
2. From the normal shock relation obtain u /u.
3. From Eq, (It-l) or the included charts obtain a new ß, choosing
the ß > (I+50 + e/2)
h. Compute a new u. from Eq, (39)
5. Repeat steps 2 ~ k until the desired degree of convergence
is reached
It may be sho-wn that if the normal shock solution is obtained by an
iteration scheme that is stopped -when two consecutive values of u /u, differ
by no more than 0.0001, a consistent degree of convergence in this method
is arrived at -when two consecutive values of ß differ by no more than 0.05
per cent, i.e., Aß/ß < 0.0005; ho-wever, this is by no means absolutely true
115
over the entire range. The rapid convergence of this method is illustrated
by three numerical examples for air given below.
It should be kept in mind that this procedure is not particularly
useful when a complete set of oblique-shock charts for a known atmosphere
is to be constructed. In such a case all that needs be done is to specify
a ß and iterate the normal-shock equations. Hhe corresponding 0 can then
be immediately calculated.
EXAMPLB3
1. Very Weak Shock
Given: w = 8000 ft/sec in air
Altitude = 82,3lt-5 ft (1956 ARDC model atmosphere) ^^
9 = 20°
Find: ß for the weak-shock solution
Solution: (a) Assume ß = 90°, then u, = 8000 ft/sec
From the normal-shock data in Ref. 20, u /u. = 0.1^35
From Iq, (kl), taking the negative sign for ß < (^5° + e/2)
ß = 23.58°
The solution is now bracketed between 23«58 and 90
(b) Taking ß = 23.58° and vising Eq. (39), u. = 3201 ft/sec
From the normal-shock data "-p/u-i = 0.2U0T
FromEq. {kl), ß = 26.98°
The solutlen is now bracketed between 26.98 and 26.16^
Contlniiing:
(b.l) v^ « 363O ft/sec
U2 ~ = 0.2198 "l
11^
ß = 26.16°
(TD.2) V^ = 3527 ft/sec
u2
~ = 0.22^3
ß = 26.33°
(b.3) ^ = 35^8 ft/sec
U2 -^ = 0.2230
ß = 26.29°
(h.k) 1^ = 3544 ft/sec
u_ —- - 0.2230 within the accuracy with idiich the charts
can be read
ß = 26.29 -vAiich is the solution compared with
ß = 26,3 obtained frcm Ref. 20
2.—Strong-Shock Solution anä Til nitration of Convergence Difficulties for the Weak Shock ö iiii:
^■ron ^ = 3000 ft/sec in air
Altitude = 175, 32^8 ft
0 = 30°
Find: ß for the strong-shotik solution
Solution: First, hunting for the weak-shock solution, assuma
ß = 90°
^ = 3000 ft/sec
From the normal-shock data
^2 r~ =■ 0.21h0
115
From Fig.'Ik for 0 = 30°, ™ = 0,2740, and ß < (^5° + 0/2)
ß = k5.6j0
From Eq. (39)
v^ = 2176 ft/sec
and from the normal-shock chart
-2 = 0.3866 u. X
But, as can he seen in Fig. 14, this is greater than
(?) \ /max
Therefore, assume
^ 'max
From Eq. (39)
^ = 2598 ft/sec
and from the shock chart
U2 ~ = 0.31^+0
U2 vhich is now less than — and the process will con-
"l max
verge to the weak-shock solution where ß < {k^0 + 9/2).
Looking for the strong-shock solution, choose ß = 90°,
1^ = 3000 ft/sec. Then as before u^u, = 0.27^0, Now
from Fig. Ik pick ß > (45° + 0/2).
ß = 74.32°
From Eq. (39)
u1 = 2888 ft/sec
116
From the aheek charts
~ = 0.2843
From Fig. Ik
ß « 73.07°
Continuing:
i^ =■ 2870 ft/sea
U2 -* = 0.281^8 T.
ß =- 72,99°
^ = 2869 ft/sec
u2 -^ =■ O.285O
ß =■ 72.96°
And since en the la«t trial
1^ = lz'9^9l2'96 = o.oooin < 0.0005
ve stap the process and take the strong-shock solution as
ß =■ 72.96°
3. Weak-Shock Solutien on Right Half of Ug/v^ vs ß Curve
Given: Mach number = 1.9 in air ^lieh is assumed a perfect gas
Wormal-shaick relations are taken from Ref. 22. 0 = 20°
Find: ß for weak shock
Solution: Clearly assuming ß = 900, J^ =1.9 viu nelä
normal
117
"2 .. (U2
"i w max
on the second iteration. Therefore assume
v 'max
From Eq. (39), using the Mach numbers instead of
velocity
K^ = I.556 n
From Ref. 23
u 1
which is greater than f^j . Therefore, the weak-
shock solution must lie whe&e ß > (kj0 + e/2).
Now, using the previously mentioned counterclockwise
solution, choose ß = (kj0 + e/2) and ' '
\ /max U2 — = O.I+9O3
JL
From the normal shock tables
J^ = 1.601* n
and from Eq. (39)
ß = 57.59°
Fram Fig. Ik
rf - O.kQQS
118
Continuing:
KL =1.608
= 57.81°
-§ = o.k88k
M. = 1.6091
ß = 57.87°
-§ = OA883
M. = 1.60914- n
ß = 57.89°
and since
f = ^^gf^ - 0.0003 < 0.0005
the process is stopped and
ß = 57.89°
compared to the solution in Ref. 22 of
ß = 57.9°
119
Appendix G
GALCÖLATIOH OT ELECTRON CXJNOEWTRATIONS AT LOW TEMFERATORES IK THE C-N-0 GASEOUS SYSTEM ~
It is tourm that rtrong interference with radio or telemetry trans-
mission from atnospherlc-entry vehicles is possible if the transmission
frequency is equal to or less than the plasma frec[uency of the medium sur-
rounding the antenna and if the collision frequency is large enough. Plasma
frequency is directly dependent upon the square root of the electron con-
centration in the surrounding medium, and it may be desired to know this
-17 concentration to as low as 10 g-mole of electrons per g-mole of gas
mixture. This concentration corresponds to a transmisBion frequency of 10
Mc through a medium, for example, at a pressure of ID2 atm and a temperature
of IGOOTC in the present tentative Venus atmosphere.
Since the equilibrium conposition of gases computed by the method of
Ref. 18 does not show the trace constituents. It is not suited to the cal-
culation of electron concentrations at low ten^peratures If the major con-
stituents of the gas mixture are known. In particular, the present calcu-
lations apply to the C-N-0 gaseous mixture but the general method may be
applied to any arbitrary mixture. The calculated electron concentration is
based on an Ideal gas mixture in thermodynamlc equilibrium vhen the major
constituents of the gas mixture are known. Since the major constituents of
the gas are generally computed on the basis of an ideal gas In thermodynamic
equilibrium, these conditions are consistent.
For a chemical reaction of ideal gases in thermodynamlc equilibrium,
the equilibrium constant may be written
120
K
n p.
n p. exp a RT
. I /F0
RT z
RT (44)
vhere 1 denotes the produets, J the reactants, and A tiie change as meaaured
by products minus reactants. frm. an exanalnatien of the reaction energies
for the C-N-0 system in Table k, it is evident that at low temperatures
the electrons may to a close approximation be considered to be produced
solely by the following reactions:
1. CO, c
2. CO
J 2
h. 0
5- C
6. NO
C02+ + e"
C0+ + e"
0g + e
0 + e
C + e
N0+ + e"
^E° = 317,731 cal/g-mole
^E° =■ 323,180 cal/g-mole
AE° = 277,900 cal/g-mole
AE° = 311;, 048 cal/g-mole
^E° = 259,842 cal/g-mole
AE° = 213,400 cal/g-mole
(24)
(9)
(9)
(9)
(9)
(9)
Using the lav of partial pressures the equilibrium constant for the corre-
sponding reactions may be expressed as
K Hi (n, pl n
CO, (£) vn/ (45)
K CO+- V C0
(I) (46)
121
3. K = '~UL±J (£) (ky Po n0
vn' Wf 2
P^ n0 ^n'' '.i*0^
nc+nv 5. Kp5 = -l-_^ (£) (,9)
6. K = (^ ) (V ) (2) (50) P6 ^o n (50)
Since it is assumed that all of the electrons coa» from the above
ionized species
V = nC0+ + "00+ + n0J + "0+ + nc+ + ^0+ (51)
Combining Eqs. (1+5) - (51), the electron concentration on the basis of n
g-moles of the mixture is
e n
0- = / n™ K + n^„ K + n K CO % nrn ^^ + no K^ + ^ K + n^ K + nTrt K - (52) Cü2 p1 co p2 o2 p3 o p^ c P5 "HO p6y lpyv:;'d',
Some equilibrium constants have been calculated arid are shown in Table 15 for a
range of 1000°*: to 6000*^. The amounts n± of the major constituents are assumed
known and may be placed in the computational Eg. (52) along with the equili-
brium constants, and the electron concentrations may be determined. It Is
12.1
suggested, however, that the accuracy obtained by this method be limited to
two significant figures at temperatures where electron concentration be-
comes significant, since the molar constituents assumed in the calculation
from Eq.. (52) do not account for the presence of electrons.
For illustrative purposes, results of some of these computations for
pure C02 corresponding to those of Ref. (1?) have been made and are pre-
sented in Table 16.
123
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12h
16. gummaiy of the Results of the Moore-Ross Flight of November 1959, Information Release by the Laboratoiy of Astrophysics and Physical Msteorology, The Johns Hopkins University, Baltimore, Maryland, March 1, i960.
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