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Calhoun: The NPS Institutional Archive
Theses and Dissertations Thesis Collection
1946-09
Investigation of the condensation shock in air by use
of the Schlieren method
Simons, Waldo Watson
Cambridge, Massachusetts; Massachusetts Institute of Technology
http://hdl.handle.net/10945/6440
lNN^]^iS.r^(^»ATIOK Of T!'(!! CONDf^NSATIONSHOCK m AIR li:Y USE OF TH£
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U. S. Naval Postgraduate School
Monterey, California
COPT FOR HEAD 07 P0STOR4DUATB SCHOOL
Library '~'
U. S. Naval Postgraduate Sciwxii
Annapolis, Md.
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IttS'^CflUSETTS INSTITUTE OF TECHNOLOOTDBparbment of Mechanical Engineering
Oarabridge 39, Maes^p UcSoAo
Room 1-202
Septembei 2U, 19h6
Captain W> lie Biiracisar
Roora 5-233Massachuaatts Insti'Jite of TechnologyCambridge 39» Maseachu-ji'Dtts
Tliosij WorJk of LT Eo Lo PERIil, USCOLT Lo W, Ao HENSHAW, USCGLCDR \Xo Wo SIMONS, USN
: ^ l>i€DR J. S. FOWEN, USN
Dear Captain Buracko;??
•flie theais bjr :-i«utenanta E> L. Parrj- and Lo W, Ao ianf h£ir
entitled "Schlieren Oboervation of Supersoric Discharg©"presents pressure mo-iisuretnents ani Schlieren photograph i? oi"
supersonic Btreams discharging into an exhaust space uncarvarious conditionso The pbotographfi show interesting d** tailwhich in general corresponds to analytical resuits c 'Rv: mostsignificant observa ;S.on '^as a comparison of two sinpersoricstreams aliko in avdirage conditions but dii'fering in th: ckne.st-
of the boundary layor^ Tii© effect of bounc^ary-lajer thlcknasji:
on the nature of th<> shO':k pattsi;a is showi clearllyo
The thesis by Lt^ Comdrs* Wo Wo Simons and J, S„ Bjmnentitled '=Inv®3tiga>:".on of the Goridenaation Shock in Aiv byUse of the Schlierea Sfct lod" presents pressure measurem^-iteand Schlioren photojjraphg of the shock patterns wtien watcjrvapor in air conden.'jtjs to form a fog of liquid or solidparticles,) It has ti.rtend.sd our kicwledge of the conditi :>n8
which control condenuatio.a and of tiie condensation shocv whichaccoiripanieo ito
From either of ihes?s theses a paper could be propaiidwhich would be publj f hed in one of the Journals c:!" theprofessional societies b,' o
lours tr-aly,
/s/ Joseph Ho Ktjenan
Joseph Ho Koenan
INVESTiaATIOH OP THE COIIDENSATION 5-HOOK IN AIR
HY USL OF THE SCHIJEREN METHOD
By
Waldo W.'' Simons, Lieut. Cotrimanrif^r, U.S. Navyb>.R., U.S. Naval Academy, 1941
John S. Bpwen, Lieut. Com.^iander , U.S. NavyS.B., U.S. Naval Academy, 1941
SUBI.1ITTOD IN PARTIAL FULFILLI.^ENT OF THE
RE'-;UIREI,iRNTS FOR THE DEGREE OF f;'ASTER OF SCIENCE
at theI.'IASSACHUSETTS INSTITUTE OF TECHNOLO^zV
1946
MASi-ACHlfSETTS INSTTTUTF OF TFOHMOLOGY77 Massachusetts AvenueCambridge, Massachusetts
September 16, 1946
Professor J. S. NewellSecretary of the FacultyMassachusetts Institute of Technology77 Massachusetts AvenueCambridge, Massachusetts
Dear Professor Newell:
Herewith we submit our thesis entitled
"Investigation of the Condensation Shock in Air by us©
of the Schlieren Method" in partial fulfillment- of the
requirements for the Degree of Master of Science in
Naval Construction and Engineering at the Massachusetts
Institute of Technology.
Very truly yours.
ACKNOWLEPgEMENT
v;e acknov/ledge with pleasure our
indebtedness to Professor J. H. Keenan
for his suggestion of the thesis topic
and method of attack. Professor E. P.
Neumann and Professor A. H. Shapiro f^ave
freely of their time in guiding the pro-
gress of our Investigation.
thble of contents
Pag;e
Index of Tables and Figures
I. Summary 1
II. • Introduction 3
III. Procedure 5
IV. Table of Symbols 9
V. Results 11
VI. Discussion of Results 32
VII. Kecommendations 49
VIII. Aopendix 50
IX. Bibliography 64
INDI';:X 0? TARTES AND Pia'fRPJS
Table I
Table II
Table III
Table IV
Table V
Table VI
Table VII
Table VIII -
Fii^ure I
Figure II
Figure III -
P'igure IV
Figure V
Figure VI
Figure VII -
Figure VIII -
Figure. IX
Figui^e X
Figure XI
Figure XII -
1
Data for P'igures V, \/I and VIIPage2o
Tsat - Ti 43
Original Data 51
Original Data 56
Original Data 58
Data of nozzle areas and arearatiosDetailed calculation across thecondensation shockDrop size calculation
59
61
62
Schematic Diagram of Apparatus 7
Location of PressureReference LinesPosition of shock VshumidityPosition of shock VshumidityP Vs distance along- -'
i-ou n H n
Taps and
relative
specific
the nozzle
S
21
22
25
tf n 26
V It tt rt ff fi 27
pQ - Pi Vs snecific ]
PoAi/at
"
numidity
ri
23
r
'V2/.V1 ^^ ^'^1 41
^sat ^^ specific humidity 44
Saturation pressures for variousdrop sizes
Figure XIII - Area and area ratio Vs distancealong the noz^.le
47
60
SUMMARY
The object of this investigation is to study the
effect of variation of moisture content in air upon '*
the formation of a condensation shock. An elector
was used to draw air through a two dimensional con-
verging-diverging nozzle with glass walls. Moisture
content of the air was varied bv a dehu.Tiidifier
.
Schlieren pictures of the flow were taken from v;hich
measurements of the position of the shock were made.
Also pressure measurements werie taken alcmg the nozzle.
The humidity of the inlet air was measured with wet
and dry bulb thermometers.
The results obtained by correlation of the data
for a nozzle of constant angle of divergence are:
(1) The pressure rise due to a condensation shock
is a direct function of the specific humidity of the
inlet air. (Figure VIII).
(2) For a given inlet temperature, the distance
of the condensation shock downstream of the throat
varies inversely as the specific humidity of the air.
(Figure IV) . From this relation, it is postulated that
the position of the area ratio at which the shock would
occur can be plotted against specific humidity for a
given inlet temperature. In this v/ay, the curve will
2,
serve for nozzles of various contours. (Figure IX).
(5) The amount of water vapor condensing out in
the condensation shock con be predicted if the inlet
specific ha.n.idity is known (Pi/rure X).
(4) The difference between the temperature at
which the water vapor becooes saturated under equili-
bruim conditions and the temperature at which the
condensation shock occurs, as the humid air passes
through the nozzle, is essentially a constant of about
110° p.
(5) The drop size of the vi^ater condensing out
in the shock is essentially/ a constant. This is shown
on Figure XII
«
INTRODUCTION
V/hen humid air is exoanded by pas3a,Q;e through a
simple convergent-diverp;ent nozzle, 'the water vaoor
remains a vapor at the saturation temperature and below
and continues to expand into the sunersaturated region.
Part of the water vapor in this meta- stable state con-
denses out rapidly at a certain point along the nozzle.
At the point of condensation a pressure rise occurs in
the stream. This phenomena is known as the condensation
shock.
A more complete understanding of the condensa-
tion shock is necessary at the present time due to
recent increase of flow speeds into the supersonic
region. Difficulty has been experienced in the design
of sunersonlc v;ind tunnels and in gas turbine research
due to the presence of condensation shocks.
Very little information on the condensation shock
is available in the literature at present. J. I. Yellott
and C. K. Plolland (1)>^ (2f have investigated the conden-
sation shock in steam. R. Hermann (3) In 1954-36 studied
the condensation shock in air. These experiments were
primarily designed for the provision of data in connec-
tion with the construction of a workable supersonic wind
tunnel. Hermann verified previous experiments by shov/ing
^ (1) (2) Refer to numbered references in the Dibliofrraphypage 64
.
that there is a unique position of the shock in the
nozzle derending upon the relative humidity of the
air. K. Oawatitsch (4) published in 1P42 a step by
step method of computing the pressure as a function
of distance along a nozzle in the region of the con-
densation shock. This method was applied to measured
data of Yellott and Holland and A.M. Binnie and M.W.
Woods (5), and his own experimental data, and excellent
correlation was obtained. As mentioned, however, the
experimental data in this field is small, and the pur-
pose of our study is to verify previous results and
obtain additional information on the laws governing the
condensation shock.
^
PROCEDURE AMD DI'SCRIPTION OF AFPAPATUS
The S^hlieren apparatus is an optical nethor"' of
indicatino; density gradients of the flow in a nozzle. -
This apparatus is adequately described in reference
(6), pp. 26-28 and pp. 33-38. In order to sto'o .'iny
posr.ible moveT.ent of the shock, the Edgerton High
Speed Flash Unit, reference (6), pn. 39-42, v/as used.
The indicated flash duration of one -five millionths
of a second .is not attained due to stray wire canaci-
tance, but the unit provides sufficient speed for the
purposes of this study.
A steam e,lector v/as used to maintain a vacuum of
about 27 inches of Hg. 'on the nozzle exhaust. This
insured supersonic flow in the nozzle. A schematic
diagram of the apparatus is shown in Figure I. Atmos-
pheric air may be drawn through valve B and dehumidi-
fied air through valve A. By ad.iustrr.ent of valves A
and B, the relative huiriidity of the air can be varied
about 40^. The maximum pressure drop through the de-
humidifier was 1.5 cm. of Hg.
Humidity measurem.ent v/as made by means of a wet
and dry bulb thermometer at the nozzle inlet. laired,
rapid reading thermometers were used.
As shown in Figure II, reference lines were olaced
c.
at the throat and at a station 7,2 inches b^low the
throat. This enabled us to accurately measure the
position of the shock and also to determine the fore-
shortening of the photograph taken. Pressure taps
v;ere placed along the nozzle at points indicated.
Pressures were measured by mercury manometers. An
additional pressure tap was located in the 2" nozzle
inlet pipe.
Simultaneous readings of pressures and of wet and
dry bulb thermometers were made after a steady state of
flow was obtained for the desired relative humidity.
Either a picture was taken or the position of the
shock measured directly from the camera screen.
7.
FIGURE I
DIAGRAMMATIC SivETCH OF APPARATUS
-\
-- 14"—
100 lb.Silica-Gel.
1 1——1
>
2 XPS
-^
5" Diam. Hoi©With Screen
'B
WetDry BulbB-ulb Thermometer
^ZT-
To Elector
TwQ DimensionalNozzle WithGlass Walls
^ cvJ
LU
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ivi
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o t^
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2o
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QC
- Area, ft.
TABLE O F SVTvTRQLS
2
Cpo - Specific heat at constant pressure of air,0o24 B.T.U./lb. Op
Cp^,,^ _ Specific heat at constant pressure of watervapor, at low pressure, 0.14 B.T.U./lb. °F
g - Acceleration of gravity, 32.2 ft./sec.
^
H - Enthalpy of a mixture of air and water vapor(plus ice or water if appropriate) B.T.U./lb.
ha - Enthalpy of air, B.T.U./lb.
hi - Enthalpy of ice, B.T.U./lb.
h^^yy - Enthaloy of water vapor, B.T.U./lb.
h" - Heat of vaporization or sublimation, P.T.U./lb.
J - Mechanical equivalent of heat, 778 ft. Ibs/P.T.U.
k - Isentropic exponent for the mixture, taken as 1.4
I/i- ivlach No.
ma - Lbs. of air
m^^^ - Lbs. of water vapor
m'*' - Lbs. of water or ice conr'lensing out in thecondensation shock
P - Pressure of the mixture, lbs./in.
^
Pg - Partial pressure of the water vapor, lbs. /in.'"
p^ - Partial pressure of the air, lbs./in.
^
F - Saturation pressure for water vapor in thermalequilibrium v.'ith drops of infinite radius, lbs./in. "^
Pgp - Saturation pressure for water vanor in thermalequilibrium with drops of radius r, lbs./in.
^
r - Radius of v/ater droplets, ft.
t - Temperature, op
T - Temperature, °P absolute
"^sat"" Teniperature at which the water viapor In the
air becomes saturated, conslderinrr a stablestate exists, as humid air flows through thenozzle, ^F absolute
V - Velocity, ft ./sec.
V - Specific volume of the mixture, ft .3 /lb.
IV - Specific volume of water at the temperatureof a drop, ft .5 /lb.
w or CO - Specific humidity, lbs. of wato* vapor perlb. of filr
X - Actual distance from throat zo condensationshock
x'"' - Distance from throat to conclensation shockas measured on a picture
(T - Surface tension of water, lbs./ft.
<f>- Relative humidity
Subscrints
No subscript - Any position in the nozzle
- State at nozzle entrance
1 - State at an infinitesimal distancebefore the shock
2 - State at an infinitesimal distanceafter the shock
a - Air
wv - Water vapor
RESULTS
Tfio data of tables HI, JV&'. (see Appendix) was
obtained in sequence. Table III gives the results of
two sets of runs taken on two different days fAnd is
arranged- in order of decreasing relative humidity.
High humidity runs 1, 2, 3, and 4 were obtained using
steam to humidify the air. The pictures &> r these
give qualitative results but the data is considered
questionable. The pictures obtained during these runs
follow and are arranged in order of decreasing specific
humidity.
Upon reviewing the above data, it was considered
desirable to obtain more pressure data in the vicinity
of the shock. Additional nressure taps were added (see
Figure II) . Table IV presents the results obtained.
Pressure measurements v/ere taken but no additional nic-
tures. Pressures at stations 8 and 4 v/ere unreliable
due to oblique shocks originating at their pressure
taps
.
Upon making the plot of x/t (defined on figures
III and IV) Vs snecific humidity, shown in Figure IV,
it was believed that x/t was a straight line function
of specific humidity for a given inlet temperature.
The data of Table V presents additional inform.ation
for a lower range of inlet temperatures than had been
obtained previously. Pictures were not taken but
v;^]ue3 of X wore measured directly *^ro""' tho camera
screen.
PICTURES
The rictures of runn 1, ?. , and 7) are taken with
aark field and the conden3at?.on shock aDoe&rs light.
All other pictures are taken v/lth light field and the
condensation shock shows up as a dark bond. Steam
was used to increase tho humidity of runs 1, , 3
and 4. x''"" is the distance from the throat to_ the :-.or
densation shock rs -^".easured on the picture, ^he uc-^ ?]
distance fron- the throat to tho conc'ensation shock -
X = X "( 1 . 7 ) .
^
1.^
Run'l, 4> = lOO'^, x"'" = 0.54 In,
Run 2, (p - 85.8^, w - 0.02118, x'"" r 0.53 in.
Run 4, c(> = 7 5^, w = 0.0171, x^"" = 0.59 in.
J4
Run 3, ^ = 77"^, -w = 0.0170, X*"' = 0.35 in,
Run 8, ^ = 56^, w = 0.01439, x'"" = 0.49 In.
Run 9,<t^ - 5lf, w = 0.01328, x"" = 0.53 in.
IL,
Run 6, (/) = oZ%, w = 0.01313, x'^ = 0.47 in.
Run 12, ^ - 47^, w = 0.01265, x'"" = 0.55 in.
Run 15, (\> - 43. G<, w = 0.01190, x^" = 0o60 in.
J.U •
Run 16, (^ = ^2%y w = 0.C1092, x'"' = 0.60 in.
Run 10, <jt)- Sl*^,' w a 0.01070, x"' « 0.57 In.
Run 20,<f)
p 39^, w = 0.01027, x'"' = 0.55 in.
17
Run 17, <^ = 42^, w = 0.01011, x'"' = 0.65 in.
Run 18, (j) = 40<, w = 0.C1004, x'"' = 0.66 In,
Run 15, (^ = 46^, w = 0.00S92, x'"' = 0.59 in.
Run 22,<f>^ 36'^, w = 0.00982, x""" = 0.55 in.
Run 25,c/>= 54^, w = 0.00964, x'^ = 0.75 in.
Run 21, (^ = city w r 0.00951, x"" r 0.65 in.
Run 2A,(f>
= 30. 5'?^, w = 0.00913, x'^ - 0o76 In,
Run 25, = 27.5'^^, w = 0.00856, x'"' 0.80 in.
Run 2G, (^ « 22'^, w = 0.00741, x'"' = 0.88 in,
PiTure III shc.va a nlot c •' tlio 'iinr^nr!^ on! gst ?'0 t i o
x/t Vs PolMlve iiLunldity . This nlot 3hov/s the same
trend as a simil.'ir lot of R. Heroiann (o). Relative
humidity is not an absolute quantity. Therefor? this
Diet represents qualitative results only.
It was decided to use sbecific humidity as a
parameter sjncf; this is an abscluto quantity. Spof^.i-
fic humidity, w, is defined as lbs. of water v&por p'-.r
lb. of air. Figure IV shows a plot of x/t Vs s'cclflc
humiditv. The inlet temnerature o^ the air for t-;aoh
uoint is indicated on the figure. It anpears from
Figiare IV that 7./t is a straifc?ht 1 j ne function o^ -v
for a /2;iven inlet tem.perature
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Data for 'aotiirr;; curves ] to B on Pi^ui'Os V, VI, and VII
was obtained as Indicated in the followinr' ttble.
TABLE I
Curve Run Table (JO U^f^v. (P9-Pl)/Pn X Symbol
1 1 TI .00581 .00581 0.044 0.83
2 2 II .00714 .00714 0.051 O.Cl A
3 4 II .00997
19 I .00975 .00985 0.062 0.70 O
22 I .00982 A
4 5 II .01082 .01087 0.073 0.65 D
»
16 I .01092 O
5 6 II .01254 .01259 0.092 0.59 X
12 I .01265 V
6 8 II .01358 .01343 0.094 •0.55 V
9 I .01328 +
7 9 II .01384 .013814 Co082 0.515 y
8 5 I .01588 .01588 0.106 0.43 G
As shown in the table, runs with about the same
specific humidity and inlet temperature were grouped,
where possible, to correlate the data.
These curves were plotted to represent an instanta-
neous pressure rise at the position of the shock as de-
fined below:
(1) The pressure ratios P/Fq » from the throat down
to the location of the shock were plotted against dis-
tance along the nozzle.
(2) Pressure ratios dov/nstream of the shock v/ere
plotted.
(3) These curves were extrapolated to the location
of the shock as determined from the photographs taken.
(4) The vertical distance between the two curves
was taken to represent an equivalent pressure rise which
could be used to represent the condensation shock in
calculations. This is the value (P2-Pl)/Po*
Curve VIII shows P2-Pl Vs specific humidity. This
curve shows that the pressure rise in a nozzle is a di-
rect function of specific humidity.
Before proceeding with the discussion of the re-
sults, the limitation on accuracy of the data presented
must be mentioned. Steady state conditions for any one
run were difficult to obtain, but relative humidity never
varied over 0c5^. This variation resulted from the rise
of inlet temperature caused b^^ the dehydrating action
of the silica-gel. Because the pressure measurement
shov/ed no variation over the duration of a run, it is
25
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. felt that the specific humidity r rnai'ned essentially
constant
.
Quick reading, large scale, paired thermometers
were used in humidity measurem.ents . Air passed over
wet and dry bulbs at 3,000 ft./min. which is well
over the 600 ft./m.in. minimum velocity required.
Relative humidity was obtained by means of reference
(9).
Specific humidity v/as read from Table VII or
Chart Xtl of reference (8) . The accuracy of the
recorded specific humidity was about t 0,00015 lbs.
water vapor/lb. air.
Measurements of x] the distance of the shock
downstream of the throat, were measured to the center
of the shock to an accuracy of about t 0.01 inch.
These distances had to be corrected for foreshortening.
The two reference lines placed 3o2 inches apart on the
nozzle measured 2.99 inches on the pictures, givine; a
correction factor of 5.2 or 1.07 which has to be2.99
applied to measurements taken from pictures.
Due to the physical nature of the condensation
shock which occurs over a relatively broad area, the
measurements were made to the point of maximum density
as mentioned above. Tt may be shown by comparisons of
pictures A and B how sensitivity in adjustment of the
Schlieren Apparatus can vary the apparent width of the
condensation shock. The condensation shock may be re-
presented as shown below.
5
It
X
Low Sensitivity Picture A,Shock of apparent width I
High Sensitivity Picture B,Shock of aoparent width II
Picture A,«*^ = 0.0168, low sensitivity adjustment
Picture P, tO= 0,015B8, High sensitivity ad.nistment
The pressure measurements obtained are considered
accurate to i 0.5 mm. Hg. Oblique shocks, observed on
the pictures, from Stations 8 and 4 were believed to
have resulted in unreliable readings and the pressure
readings at these stations were not usecj.
Upon measuring the areas of the nozzle contour,
it was found that the measured areas did not agree with
the nozzle design. Nozzle area measurements were made
by micrometer to an accuracy of .001 inch. The results
of these measurements are listed on Table VI in the
Appendix. A plot of area and area ratio Vs distance
along the nozzle (Figure XIII) follows Table VI.
DISCUSSION OP RESULTS
A3 brought out in the RESULTS, the followinp;
-correlations were obtained for the teat nozzle.
(1) For a given relative humidity, the location
of the shock can be predicted within narrow limits.
(2) For a given inlet temperature and specific
humidity, the location of the shock can be predicted.
(5) For a given specific huraidity, the pressure
rise across the shock can be predicted.
In order that the information of Figure IV iuay
be of use for nozzles of various designs. Figure IX
has been prepared. This figure shows area ratio Vs
specific humidity, where area ratio is the ratio of
the area at the position of the shock to the nozzle
throat area. It is proposed that the results of
Figure IX will hold for nozzles of various contours.
The results and conclusions stated to this point
have been derived directly from the experimental data.
It is now proposed to investigate the mechanism of the
condensation shock by analytical methods. In narticu-
lar, the following conclusions will be brought out:
(1) For a given specific humidity, the amount
of moisture condensing out at the position of the shock
will be predicted.
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(2) For any specific hurr-idity and inlet tempera-
ture, it will be shown that the difference between the
temperature at v/h5ch the air becomes saturated under
equilibrium conditions and the temperature at which
condensation occurs is essentially c. constant. (Tsat-I'l) •
(3) The order of magnitude of the drop size
formied after the condensation shock 7/ill be 'i^redicted.
V
^.
THFORI'.TTCAL ANAL^^SIS
Mathematical analysis of the flow stream across
the condensation shock can be made by the use of four
equations.
1. Energy
H] + Vi^ = H2 + Vgf. (I)
2gJ 2gJ
2. Continuity
^ 11-^IL, since -Ai = A2 (2)VI Vo
3. Momentum
Fi + Vj^ = Pg + Vg^ (3)
Vig vo,2;
or
v.. r Vi + gv. (F. - P.) (4)
Vl
The use o-" the fourth equation makes tv;o analyti-
cal p.ethods of approach possible.
(1) Either the drop size r can be assumed infinite,
hence fixing the pressure rise due to the condensation
shock, or
(2) The observed value of the pressure rise across
the shock may be used £.nci. the; VB-liie of the di'op si'^e
evaluated.
Since the second method of approach most nearly
represents actual conditions, this v;ill he follov/ed.
In addition, in ordei.-' th'it -ictual conditions at the
shock may be represented as closely as possible, the
observed value of P^/Po ^-^ ^^® position of t?ic shock
will be used in preference to the theoretical value.
The energy equation (I) may be written:
trie (tn<M - iia,-,) + m „,h , - m h _ - m^h. ^ : V^^ - V. -^^
a V 1 ?- v/vi wvi WV2 vinrp, i2 2 1
^a"^ -%vvl"^^^
Simplifying the left hand side of the equation using:
"^'^ ' ^vvl - -S/V2
^a Sa ^^1 - ^2^ ^ ^-wvl Nnrl " ^.0^2 V/2 " Kvvl " -'^n.2^^^i2 =
^a "^ ^.vvi
V '^ - V '^^2 ^1
2gJ
Re-wrltlng,
ma <=ra ^'^1 - ^2) + "Vvi ^\nil - ^12^ ' "^^2 ^Nw "^Wma + m^^2
2gJ
^i2 ~ ^.vv2 "^^
h = l'iwv2 ' ^^2
Substituting above
<5na(Ti - Tg) + m^^^ (h^^ - h^2 "^^""^
" "^2-^> .vf-
manio + "Htvi
9 9
2gJ
iHq c^q (T-j^'a "^na
f"a "*"^.vvi
- Vo - V/ (5)
2gJ
a^a Cna ('^1 " Tg) + m^^ c^^,^ (T^ - Tg) + m- h"'
^a ^ "Vvi
= ^2 ' V^"(7)
Solving for tn'"', and sotting irip s 1 lb.
m"- -(1 + w) (V^^'-Y^^i + 0.24(T2-T^) + v. (0.44) (T^-T-^) (0)
2<^3
h"-
Sai-finle calculation of m" fron icnown r)resr>"ur3 rise acrossshock.
Given:
o psi
w Z lb. /lb.
To - ^H
r'p - P-, - (measured)
X Z in. (measured)
—i. Z (measured)^o
^1 : ii ^ ^o = ^31
^^'i : ^(?i)^ - ORf o
'^
^SO - W Fq - ^3^.622(1 + v; )
.622
To find pQi .
psi - ^30 ^ ^ = PS^
^O
From perfect ^^as relation using k = 1.4 and [mowing
^lAo
T1-1 =
Vi = 49yTj_ f.'i = ft/sec
^1 =^'^1
= ' cuft/lbFl X 144
Using equation (4)"*
V2 = V^ - gvi {?2 - ?l) X 144 :: Tt/a^c
^1
V2 r Vg vi - ft^/lb
Vl
The effect of the specific volnne of the wster vapor isnot included as the correction is small.
Tg = P2 V2 X 144 =or^
R
h'"" obtained from Table 5, Reference (10).
Prom equation (8)
m''" s lb/lb
It is of interest to note that regardless of whetherh""" reorcsents heat of fusion or of sublicnatj^on, there^^is little effect on the computed, value of m". We chose"
4U.
to -nake the calculation usincr. both values of h"^, es it
will be nointed out later that it is c31 fficult to -re-
diet whether ice or water is rornied in the nhock.
This calculation was carried out fb r the eia-ht
curves on Figures V, VI, and a/II. The details cf this
calculation are shown in the Ap-.endix in Table VII.
The first values of m'", Wg, and Fg^ were obtained usin-
h"' equal to the heat of sublimation. For the second
set of values, h" equal to the heat of vaporization
was used.
For this computation based on the known pressure
rise across the shock it was found that the ratio
W2 increased as w^ decreased. W2 is defined as wn - m*.wiThis inforjnation is shown on Figure X. It is seen from
this figure that with a known inlet specific hurr^iditv,
the lbs. of water vapor per lb air after the shock can
be predicted. The m" used in the calculation of wg for
Figure X was based upon h* being the heat of sublimation
•
)
\
'h
\
1
1
rw^-K
5 2
3 3
O00 vO Ia '^ KV
cS o o o cS
(a
o
/^(T)
1?,.
It has been found th'it for steam, when exnandin.c;;
isentropically across the saturation line, that con-
densation does not occur until the V\/ilson Line ia
reached.
This line is located approximately 60 Rtu helow
the saturation line on the Mollier diap-ram. A simi-
lar relation v;as sought for humid air expanding isen-
tropically. *
For a given w and Tq, j
Pso = wPq0.622 •
Selecting a pressure ratio P/Pq,
P - P (P)^3 - 'so ^p—'
^o
From the perfect gas relation the temperature of the
mixture corresnondin'^ to P__ is
p k - 1T = To (f^) —iT"
^ o
From the calculated T and reference (10), the. saturation
pressure, Pgat* ^^^ equilibrium conditions .csh be com.-
puted. A cut and try process in which various values
of P/Fq ^^® assumed is then used to make F^z Pgat n;iv-
ing the temperature at which the air beco-ies saturated.
The results of such a comnutatlon are shown on Figure
XI. Using computed values of Ti_, the following results
were obtained.
43
TABLE II
'.V ^sat ^1 ^sat ' "^1
.00581 496 384 112
.00714 501 388 113
.00982 510 400 110
.01082 514 405 109
.01254 520 411 109
.01558 522 412 110
.01384 523 416 107
.01588 528 425 103
The results indicate that the condensation shock
occurs at T^.^^ - T^^ ^ HO^F. There anoears to be a
slight trend for the temperature difference to decrease
with increasing specific humidity.
1
I
11AA
•>^
o
«-5
V,
NV . .i
X
5
f
5
o
So,
In
so
^9\/^o C/W3X J.V?L..
45.
We define wr, - v/q^ - m" (9)
then, W2 = 0.622 Pg2 (10)
or, Pg = wgPgx «B i (11)
0.622
Values of Pgg were computed for the eight runs
for which m'"" was computed. It was found upon comparing
Pgg computed from formula (11), that Pg^ was much greater
than the saturation pressure for equilibrium conditions
corresponding to the temperature T2»
Now we will review the events occuring in the flow
of the humid air through the nozzle. At the nozzle in-
let the water vapor in the air is in a supo? heated
state. The mixture of water vapor and air is expanded
isentropically through the nozzle and at a temperature
of about 110 degrees below the temperature at v;hich
condensation should have occured, some of the v/ater
vapor condenses out. As mentioned above, the vapor
pressure after the shock is greater than the equili-
brium vapor pressure at that temperature. The vapor
ore 3 sure under stable conditions is that corresponding
to vapor in equilibrium with drops of infinite radius.
The Von-Helmholtz relation -Js- shows that if the pressure
^Reference (11), pages 434 - 457
46.
of the water vapor Is rreater than the saturation
pressure under equilibrui.Ti conditions, then the water
vapor is in equillbruim with drops of less than in-
finite radius.
_
By means of the Von-Helmholtz equation, (see
Appendix pp. 62), Pi-ure XII, which follov/s, was drawn.
Figure XII shows, for various drop sizes, the pressure
of the water vapor in equillbruim with those drops for
a given temperature. Calculated points from the eight
runs which were calculated in detail (Table VII) are
shown on Figure XII. These points are plotted using
T2 and Pgg as arguments. The circled points corresnond
to Pg2 computed on the basis that Tlif^forms and the "x"
points correspond to Pg^ computed on the 'basis that^A&i»
forms. The four low soecific .humidity points lie at a
pressure below the triple point pressure of 0.0888 psi.
Under stable conditions, ice would be formed in the shock"
at these low pressures. The vapor pressures of the four
high humidity points lie above the triple point pressure
and probably water droplets would form in the shock.
However, it is difficult to predict in this metastable
statQ whether ice or water droplets form. For the four
high humidity runs, it apnears that a drop size of about
3 X 10- feet is formed in the shock. It is nosslble that
47,
FLOT SHO\^INCrFOFiMED IN
4 70
FIGURE XDTHE S\ZE OF WATER PR0PiET5A CON OLNSATION SHOCK
Uj
Z!Wu
4^0
_ .45(9
*
44
(
430
4Z0
410
O Ci04 COS OAE a/6 OioI I
-I I L
PRF65URe FOR WATER S^APOR \H EOUILIBRUIN ^\TMl
48,
1 riet.-i stable state of subcooled water exists below the
trlnle point pressure. If this is true, it anoears from
Figure XII that for all snoclfic humidities the drop size
formln'-^ in the shock is of a radius of about 3 x 10"^ fee to*
Because of our inability to predict the exact nature of
this phenomena, both the circled and "x" points are shown.
It must be mentioned that in the computation of
drop size it was found that the various reliable authori-
ties give an appreciable variation in th? values of
Pg^ at temperatures below 32°P. These discrepencies
increase v/ith decreasing temperatures. Also values of
surface tension are not definitel7;r known at these low
temperatures.
In conclusion, it is obvious that further exploration
of this sub.iect is necessary in order to present a comnlete
picture. However, it is hoped that the results of our
experiments will contribute to the knov/ledee of this
subiect.
Cfc—
,
Rn;CQ:.!^;ENDATIONS
(1) In preference to the conventional two dimen-
sional nozzle, it would be better to use a half nozzle
contour with a sliding straight wall section fitted
with a pressure tap and a micrometer screw for accurate
longitudinal measurement of the position at which the
pressure is measured.
(2) I'.'ore accurate hamidity measurements are de-
sirable. Chemical methods can give any desired degree
of accuracy.
(3) It is desirable to cover a wider ranoie of
inlet temperatures and specific humidities.
(4) Verify the nroposed correlation that a .plot
of area ratio against specific humidity for a given
inlet temperature v/ill determ.ine the position of the
condensation shock for nozzles of various contours.
.sn
.
APPENDIX
Pacre
Table III - Original Data 51
Table IV - Original Data 56
Table V - Original Data 58
Table VI - Data of nozzle areas and area 59ratios
.Figure XIII - Area and area ratio Vs distance 60
along the nozzle
Table VII - Detailed calculation across the 61condensation shock
Table VIII - Calculation of vapor pressure 62of water vapor in thermal equi-librium with drops of variousradii.
TABLE III
Run 3
% Rel.Humo
D.B.
W.B.
P9
^8
^2
Picture
Pio/^o
^t
100
90.8
90.8
75.7
42o55
36o95
32. S5
25o20
12.80
Yes
35.8
8.'^.9
79o6
75o70
40o70
35.90
32.30
24.30
12.00
Yea
77
80.4
74.9
75.7
40o45
35oo5
32 . 30
23.90
11.70
Ye 3
75.5
80.8
74o7
75o65
40o25
35o45
32 o 40
23o75
11.60
Yes
5 UNITS
65 °P
82.7 Op
73.4 Op
75.65 crrio Hg.
40o20ti
33.85ti
32o25
23.55
11.35
.01588
Ye;
o532
o448
.427
.312
.150
o41
.84
lb 3. w. Vo/lb3« air
TABLETir (Conto 'd)
Run 6 7 8 9 10 11
% R •H* 63 61 56 51 51 50
D.B. 78 83.2 84 3 84.9 78o2 78.3
W.B. 69 73.0 72.4 71.4 65.8 65.5
Pq 76.15 75o65 75.6 75.6 76.15 76.15
PlO 40.20 40.15 40.15 40.05 40.20 40.10
fg 35.20 33.45 32.60 32.30 32 . 30 32o25
^8 30.95 32.25 32.35 32.25 30.85 30.90
P7 22.95 23.50 23 . 20 22.95 22.10 22.05
^2 11.15 11.30 11.20 11.10 10.70 10.65
OJ .01313 .01513 .01439 .0132B .01070 .01053
picture Yes Yes Yes Yes Yes -,
flo/Po .528 .532 .532 .530 .528 .526
P9/P0 .436 «442 .431 .427 .424 o424
VPo .407 .427 .428 .427 .405 o406
P7/P0 .302 .311 .307 .304 .290 .200
P2/P0 .146 .1496 .148 .147 .141 .141
x" o47 .43 .49 .53 .57
:5^t o96 .PS 1.00 1.08 I0I6
TABLBIir (Cont. 'd)
r 'Z
Run 12 13 14 15 16
% R.H.
D.B.
V/.B.
Po
PlO
^9
^8
^7
^2
CO
Picture
Pio/Po
Pg/Po
Ps/Po
P7/P0
Pp/Po
:?^t
47
85.7
70.8
75.40
39.95
32.15
38.30
22.60
10.90
.01265
Yes
o530
o426
.428
.300
.145
.55
1.12
17
46
79
64.8
70. 05
40.05
32.20
30.30
21.85
10.55
.00992
Yes
.527
.424
.398
.237
.139
.59
1.20
44
80o5
65.7
75.95
40.00
32.15
30o25
21.80
10„55
.00998
No
.527
o423
.398
.287
.139
43.6
86
59.8
75.30
39.85
32.00
3I0OO
22.05
10o6fe
«01190
Yes
.530
o425
c411
o2P2
.1415
.60
lo22
42
84.6
68.
2
7F.25
39.95
32.00
30.40
22.00
10.60
.01092
Yes
.531
.425
.404
.292
.I'H
.60
1.22
42
82.2
65.2
75.75
39.95
32.05
26.60
21.55
10.45
.01011
Yes
.527
.423
.378
.284
.138
.65
1.33
TABLEIH (Cont. *d
Run 18 19 20 21 22 23
% R.H. 40 39.5 39 37 35 34
D.B. 83.5 83.0 84.9 84o2 86 87.2
W.B. 66.5 66.2 67.1 66.0 65,8 66.9
Po 75.75 75.15 75.75 75.75 75.65 75.65
^10 39.90 39.75 39.90 39.90 39.90 59.75
Pg 32.05 31.95 32.05 32.05 32.05 31.90
^8 28.40 30.05 28 . 60 28.60 27 . 90 26o45
^7 21.60 21.65 21.65 21.65 21.75 21.70
^2 10.45 10.40 10.50 10 50 10 c 50 10.40
.01004 .00975 c01027 .00951 .00982 .00964
Picture Yes No Yes Yes Yes Yes
Pio/Po .527 .530 .527 .527 .526 .526
P9/P0 .423 o425 o423 .423 .424 .422
Pq/Po .378 o400 .378 .378 .369 .35'^
P7/P0 .285 .2P^. .286 .286 .288 .287
P2/P0 .138 .1385 .139 .139 .139 .133
-If
X .66 .65 .65 o65 o75
i/t 1.35 lo33 1.33 1.33 lo53
55
TABLEin (Cont.'d)_
Run 24
D.B.
W.B.
^8
^2
a>
Picture
PioAo
P9/P0
Pa/Po
P7A0
P2/P0
x/t
25
30.5
89.7
66.5
75.25
39.65
31,80
25.30
21.85
10 . 35
.00913
Yes
.527
.423
.336
.290
.138
.76
1.55
26
27.5
90.2
66.7
75.20
39.65
31.80
24.95
21.85
10.35
.00856
Yes
.527
.423
.332
.290
.138
cSO
1.63
22
92.4
66.5
75.0
39 50
31.65
24.45
21.40
10.25
.00741
Yes
o526
.422'
.326
.286
.136
.80
1.8
TABLE IV
Run 1 2 3 4 5 inilTS
% Rel.Hum. 24.0 27.5 34 .
5
37 .
5
41.0 Op
D.B. 82.4 84.5 85.4 85 .
5
85.0 Op
W.B. 59.8 62.9 65.8 67.0 68.0 Op
Po 74o95 75.05 75.40 75.60 75.70 cm. Hg.
^10 40.00 40.1 40.25 40.40 40.50 n
^257.95 38.00 38o20 38.30 38.40 t»
Pg 32.70 32.70 32.85 32.95 33.00 Tl
^8i27o20 27 . 30 27.50 27.90 28.65 II
Pq» 31.00 31.25 32.40 35.60 34.60 fl
^7i22o85 23.95 25.15 2^i)0 26.50 tT
P? 21.10 21.80 21.75 22.00 22.25 rt
Pe 16.80 17 . 30 18.15 18.70 18.06 r?
P4* 14.50 14.65 15.10 15.40 15.06 tl
P2 9.90 10.15 10.35 10.55 10.70 II
Pio/Po .554 .535 .534 .535 .535
P9I/P0 .505 .506 .506 .507 .507
V^o .435 .436 .435 .436 .436
^8|-/Po .363 .364 • 365 .369 .378
^7|/Po o305 .318 .333 .344 .350
P7/P0 .282 .290 .28^. .291 .294
Pe/Po .224 .231 o241 .248 .246
P2/P0 .120 .135 ol37 .140 .141
OD o00581 .00714 .00925 .00997 .01082 lbs. w. v./lbs. air
''^ Values not used
TABLE IV (Cont.'d)
71
Run 6 7 8 9 10
% R.H. 46.0 49.0 55.5 58.0 98.0
D.B. 33.5 83.6 82.8 82.0 87.0
W.B. 68.5 69.4 70.6 70.8 86.5
Po 75,80 75.90 76.10 76.10 76.10
^10 40.60 40.65 4!). 85 40.70 41.85
^9i38.50 38.55 38.55 3P.60 40.55
P9 33.10 33.15 33.35 33.60 37.30
^84 30.20 31.10 32.90 34.60 34 . 60
^8* 35.80 36.55 39.30 40.50 38.30
^7i26.90 27.30 27o05 27.00 28. SO
P7 22.65 22.80 23.25 23.30 25.10
P6 18.70 18.80 19.05 19.15 21.30
P4-^^ 15.85 15.95 I60O5 16.20 17.60
P2 10.90 10.95 11.10 11.10 12.70
Pio/Po .535 .536 .537 .535 o550
P^^o .508 o508 o507 .507 .53^
P9/P0 .436 .436 .438 .441 .490
Ps^/Po .398 o410 o432 .455 o455
P7I/P0 .354 .360 .356 .355 .579
P7/P0 .299 .300 .306 .306 .330
Pe/Po .246 .248 .250 .252 o280
P2/P0 .144 .144 ol46 .146 ol67
(a) 0OI254 .01233 .01358 o01384 .02764
* Values not iised
TABLB,
Run t Re 1. Hum. OJ D.R. .'< • B • X £/t
1 63o0 .0107 72.4 o5 .8 .49 1.00
2 57 .0099 75.4 63.2 .53 1.08
5 50 .0088 73.5 61.4 .57 1.16
4 46 .0085 74.2 61.0 .59 1.20
5 45 .0080 74.0 60.4 ^60 1.225
6 42.5 .0076 74.5 60.0 .61 lc245
7 41 .0075 74.6 59.8 .62 1.265
8 56.5 .0070 75.7 59.4 .65 1.327
Q.51 o00645 76.6 59.0 .69 1.410
10 30 .00585 77.6 58.5 .73 lo490
11 50 .00610 78.2 59.0 .74 1.510
12 28.
5
.00600 78.7 59.0 .79 I06IO
TnBLE VI
dath of nozzle areas and area ratios
Stations are spaced O.i inches ainart (see Flmjire XI"^!
following). The nozzle is 0.50 inches thick.
Station Area, in."^ A/AiJI
2 .442 1.804
3 .4135 1.690
4 .384 1.567
5 . 356 1.453
6 . 328 1 . 340
7 .2975 1.214
8 .271 1.107
Q .248 1.012
10 (Throat) .2455 1.000
11 .2455 1.002
12 .246 1.004
13 .279 lol39
14 .335 ,. 1.37
60
^ »^ 15 ^« o d § o<n
6O
\
\•A
4\
%
<vj
u2
oJ
•0
o
/iVi
»n
W* (o 14
-rf oav« V3«v
TABLE znPf TAIL CALCULATION ACROSS C0ND£N5/?T/0N ^HOCK
6/.
\ ' _J L^-!
CURH NO \ 2. 5 4 6 (. , . 8
P. 7495- 7SOf 75^f 76-7 758j
76 f - 76/ 75.^5- cn ».
a- ai9J4 K-.50 14 50 14(^4 14 65 1446 14.70 /47<7 \4a Pii
00 .(7C^5<9/ .00714 .0O^S5 .OfOSZ 0\154 .01158 .01384^ .01588
t; S4?4 544 5 546 545 5435 5418^ 54t S4Z7 •R
to SZA- S4r5 86 85.0I 83.5 szs SZO az7 *r
- w B ' 5^=1.8 U"^ &&S GSO1685 70.6 - 70.8 75.4 T
{Pf p;/^ 004^ 0.051 1 OOCZ 0.073 O.ORd 0.094 0.08Z 0.f06 gar
a-p 0-6:53 74- 0.<^7 107 1348 1380 IZOS l.$5Z P5C
X 85 0.81 0.70 ai>s O.S'^3 0550 0.515 0A3O /w-
<. R/ft. 0300 0.507 0.340 0.35Z 0375 0.38Z OJ^S 415/^'^^^ a 70^' 0.7f3 0.734 0.742 0.75^ 0.760 0767 a 763 !
38^ 383 400 AOS All 4(2 41b 425*
R 4.35- 4.4F 4.98 . 515 5.50 561 5.80 6.ZZ 1 ftc
B 4*?^ 6^*^ ' S.sa 6Zi \ 665 699 701;777 Psc
0134- 0165 0ZZ8 0Z5t aZ8^ 0.3iZ 0.3ZO 0.364 PSC
Pu'H^-R/fi 0.0402 0050lif\ 00775 0088 010^5 O.iZO 0.IZ65 0.155O ftc
M, 1453: 1.417 1343 1.318 1.27/ IZ57 I.Z33 urr 4*» <*M'
v.^Wf-n IJ78 15^7 /5/6 1300 IZdZ IZ60 IZ3Z //a9 i''/.r. !
%-S/z. 10 5f ^5 \l\ 137 139 iZ\ I5Z ^/uc
^rT,';37/p, 3Z6 3^2 Z9.7 2*! / 27^ 272 266 152 rury^
i4=w-^/'-s-'?;,"* 1308 t/2S6 IZZ3 (ifl^ IIZ5^ Hil nil ioyfA '^src
Wi^ gi-^ JO.^ SOZ .27ri Z6.6 Z4 6 24.2 Z4.0 12.0 «7/»
Tj./itf-^.? 416 4E3 438 446 45? 457 45S 461 S^ i
^->^;^K^V'J/£wo J76 4?9 4.81 5.S2 6S5 655 566 677 Biu/A
>,^ l^^l^ \ZZiZ IZZIZ i^ZiZ itVZ IZZt. 2 122(2 IZZI.Z Biulfb
t^-t; ~3Z 3S 33 4h-- 44 ' 45 ^Q %^ - r'f'
On y*
'5.0d -4.16 -4 75 -557 -6.60 '648 -560 '6.30 dU/H,
QU(Ti-T} 7.68 8.40 9/2 954- \0 55 laao 935" 865 BU/lk
144a)(Tz-Ti} aod O.H O.lb ^ 0.20 az4 0.27 024 0.25 QH^V
^z ;•• -n 2 74n 4.55 4.53 447 4./9 4.59 399 2.10 BL^Jk
*^ /-^^^- .00?i4- .00556 00571 .00367 .00343 .00376 .00327 .oonz' CA^j-Oj,--^* .0055-7 .00558 .00il4 \
00715:
1 11
.OO'^ll\— ,
.00*^81 .0/057 .O/V/6
1] IHHHHHii(
n> .0Zt71
.0715" .loec. uoo .11^0 .179 Fsi
If i-C lOIT 1077 1077 10 77 1077 1077 1077 1077 \nti>Hb
.ooisi' eoHoS .ooHll .ooiiL ooiio .00427 1.00572 \.oolH ^ihlli>
"
§\u;j,a>,-/w* .OOJZC OOlOZ .06S^^S\.ooC6C \,00ti*( oont ,01011 .oi3^Z
\ lblib
1Pst .6ZCX .015*1 oSi'i .OUC .09SZ ./04S- JI4.1 .1741 fi*
$^ .—
I
! 1
1
62.
The calculation of tho vapor pressure of water
vapor in thermal equilibrium v/ith drops of various
radii for various temperatures
j
Ln Pgj, r 2 0" v^
Psoo rRT
Where R r 85.8 and (T is the surface tension in
dynes/cm. f^iven by:
Cr ^" = 75.64 - 0.1391t - c0003t^ (t in ^C)
^soo ^3 given by the equation below:
^^SlO ^tto' 21.075 - 2903.59 - 4,71734 logjQTT
Where T is expressed in ^Kelvin.
^soo ^s ^^ psi.
-" Reference (11) pp.447»-•«• Reference (7) pp.569
TABL^ VIII
-9
T 0" X 10^ v^ ^300X "- J- . w
p ~/t—
'
> sr
X — J- • CJ^
(^F) lb/ ft cuft/lb nsi
470 5.29 .160 .0354 13.8 0.488 9.1 .322
460 5.34 .160 .0224 15.01 0.336 9.8 .220
450 5.39 .160 .0139 16.25 0.226 10.45 .145
440 5,43 .160 .00841 17 . 80 0.150 11.30 .095
450 5.48 = 160 .00496 19.50 0.0965 12.20 0C6O5
420 5o52 .160«
.00284 21.40 0.0608 13.20 .0575
410 5.56 .160 .00158 23.50 0.0372 14 . 30 .02:^6
T
r=2.1xl0"^p yp ^
"3r 300 3r
ri5 xlO"^P /?
sr
470 7.40 .264 2.32 .082
460«-
7.88 .1766 2 . 38 .0533
450 t* 9.40 .11138 2.44 .0340
440 8.93 .0755 2.51 .0211
430 9.65 .0478 2.58 .0128
420 10.35 .0204 2.65 .C0755
410«
11.15 .0176 2.74 .00433
'j\
HIBLIOGRAPfTY
1. J. I. Yellott, Jr., "Supersaturated Steam",A.S,.:.R. Transactions , Vol. 56, 1954,pp. 4 11- '130.
2. J. I. Yellott, Jr., and C. K. Holland, "TheCondensation of Flowing Steam", A.S.M .K .
Transactions , Vol. 59, 1937, Dp. 171-183.
3. R. Hermann, "Condensation Shock Waves inSupersonic 'Wind Tunnel Nozzles", Luf tfah-rtforsehun.T , Vol. 1?, No. 6, June 20, 1942,pp. 201-209.
4. K. Oswatitsch, "Condensation Phenomena inSupersonic Nozzles", Z .A .i:.]\'' . , Vol. 22,No. 1, February 1942, pp. 1-14.
5. A.M. Binnie and M.W. Woods, "The PressureDistribution in a Converging-Diverglnr:Steam Nozzle", Proc . Inst. Mech. En^^rs *-*
Vol. 133, 1938, pp. 229-231.
6. L. A. DeFrate, "Investigation of SupersonicFlow in Nozzles and Tubes by the SchlierenMethod", Master's Thesis, !':iT, 1943. ( M.E.Department)
.
7. C.F. Powell, Proceedings o^ the Royal Society ,
Vol. 114 (A), 1928, Pp. 553.
8. O.T. Zimmerman and Irvin Lavlne, "Psychrome-tric Tables and Charts" Industrial ResearchService, Dover, New Hampshire.
9. U.S. Dept. of Commerce Bulletin, ';7-P 235 ,
U.S. Govornment Printing Office 1941.
10. J.H. Ke'^nan and F. G. Keyes, ThermodynamicFroperties of Steam , New York, 1944
o
11. J.H. Keenan, rhermodynamics. New York, 1941.
QMS ITDl
Ml n E "^ Y
1 1 W05
S495
11666SimonsInvesti^ration of the conr'.eTisp.
tion shock in air by use of theSchlleren method.
Library
U. S. Naval Postgraduate SchoolMonterey, California