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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Monterey, California

COPT FOR HEAD 07 P0STOR4DUATB SCHOOL

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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

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-- 14"—

100 lb.Silica-Gel.

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2 XPS

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5" Diam. Hoi©With Screen

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TwQ DimensionalNozzle WithGlass Walls

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- 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

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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