GL-TR-90-0246

AD-A230 729

ONE-DIMENSIONAL ANALYSIS OF A RADLAL SOURCE FLOW OF WATER PARTICLES

INTO A VACUUM

A. Setayesh

Radex, Inc.Three Preston CourtBedford, MA 01730

August 31, 1990

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One-Dimensional Analysis of a Radial Source Flow of Water Particles into a Vacuum

~ESNLAUTHOR(S)A. Setavesh

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'6 SUPPLEMENTARY NOTATION

"7 COSATI CODES 18 SUBJECT TERMS'(Conrmue-or-reV~rfe if necessary and identify by block number)FIELD {GROUP SUB-GROUP I flow models, water release, space physics, particle temperature, particle size,

particle freezing, vapor pressure, particle velocity 'T f11 (.

9 ABSTRACT (Continue on reverse if necessary and identify by bloc.': number)

,:n analytical investigation is made of a one-dimensional radial source flow of water into a vacuum. 1he equations thatde!scribe this model are a sixth-order, nonlinear, two-point boundary value problem. The problem is solved by theTshooting/splitting" technique. The results of this investigation are compared with the other works in the literature and the

comparisons generally are in good agreement. Engineering parameters for the release of water into space are used to givesome indication of water flow characteristics into a vacuum environment; these parameters were chosen to resemble thoseof a recent water release experiment from the space shuttle. i

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ACKNOWLEDGEMEN'IS

The author wishes to acknowledge Edmond Murad, G[IPIHK, for

initiation of this work. and for his helpful advice throughout the course

of this investigation. The acknowledgement is also extended to Irving

Kofsky and David Rail, Photometrics, Inc., and Jim Elgin and Alex Berk,

Spectral Sciences, Inc., for helpful advice and providing some of the

references used in this work.Aceess1.en For

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TABLE OF CONTENTS

1. Introduction ............................................................ I

2. Description of the M odel ................................................... 1

3. M ethod of Solution ....................................................... 7

4. D iscussion of the Results ................................................... 7

5. Summary and Conclusions ................................................. . 24

R eferences ............................................................... 25

Appendix A. Derivation of Governing Equations ................................. A-1A.1 Continuity of Vapor and Particle .................................... A-1A.2 Momentum Equation for Vapor and Particle ........................... A-2A.3 Energy Equation for Vapor and Particulate Matter ....................... A-4A.4 Constitutive Relations ............................................ A-9

A ppendix B. G lossary ..................................................... B-1

Best Available Copy

v

LIS'l'OF FIGURES

Figure 1. Effect of Initial Particle Size on Particle Temperature ............. 8

Figure 2. Effect of Initial Particle Size on Particle Velocity............................9

Figure 3. Effect of Initial :'article Size on Vapor Temperature........................... 10

Figure 4. Particle size as a function of the distance.................................... 11

Figure 5. Pitot-Pressure measured at the vapor expansion flow radial to the jet ............... 12

Figure 6. 'llie Effect of Initial Particle Size on Particle T1emperature ..................... 16

Figure 7. Particle Size as a Function of Distance ................................... 17

Figure 8. Th'le Effect of Initial Particle Size on Particle Velocity ......................... 18

Figure 9. The Effect of Initial Particle Size on Vapor Velocity ......................... 19

Figure 10. illie Effect of Initial Particle Size on Vapor Density ......................... 20

Figure 11. Thle Effect of Initial Particle Size on Vapor Number Density ................... 21

Figure 12. T1he Effect of Initial Particle Size on Particle Number Density .................. 22

LIST OFTIABI.F.S

Table 1. Parameters Used for Figures 1-4 ........................................ 13

Ta~ble 2. Paramee Used for Figuire 5............................................ 14

Table 3. Parameters Used for Figures 6-12.......................................... 23

vi

1. INTRODUCTION

'llie flow of a liquid jet into a near vacuum envircnment has been studied by many investigators. Manyof these studies are experimental but also provide some theoretical interpretation of the experiments.For example, see the works of Johnson and I lallett 11 908S,Xkassal 119741, Fuchs and Legge 119791, Curryet al. Il SSXa,bJ, and Muniz and Orme 119871. Johnson andl Ilaiicit 119081 investigated the behavior ofsupercooled water drops at various conditions, such as in carbon dioxide below 0.05 atmosphericpressure. Kassal's 119741 experiment showed that the spherical particles break in half during the freezingprocess and form bowl-shaped configurations in the vacuum chamber. Fuchs and Legge [1979]concluded that the bursting effect caused by the sudden boiling of the water produces vapor bubbleswhich decompose the jet. They stated that the formation of vapor bubbles was caused by a superheatingof the water due to sudden pressure drop and cooling at the surface of the jet. Muntz and Orme 119871discussed the possible events when liquid streams are released into a vacuum. They stated that for fluidshaving vapor pressures up to several torr in streams with a few hundred microns in diameter, it ispossible to maintain stable and controllable streams. However, fluid streams having vapor pressures inthe of tens of torr range with one millimeter and greater diameter are expected to burst.

Curry et al. [1985a, b] performed an experiment in a vacuum chamber (10- 5 Torr), wherein water wasreleased from an orifice with a diameter of 1.6 mm into the chamber. The vent plume was studied usinga Mie scattering polarimeter and a high-speed flow visualization camera. They concluded that there aremainly two classes of partick.s which can be found in the water plume, viz. the large particles withdiameters close to the orifice diameter (1.6 mm) and a cloud of very small particles whose diameters wereabout 0.15 p.m. Pike et al. [19901 showed that the particle sizes as measured from the recentgroundbased and onboard video images of a sunlit Shuttle Orbiter water dump are similar to those notedin vacuum-tank experiments (Curry et al. [1985a,b]).

Some mathematical modeling of such flows has also been done and compared with experimental data(Gale et al., 1964, Glenn 1969). Gale et al. [1964], in their mathematical study of liquids (includingwater) released into a vacuum, analyzed the effects of initial temperature, droplet diameter, and ambientpressure on the time required for cooling to the triple point and for subsequent freezing. Part of theirmathematical analysis was supported by experimental data. Glenn [19691 studied a radial source flowof uniform-size liquid particles into a vacuum. A numerical procedure for the integration was developedfor the models, and he discussed parametrically the effects on the flow structure of such variables asparticle size, velocity, and temperature. The present investigation is inspired by the work of Glenn [1969]in which hc assumed a one-dimensional jet flow of spherical particles of liquid water into vacuum.Glenn's [1%91 work has been extended, and some revisions made to his work which are included in thisreport.

2. DESCRIPTION OF TIlE MODEL

"[he problem considered here is the mathematical modeling of stationary radial source flow into avacuum. The equations describing this model are the one-dimensional continuity, momentum, andenergy for the vapor and particles. The analysis constitutes a sixth-order, nonlinear, two-point boundaryvalue problem. A number of assumptions have been made to simplify the problem. These include the

I

following: Ilie particles are liquid, spheric'il shaped, with uniform size and temperature. lie wake ofthe particles does not influence the nearby particles. Re-condensation of vapor as a result of anysubcooling during the expansion process is neglected. The heat transfer between particles and vaporoccurs only by convection and radiation. The fluid is considered to be inviscid, except the part that canproduce drag. From the above constraints, one cn assume a simple liquid jet breakup into the vacuumenvironment.

The derivations of the equations that describe this model can be found in Appendix A. Glenn's work119691 has been extended and the equations have been modified to investigate the behavior of particlesafter freezing, and to include heat radiation from the particle. The Clausius-Clapeyron equation is usedfor calculating the local vapor pressure in all cases considered. The problem has been solved with theengineering parameters for the release of water into space. These parameters were chosen to resemblethose ot a recent walter release experiment from the space shuttle (Discovery, March 1989). Nevertheless.it should be mentioned here that the mechanism for water release and its breakup into space is anextremely complex process which cannot be fully described by this simple one-dimensional model.I lowever. it may give some indication of the factors involved. Future work will include two, and perhaps.three dimensional models.

The governing equations that are used in this study are as follows:

d ( -- , a r2 (I)dr

dIr2 pp = -rh a r 2 (2)

dr

/ p/

P ,, 0 l u , V / 4 ( 1 -- ,,) - p Z I t h C o ( u p - % ) ( 3 )

PP( up + a + af = 0 (4)

p~ I-c) k C I/ - UV~ (]-a) P1 =K q + U fp, (Upt,-u) +W (-) U, CIN. IV P

(5)

Ii U I Cv (T T T) - (up- ) 2

2

dI X 4 X P/ up aP /

d T, Jj(6)U ' qP. - ViZ aI[~+ 1

Lquations (1-6) constitute a sixth order system in six unknown quantities; vapor density p,, vapor velocity

u . particle velocity Up, particle temperature Tp, vapor temperature T,, and particle volume fraction u,and five addition2i variables which will be defined later.

The above equations are subject to the following constraints:

T> T.; 1, 13=0

Tp = T.; T'=0 (7)

Tp < T.,; 0= /=O

Where T.. and T. are the triple point and initial temperatures respectively, and 13 represents the particleliquid volume fraction.

'lie five additional unknowns, mass rate m, particle radius (T, local pressure P, drag force fp, and totalheat transfer by radiation and convection qp, are defined as follows:

vh= ( ) [Po (Tp) - P] (21r R TP)- 1"2 (8)

"(9)

P =pR T, (10)

3

fp CP3 ( ukv)u. U U1

qp = 3 p (p 1U) T1 -I) +

(12)

3 E TsB (4P - TUV

For convenience, the equations can be normalized by the following nondimensional barred quantities:

= u U. UV a ,t V= TV T.

[IVUP. P 0 P2 r = r', r

p = -. P PO = PO ( 7 ) PO P.

Tp = Tp T, pp = p. pp con8t

"llie following quantities, based on the initial conditions, can be introduced in the problem:

4

2ro,--1- 2 j ' 02 -,.

1 1

* 04 = 3 (2,a) 2 E (to

p.1? '.

1 1

0 " l) , 0 - (v (N ut,)-, 3

C1,), (To U(

cl , 7 '.07 - (14 )

U*

d , , f

00 = , -

2 2U U*

1

3 SB 'r T

3p, U,

Then the results can be written in matrix form:

U(I-a) p,(1-a) 0 0 0 -pu pAl

0 0 ppa 0 0 [)pUp Uv A 2

2- -- 2-0 1T v Pv %u 0 0 0

1P v 0 -! A 3= (15)

2-= -- 2-7. 0 pup 0 01p 0 "- A4lv

p- o1Ut 7 0 0 0 t)vUv(O7 -t) 0 f f -Av A6J

20 2p~i 0 -0--0 - - 2-- -A-01 uT v 0 0 )pupO -0uppv 0 Q5

, 5

The 0 in the sixth row and fourth column of the matrix is equal to 08, except at the triple point Tp -

T.., where -0 0,,; and parameters A, throl.h A6 are defined as follows:

0 1 02 04 ( 0)0 [03 P ( 2 p; u- ) (16)

A I )13 (jTI,)1/2 (- 2 jT

A2 _ 0204 6 103 - p T_ P up (17)

() 2 ( , - 2A,)

(r) 23 U) (1 - a)(18)

° 1- 0 2 0

f02 05 Pv1UP + -12 03 Po TP) -Pv ]

02 05A,4= - [i Iu(P-;V)I P- iA (19)

( )2/3 (Up1/3 ( )1,,

A5 = ( )21f 0 02 04[0 (TO - v Xl~ ~ ~()f (;- 113 112+o,+<:.-]

S(P )2[)2 - (20)

07(PT - I v +[0 5 (1 vup-;UJI

02 06 (TP - ) --17- 0201 - -3 1)

r) ( p)

A6 _1 [01 62 046 ()1/ " ()2J3 (p)1/

3 (T)l/ (21)

[03 P (T) 1 (T - + 02 01 (T 4 - 7$)

Here, the locad vapor pressure as a function of particle temperature can be stated in terms of theClausius-Clapeyron equation

09

P(Tp) - (22)

3. METHOD OF SOLUTION

The finite difference and shooting methods are two of a number of methods available in the literatureto solve the two-point boundary value problems. Here the shooting method was chosen. Glenn [19681used the simplest form of shooting method, called the " straight-shooting" scheme. In this method(which is also common in all other shooting methods), the two-point boundary value problem can bereduced to an initial value problem. The differential equations are integrated as an initial value problem,and the initial conditions are adjusted until the conditions at the other end point of the interval aresatisfied. Further discussion of this method can be found in Glenn's work [1968].

4. DISCUSSION OF THE RESULTS

Figures 1 through 5 present comparisons of the current work to the work of Glenn [1969], and Fuchsand Legge [19791. Figures 1 to 4 show the variation of the particle temperature, particle velocity, vaportemperature, and particle size (evaporation rate) with respect to radial distance r from the source fordifferent sizes of the particles. Most of the initial conditions in these figures are similar to those whichwere used by Glenn [19691. For example, the initial temperature, the initial velocity of the stream, andthe source diameter are taken to be 93.65 "C, 15.24 m/s, and 6.35 mm, respectively. The parameters usedfor the purpose of comparison with the work of Glenn [1969], and Fuchs and Legge [1979], can be foundin Trables 1 and 2, respectively. Figure 1 compares the temperature variation for different initial sizeparticles. In Figure 2, the velocity changes are investigated and compared to the work of Glenn [1969].Since the vapor velocity increased very rapidly after injection of the water into the vacuum, the smallerparticles are most likely affected by this increase in vapor speed and their velocity will increase according

7

7.

Lr

4J L..O. - -

CE0 LO) (N G)9-

t- ) N

k C

0 -0

0- 0 0 0 0 .

0 0S 0 0

*- C* CD -- 0)

5'S r 0)C C *,

di @ I

.T~CN" 0

(\J C:)rI'. (~) F

0

CD 0

0 Ln CD0 0n

(S/UT d 0 1JDcA AIPO0

ci

o 00

C CD

0O(0 L6 6 6HU

[C)0

I0

-00

'4--

S0

*C D

0-

P@ZIIW'DU)

00

000--

4 --, 4- CI:::D.0

JLEEE[ aa

0Q) oU) Qa

0Y 0 00

C:- E 0

jOO) 00) U

0 U C) C o(- U C) 0V

a C))

4 0 0

IC))

C- IC) 0 IC) CD Lf) o) C) i0 r-- Ln C 0 1-- CN 0

DiflwJOJ 104 ql;5!IADNJ UO P@SDF9 ()OI) 'd 9.)'SS@Jd_

12

Table 1. Parameters Used for Figures 1-4

i -

Initial lcntllerattire (OK) 360.65

Initial Velocity (m/s) 15.24

Initial Source Radius (m) 3.175 x 10-'

Initial Particle Radius (gm) 12.5, 25, or 5'

Initial Vapor Pressure (N/m 2) 8.0 x 1,

Initial Vapor Density (kg/m 3) 0.47916

Initial Vc-jume Fraction 0.74

Evaporation Coefficient 0.05

Drag Coefficient 0.9

Particle Nusselt Number 2.73

Vapor Thermal Conductivity (W/m OK) 0.024403

Vapor Specific Heat (J/kg 'K) 1863.172

Heat of Evaporation (J/kg) 2.2741 x 106

Particle Density (kg/m) 962.8

Gas Constant for Water (Ji'kg 'K) 461.94

13

Table 2. Parameters Used for Figure 5

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

Initial Temperature ('K) 300.15

Initial Velocity (m/s) 12

Initial Source Radius (m) 1.0 x 10,

Initial Particle Radius (tim) 5

Initial Vapor Pressure (N/m 2) 4.0 x 10'

Initial Vapor Density (kg/m3) U.025

Initial Volume Fraction 0.74

Evaporation Coefficient 0.05

Drag Coefficient C.j

Particle Nusselt Number 2.73

Vapor Iliermal Condu.. aty (W/m °K) 0.024403

Vapor Specific Heat (J/kg "K) 1863.172

Heat of Evaporation (J/kg) 2..&6136 x 106

Ileat of Fusion (J/kg) 3.340777 x l0

Particle Dcnsily (kg/in) 987.l

Gas Constant for Water (Jikg 'K) 461.94

14

to the particle size. For example, tile larger particles will be less affected by the vapor velocily. "lbereis a further discussion with respect to the velocity increase of the particles in tile later part of this work.Figure 3 shows the vapor temperature variation. "Die temperature drop in vapor is somewhat differentfrom that found in the particles. Figure 3 shows that the vapor temperature decreases more rapidly forthe larger particles than for the Smaller particles. Figure 4 shows the size reduction of the particles dueto evaporation, as a function of the radial distance. This figure indicates that the smaller particlesevaporate much faster than larger particles near the source. As can be seen from Figures 1 through 4,and in particular Figure 2, the present results are in good agreement with the work of Glenn [19691.

Figure 5 investigates the changes in vapor pressure as a function of radial distance R, based on Equation(23). This equation relates the radial distance R to the Mach number and the specific heat ratio ofvapor. The r. is the fictitious sonic radius defined by Fuchs and Legge [19791. Since the vapor flow issupersonic, the vapor pressure is calculated based on the Rayleigh-Pitot formula (Equation (24), wherePO is defined as a critical pressure at sonic ro ) which is considered by Fuchs and Legge 119791. In thisfigure the experimental measurement of Fuchs and Legge [19791 compares very well with the calculatedresults of the present work. As can be seen from Figure 5, the curves start at some distance from thesource, the distance thi the vapor needs to travel to reach supersonic speed.

( , (y+ )

R . 2](A-1)M 2(Yy- ) (23)ro M +1" J

-y +1)(y.)[ (y+)M ) Y 2Y)(Y..0 (24)

2 J2 2+(-y-1)M 2 -+

Figures 6 through 12 show water flow characteristics near the source of release, into a vacuumenvironment, for the parameters listed in Table 3.

Figure 6 shows the variation of particle temperature as a function of initial particle size for three differentparticles. The particle sizes are well within the estimates for water release from the space shuttleDiscovery for the flight of March 1989 (ranges within 0.25 prm to 1 millimeter sizes). The initialtemperature of the liquid water is-70 °C (343.15 °K), which leaves with an initial velocity of 15 m/s. Thesmaller particles, such as 7.5 ptm, freeze only about eight centimeters from the source, while the largeparticles, such as 250 p.m, take much longer distances to freeze.

Figure 7 shows the evaporation rate of the particles as a function of distance from the source. 'lhe initialconditions of these curves are the same as for Figure 6. 'Ilie particles with smaller radius tend toevaporate faster.

15

C

N C

0

T-rElT - - - f -I I - -rl CD 0

C~j (N

_L jrqa o- d GL 10.1JC0

1()

toD

C3 U)4--

C14 0

C) U)

LO (N

0)00 0 0

s~njp) 0,-)IJDCJPOZ1C("O

176

IC) .9

LO.

-00

Ln 0)

C) %4

too LOf0

(S/W AI!01@A @101-)0

18.

4,

* ------- 7-.~u

L0 0C) 0

ZLZ 1

In U') N~~

Ula)

C) 0

00

(00

19

LC)-N a

60Cl.

000

C

0

N

00E F iE CL)

-0-

s-0.00

6u-00

LUJ

-0kn)

V-I-

Al~ug- cocAL

Cl)

0

4-J

to3

-00

Q)0

C) 0E u

.0

212

---o ---. (1)

HH

0 0 0 0 0 0 -)

(w) ^u /~tsu9Q ieqwnN JodoA "

ail

J) 0)

000

IC)

00

LO)

LC) u

EE E

4- Ur C

+O u

04+

++1

IC)

0 00+ 04- 00

i it) if :)( to n nj c"' I C

22

Table 3. Parameters Used for Figures 6-12

Initial Temperature ('K) 343.15

Initial Velocity (m/s) 15

Initial Source Radius (m) 7 x 104

Initial Particle Radius (jLm) 7.5, 25, or 250

Initial Vapor Pressure (N/m 2) 3.119 x I04

Initial Vapor Density (kg/m3) 0.19833

Initial Volume Fraction 0.74

Evaporation Coefficient 0.05

Drag Coefficient 0.9

Particle Nusselt Number 2.73

Vapor Thermal Conductivity (W/m °K) 0.024403

Vapor Specific Heat (J/kg *K) 1863.172

Heat of Evaporation (i/kg) 2.33.382 x 106

Heat of Fusion (J/kg) 3.34077 x 10

Particle Density (kg/m3) 977.7

Gas Constant for Water (J/kg °K) 461.94

Stefan-Boltzman Constant (W/m2 OK4 ) 5.669 x 10-8

Emissivity of Particle 0.9

23

Figure 8 shows how particle velocity changes as a function of distance from the source. For the largeparticles, the velocity changes very little from its initial velocity of 15 mis. However, for smaller particles,the velocity changes very rapidly near the source and becomes almost constant far from the source. This 9phenomenon can be explained by the forces that influence particle speeds. As explained earlier, thevapor velocity increases very rapidly after ejection from the source. This can be seen in Figure 9 where,in less than one half millimeter from the source, the vapor velocity reaches sonic speed. Therefore, this-sharp increase in vapor velocity can affect the velocity of the particles, especially the smaller ones. 'Ibissituation can be seen in Figure 8 where the 7.5 gim particle velocity increases from 15 m/s to as muchas 60 mr/s. In contrast the 250 pim particle velocity increases very slightly ( from 15 m/s to about 16 m/s).

.Glenn f 1969] explained another factor, which is worth mentioning hcre. In Figure 7, it is clear that theevaporation rate of the smaller particles near the source is much faster than for the larger ones.Therefore, it is possible that the local back pr%.4ure which is created by evaporation, and as earlierdiscussed, vapor drag, could increase the velocity of the smaller particles.

Figure 10 shows that vapor density decreases very rapidly from about 0.2 kg/m3 to 0.00001 kg/m3 in only0.25 meter distan:e from the source. Figure 11 is similar to Figure 10 except that the vapor density ispresented in terms of vapor number density. Figure 12 shows particle number density variations for 7.5,25, and 250 p.m as a function of radial distance r. Figure 12 also shows a representation of theexperimental measurements of Curry et al. [1985b] for a 0.13 p.m diameter particle. As can be seen fromFigure 12, the particle number density calculations of the present work are in good quantitativeagreement with the experimental work of Curry et al. [1985b].

5. SUMMARY AND CONCLUSIONS

An analytical investigation was made of a one-dimensional radial source flow of water into a vacuum.The particles were assumed to be spherical, with uniform size and temperature. The heat transferbetween particles and vapor was assumed to be by convection and radiation only. The equations thatdescribe this model are a sixth-order, nonlinear, two-point boundary value problem. The problem wassolved by the "shooting/splitting" technique. The results of this investigation were compared with theother works in the literature, and were generally in good agreement.

Data similar to those of recont shuttle orbiter water dumps into space were used to give some indicationof water flow characteristics into a vacuum environment. Since the results of the present work arelimited to a one-dimensional model, for a better understanding of the problem, the following goals aresuggested for future study. A two.dimensional continuum model of analysis for a source flow of liquidwater jet into a vacuum (for distances close to the source) is the next logical step, In order to investigatethe complete water flow structure as it is released into space, kinetic models would be necessary.

24

REFERENCES

--- Curry, B. P., Bryson, R. J., Seibner, B. L, and Jones, ]. H., "Selected Results from an Experiment onVenting an H20 Jet into Vacuum", Technical Report AEDC-TR-84-28. Arnold EngineeringDevelopment Center. Tullahoma, TN, January 1985a.

Curry, B. P., Bryson, R. J., Seibner, B. L, and Kiech, E. L, "Additional Results from an Experiment onVenting an H20 Jet into Vacuum", Technical Report AEDC-TR-85-3, Arnold Engineering DevelopmentCenter, Tullahoma, TN, June 1985b.

Fuchs, H. and Legge, H., "Flow of a Water Jet into Vacuum", Acta Astronautica, Vol. 6, 1979, pp. 1213-1226.

Gayle, J. B., Egger, C. T.. Bransford, J. W., "Freezing of Liquid on Sudden Exposure to Vacuum",Journal of Spacecraft and Rockets, Vol. 1 No. 3, May-June 1964, pp. 323-326.

Glenn L. A. "Stationary Radial Source Flow of Liquid Particles into Vacuum", Ph.D. Dissertation,University of Southern California, Los Angeles, January 1968 and AIAA Journal, Vol. 7, No. 3, March1%9, pp. 4,43-450.

Johnson, D. A. and Hallet, J., "Freezing and Shattering of Supercooled Water Drops", Quarterly Journalof the Royal Met. Soc., Vol. 94, No. 402, October 1968, pp. 468-482.

Kassal, F. T., "Scattering Properties of Ice Particles Formed by Release of 1-120 in Vacuum", Journal ofSpacecraft and Rockets, Vol. 11, No. 1, January 1974, pp. 54-56.

Muntz, E. P. and Orme, M., "Characteristics, Control, and Uses of Liquid Streams in Space" ,AIAAJournal, Vol. 25, No. 5, May 1987, pp. 746-756.

Pike, C. P., Knecht, D. J., Viereck, R. A., Murad, E., Kofsky, I. L., Mris, M. A., Tran, N. H., Ashley,G., "l.wst, L., Gersh, M. E., Elgin, J. B., Berk, A., Stair, Jr. A. T., Bagian, J P., and Buchli J. F., "Releaseof Liquid Water From the Space Shuttle", Geophysical Research Letters, Vol. 17, No. 2, February 1990,pp.139-142.

25

APPENDIX A. DERIVATION OF GOVERNING EQUATIONS

-Under the assumptions stated in the text, and using the control volume analysis, the system of equationsgoverning the flow are derived as follows:

A.1 CONTINUITY or VAPOR AND PAR' ICLE

The net flux of vapor out of a spherical control volume is equal to the rate at which mass is generated.by evaporation from the particles. With this analogy, the continuity equation of vapor can be stated inintegral form as

- J (1-) dA- fffm a dV (Al)

Where p, a, u, and rh are the vapor density, the particle volume fraction, vapor velocity, and mass rate

evaporated per unit volume of particles, respectively. n represents the unit vector.

Applying Gauss' theorem to the left side of the Equation (AI) yields:

ff pv (I-ax) t, dA r: f f [f t (1-c) tj dV (A2)

therefore, Equation (Al) can be stated as

fffa[p, (1-) j] dV- fffh . v (A3)

In the spherical coordinate system, with motion restricted to the radial coordinate,

d IP , ( 0 -a ) Ul 2 [p , ( I -C ) U l -, h a ( A 4 )dr r

or simply:

d Ir 2 (ltw) ' a r2 (A)

dr

A-1

-= =- ~ - = - -~ = --= ~ -

2

By analogy with Equation (A), the continuity equations for particles can be stated as follows:

f~ pp pI dA = afff d V (A6)

where p and p are particle density and velocity, respectively.

Proceeding as before, - -i

EfPP u, rdA ff 'jj Ipp p dV (A7)

Thus, equation (A6) can be written as

'f± a u I dV --fffh a dV (A8)

or, in one-dimensional spherical coordinates

d12 pa UPI _, a 2 (A9)

A.2 MOMENTUM EOUATION FOR VAPOR AND PARTICLE

The net force acting on the vapor within the control volume, plus the rate at which momentum is addedto the vapor by the evaporating particles, must equal the net rate of momentum outflow. The momentumequation for the vapor can be stated as

-ffP (I-a) d dA + ff[a p + P a (1_) dV + (A1)

fff; a up, d V ffpi (1-a) ,, tiI dA

when f is the total drag force exerted on the vapor per unit volume. P represents the local pressure.

Again, applying Gauss' theorem twice in Equation (A10), the result becomes:

A-2

1 -a) aI I ab a dV -

I a (All)

f JJ 1 (-a) V .U + U, "-' (1-a) tk dVA

Using the vapor Equation (A3) and substituting it into Equation (All), the result then becomes

-pf 1o, dV

(A12)

Restricting motion along the radial coordinate. (A12) reduces to:

P,(1-a) UU (I-a) P, -af = i a (U - U) (A13)

"11le momentum equation for the particle can be formulated in a manner similar to the analogy used forthe vapor momentum equation and one obtains:

-ff P a 9 dA -J ' ff - dV-fJ rhau dV-f ff IdVz (A14)

- f f pp a u, upv dA

Using Gauss' theorem and employing the particle continuity equation (AS), the momentum equation forthe particle in one-dimensional radial coordinates becomes

/ P/ (A5)p'au u ap +afr (A0

A-3

A.3 ENERGY EQUATION FOR VAPOR AND PARTICULATE MATER

From the first law of thermodynamics, the rate of heat addition to the vapor, plus the rate at which workis done on the vapor, plus the rate of energy addition due to the evaporating particulate matter, must -

equal the rate of energy outflow of the vapor,

ff q, u dV +f.f , dV -ff P I-Ix ,

f P ( 1+ - )t

rrr 1fff Tn 2"P + £ (A16)

II

ff p )( , + 2 U wUVu, dA

%%hre q is the total rate of heat transfer per unit volume by convectiun and radiation from the particles

to vapor, hT P is enthalpy of the vapor at particle temperature and e. is internal energy of vapor

at Napor temperature.

Equation (A16) can be written as

+f- up, ui dVT qP +( 2 (A17) 31

ff P, I'(", + V ivi U',) dIAa'V 2 I

I

where h e + Pip. Again, using Gauss' theorem, the right hand side of equation (A17) becomes

A 4

1

ff(pv-(1 (A dA2Y

(A18)

f f f (I- o [ O o ( + I I u 111) +

( + I / 4YJ

Using (A3)

aPv (1 U a (A3)

and substituting the above equation into (A18), we have

fffa I+, 9 s, + ii (A +. ,I s,.- h -7 t, U,) d -P T2IITV 2

(A19)± 1fjjfp(1 a) i (h. r + 41 ~j ) dV

If the vapor is now assumed to behave as a perfect gas, so that

k - f c,, dT

where Cpv is specific heat of the vapor and considered to be constant, then Equation (A19) can be

reduced to the following equation

A-5

0 t) t v CT 1 + (1 -a ) u u-

(A20)[ qp + fp upI + th a [IC ( T - T,,) + "I (u 2 U u2) .. . .. :. _

P P PW1 2 P 2~

Now, multiplying the vapor momentum Equation (A13) by u, and subtracting the result from (A20)yields:

01 qp + o f f - tP() + rh [ Cpv (Tp - ) + 1 (A 2)

t 2

The energ equation for particles by similar analogy with Equation (A16) can be formulated as follows:

-fff a , dV- fff a4 u, dV- ff P o, u, I, dA -

ffy oa rh(hl, - ,+ I Upi) dr={Ja F1i + 2 + UPI Pid

~(A22)

f f PP at 01 (Ct~ + (A22),)

(1 +Tp + 2 pi upi)] up, I dA

where eLf P and erP , and 13 are the internal energy of the liquid and solid phases at particle

temperature, and liquid volume fraction of the particles, respectively. Xv is latent heat of vaporization atand defined as

X,, - hv - h. (A23)

h'lien Equation (A22) becomes

A-6

fJJ -f ,P + f( , ' + + , Up) dV

-P 2f PP (h~ + 16p up) +(A24)

(1 - ( I "+ u) up, I) dA

where ht. p and hsrp are the enthalpy of the liquid and solid phases of the particles at particle

temperature, respectively. Using Gauss' theorem, the right hand side of Equation (A24) can be statedas

Jf PP (h _ ,_ up +

(1- + 1 t A) u, 9 dA .

(A25)

fff a [ (p a+ 1 h, .p,) +

(1.. P)(hs, + I si up)Jtspj dV

Substituting (A25) into (A24), and with some mathematical manipu'ation, (A24) becomes

111 f - uf p i+ t h OI(h + + I 4pi up) d Vof -" (1 21)x 1

fff ppa upi (htr X + U (A26)

1 a d

(hu - (1 - 03) Xf + I upi u,) a pp au dVp 2 35A-7

A-7

Where Xf is heat of fusion and defined as

Employing the particle continuity relation (A),

JfJ-( , + 4 -( ,I [ ( t-)f U, } Nv

(A27)a + IdV

j.ip P p (1 -, ) 2 J +v

Equation (A27) reduces to

P a Tp cp p - X 2 + PP a / 2 1P P vUP (A28)

- otIl + f up + YA (Xv + (1I ~ Xf)J

and, as before, multiplying the particle momentum equation (A1S) by up and subtracting the resull from(A28), yields

PP a up -P 6J 1 / (A29)

- a - [x+13 - , + X -

Equation (A19) is subjected to the following constraints:

Tp > T. ; l3 =1, 13'0

TP =T.. ; T'o (A-10)TP < T..; -3- 0

where T. and T.. are the initial and triple point temperatures respectively.

Eqiations (A5), (A), (Al3), (A15), (A21). and (A29) constitute a sixth order system in the ten unknownquantities p, a, u, m, up. P, fp, T1", qp, and TP. Therefore four constitutive relations are required.

A-9

A.4 CONSTITUTIVE RELATIONS

The rate of mass evaporated per unit volume of particles is assumed to be defined by the Hertz-Knudsen- equation: .

4 rrcy2 f(PO-P)( M 122 - R Tp 3 e 12 (A3 1)

4 TrrT3 a 2 7 R Tp3

where i, R, M, and P0 are evaporation coefficient, universcl gas constant, molar mass, and vaporpressure, respectively. The particle size, a, can be related to particle velocity and volume fraction byassuming that no particles are either created or destroyed. The number density of particles n. (numberof particles per unit volume) can be written:

C,

4 tro3 (A32)

3

and applying the aforementioned conservation theorem:

d (r2n,)=0 (A33)

substituting (A32) into (A33) and integrating. Equation (A33) then becomes

r2 a upr = (A34)

4 33

where the constant C can be defined in terms of the constants at the source, "llen (A34) becomnes

2 2

4 4 3 (A35)

3 3 6

where the subscript (*) refers to conditions at the initial source radius, r = r..

A-9

Equation (A35) can be written as

or3(r ) a(-) -~ (A36)(7, r, a* Up*

Next, assuming the vapor behaves as a perfect gas, the equation of state is:

Ppv RT (A37)

and the drag force per unit volume of the particles is taken to be of the form:

3 .. p )(A38)

where CD is the drag coefficient and is, in general, a function of particle Reynolds and Mach numbers.

qpqc+q, (A-39)

Finally, the heat transfer rate by convection and radiation is given by qc and q, respectively, and can bedefined per unit volume of the particle as

qc- 43 (A40)

3

Here the heat transfer coefficient h can be defined as a function of the Nusselt number, thermalconductivity, and radius of the particle

h= NuPk (A41)2ar

therefore Equation (A40) becomes

A-10

qc-3 k Nup(Tp - T)(A221 (A42)

and

4 ra 2 cr'SB 4 4

3

where us, and e, are the Stefan-Boltzman constant and emissivity of the particle. Equation (A3) issimplified to the following equation:

qr= - Et SB (A)a

Equations (A5), (A9), (A15), (A21), and (A29), together with Equations (A31), (A36), (A38), (A42). ,2'd(A44) constitute the complete system. Therefore, they can be consolidated into a sixth order system inihe unknowns pv, uv, up, T., T,, and x.

A-11

AI'PENDIX B. GLOSSARY

A Area

AI-A 6 Intermediate parameters defined in Equations (16-21)

C'. Specific heat of vapor

C1) Drag coefficient

e Specific internal energy

fp Drag force per unit volume

h Specific enthalpy

k Iliermal conductivity of vapor

Ph Mass rate evaporated rer unit volume of particles

M Molecular weight

np P Particle number density

Nu Particle Nusselt number

P Local pressure

P0 Vapor pressure

q, Convection heat transfer rate per unit volume of particle

q Convection and radiation heat transfer (qp = q, + q)

(Ir Radiation heat transfer rate per unit volume of particle

R Universal gas constant, Radial distance in Equation (23)

r radial coordinate

T Temperature

u Velocity

V Volume

B-I

a Particle volume fraction; particle volume/total volume

Particle liquid fraction; liquid particle volume/total particle volume

^f Ratio of specific heats

C Evaporation coefficient

Cf Emissivity of the particle

. 01 Dimensionless gas constant parameter . ...

02 Dimensionless initial radius ratio

03 Dimensionless initial pressure ratio

04 Dimensionless evaporation parameter

65 Dimensionless drag parameter

06 Dimensionless heat transfer parameter (convection)

07 Dimensionless vapor specific heat

08 Dimensionless parameter [C~ T.

P #

09 Dimensionless heat of vaporation

010 Dimensionless heat of fusion

Oil Dimensionless heat transfer parameter (radiition)

X, Heat of vaporation

X I. leat ol fusion

p Density

a Particle radius

UsH Stefan-Boltzman constant

B-2

* Initial condition

T riple point

v Vapor

p Particle

L Liquid

S solid

B-3