Condensation of water vapour with or without a carrier gas in a shock tube

CONDENSATION OF WATER VAPOUR WITH OR WITHOUT A CARRIER GAS IN A SHOCK TUBE

by

Jean Pascal Sislian

November, 1975

TECHNISCHE

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UTIAS Report No. 201 CN ISSN 0082-5255

..

..

•

CONDENSATION OF WATER VAPOUR WITH OR WITHOUT A CARRIER GAS

IN A SHOCK

TUBE

by

Jean Pascal Sis1ian

Subrnitted November,

1975

Q

Acknowledgements

I should like to express my gratitudeto Professor I. I. Glass,

rrry supervisor, for his helpful advice, continued interest and encouragement throughout the course of this investigation.

Many helpful discussions with Dr. S. P. Kalra are acknowledged wi th thanks.

MY

thanks are also extended to Mrs. Winifred Dillon for typing the manuscript.

This research was financially supported by the Atomic Energy of

Canada Ltd. (AECL), Chalk :B.iver, under Contract and by the

u.s.

Air Force Office of Scientific Research, under Grant No. AF-AFOSR 72-2274. Their support is grate:t'ully acknowledged.

•

Abstract

A detailed theoretical investigation has been made of the condensati on of water vapour wi th or without a carrier gas in the nonstationary rarefaction wave generated in a shock tube. All possible si tuations such as frozen,

nonequilibrium and equilibrium flows have been considered.

In the case of equilibrium condensation, analytical expressions were

obtained for all the flow variables in the rarefaction wave for the cases of pure water vapour and water vapour /nitrogen mixtures for gi ven ini tial condi ti ons

in the shock tube. It was found that equilibrium condensation affects mainly the temperature and hence the equilibrium sound speed and Mach number, i.e., the energy released by condensation simply heats 'the gas in the driver section

and has no appreciable effect on the pressure and densi ty profiles . The effect of condensation in the driver gas on the shock wave generated by burs ting a

diaphragm in a shock tube is negligible for finite values of the diaphragm pressure ratio P4l. It is appreciable only w hen P4l -700.

In the nonequilibrium flow case when condensation takes place by homogeneous nucleation, the equation of motion together with the nucleation rate and the droplet growth equations were solved numerically by the method of

characteristics and Lax ' s method of implicit artificial viscosi ty. I t is found that for the case considered the condensation wave formed by the collapse of the metastable nonequilibrium state is followed by a shock wave generated by the i:b.tersection of characteristics of the same family. The expansion is practically isentropic up to the onset of condensation and the condensation

front is concave to the oncoming flow. The results of the computations for a chosen case of water vapour/nitrogen mixture are presented by platting variations

of pressure, temperature, supersaturation, nucleation rate, number density of critical clusters, condensate mass fraction, droplet radius and cri tical-cluster radius along three particle paths and at four time levels, which illustratethe detailed structure of the resul ting wave system.

Some consideration is gi ven to homogeneous condensation experiments conducted in the shock tubes by a number of experimenters • Although a direct comparison of the present theoretical work and these experiments is not

..

PART I TABLE OF CONTENTS Acknowledg emen't ii Abstract iii Table of Conten'ts iv No'tation vi

INTROD UCT ION 1

EQUILIBRIUM CONDENSATION

4

1. Thermodynamic Equations

4

2. Formulation of Problem 10

3.

De'termination of the Onset of Equilibrium Condensation 11

4.

Solution of Problem 12

4.1 Calculation of Equilibrium Condensate Mass Fraction 12

4.2 Equilibrium Speed of Sound 13

4.3

The Equilibrium Isentrope in the p,T-plane

15

4.4

Relation Between the Total Pressure and Temperature

16

4.5

Determination of the Nonstationary Parti cle Velo city

17

4.6

Relation Between Characteristics Slope and

Temperature

5 . Strength of the Transmitted Shock Wave

6.

Discussion of Resul'ts

18

19

20

PART II NONEQUILIBRIUM CONDENSATION 22

7.

Nucleation and Droplet-Growth Equations

8.

Basic Equations of Motion

9.

Characteristics of Equations of Motion

10. Numer ical Solution - Method of Characteris tics 11. Numerical Solution - Laxts Method

12. Discussion of Results

13. Comparisons wi th Experimental Data PART III CONCLUSIONS

22

29

34

39

45

52

56

58 REFERENCES

62

TABLE I NUMERICAL CONSTANTS

TABLE Ir INITIAL CONDITIONS IN THE SHOCK TUBE

APPENDIX A: NUMERICAL INTEGRATION OF RATE EQUATIONS

A.l Implicit Integration of Rate Equa:tions in the Method for Characteristics

A.2 Implicit Integration of Rate Equations in Lax' s Method

APPENDIX B: NUMERICAL METHOD FOR CHARACTERISTICS B.l Ini tial Concü,tions

B.2 Nonequilibrium Flow Field Calculations and Error Con'trol

'

.

A B

c

E G H I L M

N

R T

u

v

w

a b c c _r e

NarATION

rate variab1e, Eq. 8.2; constant in Eq. 1.4; nondimensiona1 constant in Eq. 4.5

nondimensiona1 constant, Eq. 8.17

ra:te variab1e, Eq. 8.5; constant in Eq. 1.4

rate variab1e, Eq. 8.7; integration constant in Eq. 1. 3 tota1 specific energy, Eq. 11.11

coefficient of characteristic re1ations, Eq. 10.19 dimensiona1 constant, Eq. 8.21

nondimensiona1 constant, Eq. 8.30

coefficient of characteristic re1ations, Eq. 10.20; tota1 entha1py nuc1eation rate

latent heat of vaporization Mach number

nondimensiona1 slope of characteristics; number of molecules Avogadro number

characteristic slope, Eq. 10.2

universa1 gas constant; characteristic slope, Eq. 10.6 tempera:ture

interna1 energy; nondimensiona1 velo city , Eqs. 4.34 and 5.2 volume

nondimensiona1 constant, Eq. 8.24

speed of sound; coefficient in Eq. 10.7 L ~

IR,

Eq. 4.9; coefficient in Eq. 10.7

v

specific heat; Lagrangian coordinate; coefficient in Eq. 10.7 rate variables, Eq. 10.29

characteristic Lagrangian coordinate, Eq. 11.8 specific interna1 energy, Eq. 11.7

f "production" function, Eq.

7.14

"production" functions, Eqs.

9.24

to

9.27

g condensate mass fraction

h specific enthalpy

k Bol tzmann cons tant

t

characteristic length

m mass, quantity in Eq.

10.14

n quarrtity in Eq.

10.14

p pressure

q Lagrangian vector variable, Eq.

11.20

r radius

r

1

dimensj,onal constant, Eq.

8.19

s supersat~ation

t time

u velocity , specific internal energy

v specific volume

w Lagrangian vector variable, Eq.

11.20

x ab~cissa·

y; Lagrangian coordina:te, Eq.

11.18,

nondimensional variable in

Eq.

4.7

z nondimensional temperature, Eq.

4.7

r

isentropie index, Eqs.

4.15

and

9.14

e

nondimensional constant, Eq.

11.15

8(z) function in Eq.

4.20

thermodynamic pot ential

a

condensation coefficient

R/CpO~' Eqs.

4.5

and

4.15

.,

specific heat ratio

nondimensional variable, Cp

TIL

in Eq. 4.5; nondimensional constant,

Eq. 8.14 0

molecular weight, constant in Eq. 11.24 constant in Eq. 11. 31

p density

cr

surf ace tension

,.

_{characteristic time, Eq. 8.35}

relative humidity; thermodynamic potential

Cl> specific humidity

Subscripts

o

condition in the driver section of the shock tube

1 condition in the chamber of the shock tube

v vapour

i inert carrier gas

j finite-difference Lagrangian coordinate position, Fig. 25

e equilibrium.

f frozen

d droplet

c onset of equilibrium. condensation

liquid

s saturation state

D droplet

1,2,3,4,5 points in characteristic mesh, Fig. 23 Superscripts

n finite-difference time level, Fig. 25

*

critical

nondimensional

INTRODUCTION

Problems of the mechanics of multiphase and multicomponent media have long retained the attention of researchers • Of special interest are problems of two-phase and two-component systems. This interest is not incidental. In nuc1ear-reactor, chemical and meta11urgical technologies, in aeronautics, in steam turbine and other fields of technology, we have to deal with processes which are accompanied by the formation of liquid-gaseous systems during stationary or nonstationary changes occurring in these systems.

The most essential physical aspects of the mo'tion of two-phase media are the occurrence of phase changes and the thermal and f1uid-dynamic interactions between phases. The motion of two-phase systems is further comp1icated by the presence of metastable nonequilibriilln states such as. supercoo1ing or superheating, condensation or vapourization shocks and so on.

The present investigation is concerned wi th a particular type of compressible two-phase system, namely, water-vapour condensation, which has been intensively investigated in the past in diverse fields like meteorology, supersonic-nozzle flow, steam-turbine applications, cloud chambers, etc. The exothermic character of the condensation process places it in close relation to flows involving chemical reactions.

The theory of condensation phenomenon in a chemically and physically pure gaseous medium, i.e., the theory of homogeneous nucleation, has been developed by Volmer and Weber (Ref. 1), Farkas (Ref. 2), Becker and Doring (Ref.

3),

Frenkel (Ref. 4), and Zeldovich (Ref.

5).

More recently, several authors have tried to improve this classical theory based on a semi-phenomeno-logical model, by treating homogeneous nucleation from a statistical mechanical viewpoint (see, for example, Refs. 6,7,8). Parallel to this work, extensive experimental and analytical investigations of s'team condensation primarily in steady supersonic-nozz1e flows were carried on, both to check the validity of various available nucleation theories and to solve gasdynamic problems with condensation of interest in different areas of engineering.

Until recent1y, almost all of the analytical studies of gasdynamic prob1ems involving condensation were performed within the framework of one-dimensional steady flow. The first attempt at a more detailed two-dimensional investigation of the steady supersonic-nozzle flow field with steam condensation was done by Bartlma in 1964 (Ref. 9), who applied the method of characteristics together with a simp1ified linearized condensation rate equation wi th constant relaxation time. Tkalenko et al (Refs. 10, 11) studied both experimentally and analytically the water-vapour condensation in a steady two-dimensional Prandt1-Meyer flow around a sharp corner by the method of characteristics and using the c1assical homogeneous nuc1eation and drop1et-growth theories. Smith (Ref. 12) investigated this problem experimentally. Later, Tka1enko (Ref. 13) and

independently Davydov (Ref. 14) used the same approach in the analytical investigation of steady two-dimensional and axisymmetric supersonic nozzle steam flow with condensation.

Along with the techniques of steady nozzle flow, nonstationary expansion in a shock tube, as used in condensation studies, provides one more _I

possibility of application of experimental and quantitative methods of

gasdynamics to the resolution of fast processes of interest in various areas of technology. It may prove i ts value by providing quant i tati ve experimental and analytical results for these processes in nonstationary flow situations.

The application of a shock tube to experimental investigation of moist air condensation was initiated by Wegener and Lundquist (Ref. 15). First streak photographs showing condensation zones in the expansion fan were obtained by Glass and Patterson (Ref. 16). Courtney (Ref. 17) used the shock tube for the experimental investigation of condensation of water vapour, methyl alcohol and carbon te'trachloride. Barschdorff (Ref. 18) experimentally investigated the carrier gas effects on homogeneous nucleation of water vapour in a shock tube. Similar experiments in a shock tube were performed by Kawada and

Mori (Ref. 19). Lately, at the Institute for Aerospace studies, University of Toron to, S. P. Kalra (Ref. 20) conduc ted experin:en ts on homogeneous nucle a:tion of water vapour in a shock tube. I't was hoped that his results would make possible a comparison of theory and experiment.

In the present inves'tigation a detailed analytical study is made of condensation of water yapour with or wi thout a carrier gas in the nonstationary rarefaction wave generated in a shock tube. Quanti'tative analytical and

numerical methods are developed to predictthe effects of condensation on the flow variables in nonequilibrium and equilibrium flow situations.

In Part I 'the simpler equilibrium case to which the nonequilibrium flow tends asymptotically for t ~ 00 has been considered. The overall mass

conservation, momentum and energy equations together wi th Clausius-Clapeyron equation for the equilibrium between phases were solved for given initial conditions to yield analytical e2ÇPressions for the condensate mass fraction, 'temperature, sound velocity, particle velocity, Mach number, pressure and density in the rarefaction wave for the cases of pure vapour and water

vapour/carrier gas mixtures. The shock-tube problem has also been investigated and the strength of the shock wave vs the diaphragm pressure ratio has been calculated for a driver with condensation.

In Part II the nonequilibrium flow situation resulting from homo-geneous condensation is considered. In this case the Clausius-Clapeyron equation is replaced by the condensa:tion-rate equation based on nucleaticn and droplet-growth theories, which relate the condensate mass fraction 'to t he other flow variables . The basic equa:tions describing such a motion are m t solvable analytically and we must resort to numerical methods of integration. Two such methods were applied. In the first, the hyperbolic character of

the basic partial differential equations was used to reduce them to a system of ordinary equations for the characteristics, which describe the laws of propagation of disturbances in the nonstationary flow field. These equations were then solved numerically s'tep-by-step, subject to ini'tial conditions along the frozen rarefaction wave head and at the centre of the rarefacQon wave. Computations by 'this method were performed only in the rarefaction fan of the flow field where most of the condensation occurs.

The numerical method of characteristics used in the present

investigation for the solution of the problem of homogeneous nonequilibrium condensation in a nonstationary rarefaction wave is valid only for smooth flows, i. e., the dependent variables are all continuous over the whole

computed flow field. However, flow variables may have discontinuous values, as for· instance, across shock waves. In order to handle such discontinuities in the flow field a second, Lax's method of artificial viscosity, was used in the present investigation. Lax' s method requires that all partial differential equations of motion be in conservative Lagrangian form. These equations were integrated numerically step-by-step subject to initial and beundary conditions along a wall at -42.26cm from the diaphragm of the shock tube and along the contact surface • The rate equations were integrated "Qy an implici t integration scheme to avoid instabili ties in the computations due to the rapid, sudden approach of the flow to equilibrium af ter the condensation zone.

The numerical computations by both methods were performed on an IBM370-l65 at the Insti tute of Computer Science, Uni versity of. Toronto. The results of the computations for a chosen case of water vapour/nitrogen mixture are presented by variations of pressure , temperature , supersaturation,

nucleatien rate, number densi ty of cri tical size droplets , condensate mass fractiEm, droplet radius and cri tical cluster radius, along three particle paths ini tially at -5, -10 and -20cm from the diaphragm and at four time . levels, 0.48, 0.72, 0.96 and 1.2msec.

It is found that equilibrium condensation affects mainly the

temperature and hence the equili bri um sound speed and Mach number, i. e., the energy released by condensation simply heats the gas in the driver section wi thout appreciably affecting the pressure and densi ty profiles • The effect

ofcondensation in the driver gas on the shock wave::generated by bursting a diaphragm in a shock tube is negligible for finite values of the diaphragm pressure ratio, P4l. It is appreciably only wh en the diaphragm pressure ratio is very large, P4l ~ 0::1.

In the nonequilibrium flow when condensation takes place by homo-geneous nucleation, i t is found that the condensation wave formed by the collapse of the metastable nonequilibrium state is followed by a shock wave generated by the intersection of characteristics of the same fa.m:i.ly comprising the rarefaction wave. The leading front of the condensation wave is found to be concave with respect to the oncoming flow. The rise in pressure and

temperature due to condensation is higher for particle paths farther from the origin and the width of the condensation wave increases with increasing distance fram the origin withinlimits.

An exa.m:i.nation of the experimental pressure profiles ebtained by several authors at observation stations relatively close to the cliaphragm, shows that the rarefaction waves generated in shock tubes at these distances arenen-centred, and 1ll9.y even be non-planar, and that the rates of expansion are lower than thosè in the calculated centred-expansion waves assUIDyd to start at the origin in the present analytical investigation • . The homogeneous condensatien process is very sensitive to rates of expansion and cooling. The degree of supersaturation increases rapidly with an increasing rate of

expansion. The degrees of supersaturation obtained analytically in the idealized nonstationary centred-rarefaction wave are higher than the experi-mental values, where the imaginary origin of the wave has occurred miich earlier in time, thereby resulting in lower condensation rates. Experiments on homo-geneous condensation in a shocJ;ç. tube conducted at UTIAS give a degree of supersaturation ~ 14, at a distance of l7.lcm from the di aphragm , whereas

the theeretical value obtained in the present investigation is ~ 25 for the same distance. The line of the onset of condensation calculated from experimental data lies cIos er to the rarefaction wave head than te the tail of the wave, whereas the theoretically obtained line of the onset is closerto the tail of the rarefaction wave. This result can be reconciled on the basis of the different cooling rates and values of surface tension.

PART I. EQUILIBRIUM CONDENSATION

1. Thermodynamic Equations

In this section, the definitions of the thermodynamic quantities and thermodynamic equations for a condensing mixture of an inert carrier gas and water vapour will be summarized •.. The detailed derivations of the thermodynamic equations can be found in textbooks (for example, Ref. 21) or in the literature (Refs. 22 and 23). In writing the basic equations, the fOllowing assumptions mIl be made in accordance with the thermodynamic theory of condensing gas mixtures:

(i) The molecular transport effects leading to viscosity, heat conductim, and diffusion are neglected.

(ii) Up to the actual appearance of condensation, the gas mixture may be treated as a thermally and calorically perfect gas, i.e., the phase change occurs sufficiently far from the so-called critical point on the phase diagram (see Fig. 1) •

(iii) The density, enthalpy, etc. for the mixture are the weighted sums of the corresponding properties for the single system.

Consider a liquid-vapour system in thermodynamic equilibrium. . The isotherms on the phase diagram (V ,p) in Fig. 1 for such a system are obtained in the fOllowing way: at constant temperature the volume of the vapour is increased. Then for the equilibrium to be maintained between the phases, a certain ·amount of liquid should evaporate; this keeps the vapour pressure constant •. Therefore, as long as there is a sufficient amount of liquid, the vapour pressureremains constant when the volume of vapour is increased. The isotherm for a liquid-vapour mixture is, therefore, a line of constant pressure parallel to the V-axis (see Fig. 1). Af ter the whole liquid is evaporated, the pressure decreases with increasing volume of the vapour.

During the reverse process of compressing the system at constant temperature , the pressure increases until the saturation pressure for ih at temperature is reached, then as the vapour condenses its pressure remains constant, and af ter the whole vapour is condensed into liquid the pressure rises sharply due to the small coItlPressibility of the liquide

Several isotherms of the kind des cri bed abeve are plotted in Fig. 1 for different temperatures (lines a, b, c, d). It is readily seen that the length of the horizontal segment of the isotherms, i.e., the volume interval for which the liquid and vapour, for a gi ven constant temperature , are in

thermedynamic equilibrium, decreases with increasing temperature and becoIres

. infinitesimally small at the point of inflexion • . The isotherm containing the point of inflexion is called the critical isotherm and the corresponding

temperature T*, volume V*, and pressure p* are called the cri tical telI!Perat ure, volume and pressure, respectively. This critical state of the system is a

- - -

---The isotherms for temperatures higher than the critical are monotomic decreasing functions of the volume and for very high temperatures they approach equilateral hyperbolas, as for an ideal gas.

The dotted line in Fig. land the critical isotherm divide the (V,p) plane into four regions: in region L, the system is in a purely liquid state; in region L,V, the system is a mixture of liquid and saturated vapour; in region V, the system is in a vapour state, and in region G in a gas state.

Let

Vt,

vv' Ut,

Uv

be the specific volumes and specific energies (i.e., volumes and energies referred to unit mass) of the liquid and vapour phases respectively in the L,V region on the (V,p) diagram of Fig. 1. The quantities p,

Vt,

vv' Ut,

Uv

are functions only of temperature. If mis the total mass of the substance under consideration and mt and

IDv

the masses 0 f

the liquid and vapour phases, then

m = mt + mv

The total volume and the total energy of the system are

Consider now an isothermic process in this system during which a certain amount, dm, of substanee passes from the vapour to the liquid state; this transformation changes the total volume and the total energy by dV and dU. At the end of this process there will be mt + dm grams of liquid and

IDv -

dm grams of vapour sa that thetotal volume will be

or

Similarly, the total energy of the system will change by an amount

From the first law of thermodynamics we have

or

E~uation 1.1 represents the heat released during the condensation of one gram of vapour at constant temperature and is called latent heat of

condensation. The quantity L, depends on the temperature and pressure and

has different values for different vapours. For water vapour at normal pressure and temperature around the condensation point L

=

623 cal/go

For an isothermal process we can write

or by (lol)

Using the well known thermodynamic relation (Ref. 21)

dU

d-p

dvj

=TöTl-p

·T V

and taking into account that the pressure in the system considered is a function only of temperature, we finally obtain

(1.2)

This is the Clausius-Clapeyron e~uation for e~uilibrium between li~uid and vapour phases. By analogy with reacting flows, it may be viewed as the law of mass action for the equilibrium balance of condensate and vapour. Ordinarily, we may assume tha't the liquid volume is negligible compared wi th the gaseous one so that Vv - vi, s:::f. vv' and furth~r that the latent heat of

condensation is constant in an appreciable temperature range. If, as assumed, the vapour satisfies the e~uation of state of a perfect gas .

R

P = - p T

v Ilv v

where Ilv is the molecular weigh t of the vapour, then E~. 1. 2 may be in tegr a ted to give lnp s L

= - - - + c

_R

- T

Ilv where C is the constant of integration, or

=

B _ A T

(1.3)

For water vapour, the constants to be used in Eq. 1.4 throughout 'the present investigation are given in Ref. 23, and are, B = 6.064 atm, and A = 2263.0oK. These values are obtained by calculating the best fit to Eq.

1.4

for water vapour pressure data tabula'ted in standard handbooks.

, The 'total enthalpy H of the mixture of ire rt gas (subscript i), water vapour (subscript v), and condensate (subscript ,g) in terms of the

en'thalpies of the separate components hi' hv and ht, is

(1.5)

where, m is the total mass of the mixture: m =

lIl:i.

+

Illv

+ m,g. At thermodynamic equilibrium, phasetransformations take place at constant pressure, hence from 'the definition of enthalpy, we have

db = du + pdv or

or

The difference v,g(T,g) - v,g(T) is small compared with v,g(T) - vv(T) and may be neglected. Taking account of Eq. 1.1, we obtain

(1.6)

where, c,g is the specific heat and T,g, the temperature of theliquid phase. Inserting Eq.

1-9.

into Eq.

1.5

gives

H mi m m,g h=~=-h. _{m m ] .} +-2h +-m [hv-L+Cn(Tn -T)] _{m v} ) ( J ) ( J or m. m + mn h = ~ h. + v )(J h m ] . m v (1. 7)

Condensate appearance at the expense of vapour depletion leads to dm,g =-d.mv or ~ + llIv-

=

llIv-o' where, ~o is the initial amount of water vapour present (prior to condensation). Denoting

m.

and = ~ + .W

m 0 (1.8) where, Wo is called the initial specific humidity and g the condensate mass fraction, and taking the inert gas and water vapour individually to be thermally and calorically perfect gases, Eq. 1.7 becomes

(1.9) where,

c....

=

(1 - w ) c... + w cp

.I! 0 o.l!~ 0 V (1.10)

where, Cp. and cp are the specific heats at constant pressure of the inert gas and vapouF, respe~tively.

For condensation in inert gas expansions, full-temperature accommoda-tion of the condensate particles may be assumed. The particle is hit about a thousand times by the inert molecules before an H20 molecule impinges again (Ref.

23).

This process provides for extremely rapid temperature equalization. For condensation of pure water vapour (steam) the third term in the right-hand

side of Eq. 1.9 can be neglected compared with the first. Typically, at the start of condensation,

after precipitation of a sUbstantial quantity of condensate,

Since Cpo "" c~ then

Consequently, the expression for the enthalpy of the gas-liquid mixture takes the form

h =

where, Cpo is given by Eq. 1.10.

(1.11)

The equation of state for a mixture of two thermally and caloricaJ..ly perfect gases (inert gas and water vapour) is

(

P. P)

p=R .2:.+~ T

!-Li !-Lv (1.12)

where, R is the universal gas constant, and !-Li and !-Lv are the molecular weights of the corresponding gases. Introducing the molecular weight of the mixture we can write

Then Eq. 1.12 becomes

(

P' + P )

P

=

R i

~

v

RT (1.13)

Consider a fixed mass m of the mixture. The sum of the vapour and liquid mass is constant end equal to the initial vapour mass Illv. Then from

I 0 m= m. + mv +

mt,

mv +

mt

=

_Illv

_o (1.14 ) ~ we ha.ve m -

Inv

_o , -1

Inv

o 1 ₊

mt

1 (1.15 ) ~ m -

mt

_~i m - m,e _~v or from E<l' 1.8 1 - W W _{- g} 1 0 1 ₊ 0 1 (1.16) = ~ 1 - g _~i 1 - g _~v

From Eq. 1.14 the mixture density is

(1.17) where, pi is defined as the condensed mass, referred to the same volume of

inert gas and vapour as the other density terms. The assumption that the liquid volume is much less the gas volume is implied in this definition.

Using Eq. 1.17 we cen write Pi + Pv in Eq. 1.13 in terms of the total density P end mass condensate fraction to obtain

or Using Eq. 1.16 p

=

P(l - g)

~

T ~ ( 1 - Wo W - g ) p

=

+ 0 RPT ~i ~v ( 1.18) (1.19 ) Equations 1.18 and 1.19 are two different forms of the equation of state for a condensing mixture of gases. It is readily seen that the mixture

of condensate, inert gas and vapour does not obey the thermally perfect gas

equation, since, in general, 0

<

g <~:1. Equation 1.18 is analogous to the

equation of state of a dissociatinggas, where the degree of dissociation

takes the place of g, and ~ refers to the undissociated species.

The differential form of the equation of state, Eq. 1.19 is

dg

1 - g (1.20 )

2. Formulation of Problem

The problem to be considered in the present investigation is the following:

steam (water vapour) or a mixture of water vapour and an inert carrier gas at a certain total pressure Po and temperature To is initially at rest in the driver section of a shock tube. At time t

=

0, the diaphragm separating the driver sedion from the channel (low-pressure section) of the shock tube is removed, andthe mixture is suddenly expanded into the channel. The vapour contained in the mixture (point 0, Fig. 2) i s i sen tropi cally

expanded by a rarefaction wave and cooled until it becomes saturated (point C, Fig. 2) af ter which condensation starts. The problem is to determine the

effect of condensation on the nonstationary rarefaction wave and how it effects the strength of the shock wave. The wave system following diaphragm rupt1.r e is illustrated in the time-distance (x,t)-plane on Fig.

3.

It can be seen that in addition to the shock wave and rarefaction wave a contact surface also forms, separating the original gases. Particle paths are also shown as well as the head and tail of the rarefadion wave.

In this section the following simplifying assumptions are made in order to solvethis problem:

(i) The effects due to viscosity, heat conduction and diffusion are neglected.

(ii) The condensate moves at the same flow speed as the inert gas and vapour .

(iii) The condensate and the mixture of inert gas and vapour are at the same temperature.

(iv) Condensation of water vapour starts immediately at the saturation line, i.e., condensation always occurs at thermodynamic equilibrium.

Since the nonstationary flow during such an expansion is always in thermodynamic equilibrium, the flow, af ter reaching the saturation line

(point C, Fig. 2), continues to expand along a particular saturated isentrope C - 11. In the absence of any characteristic length or time the flow is self-similar and the wave pattern of the flow is as shown on Fig.

3,

where c

denotes the characteristic of the rarefaction wave along which saturation is reached.

The gasdynamic and thermodynamic equations describing such a flow are:

Continuity: dP

_dt

₊ d(PU) _êïx

=

0 (2.1)

Momentum: _(TI; dU _{+ u}

_di

dU 1

~

(2.2)

P

Energy: (adiabatic flow) _dh

_=!

_~

(2.3) dt P dt

Clausius-Clapeyron:

(equilibrium between phases)

Equation of state:

dp Lp

s

v

-- =--

_{dT T}

(2.4)

,

Here p ~'Pi

+

Pv

+

Pt, p = Pi + Pv; h is the enthalpy of the gas-liquid mixture and is given by Eq. 1.11. We have thus five equations, Eqs. 2.1 to

2.5,

for the fi ve unknown functions g, P, p, u, and T of the independent variables x and t.

3.

Determination of the Onset of Equilibrium Condensation is

The equation of the isentrope passing through the point 0, Fig. 2,

ro

E...-

=

(~

)7

0

-1

(3 .1)

Po 0

where, 7₀ is the ratio of specific heats of the mixture before condensation takes place. As long as no phase change occurs, the partial pressure ratio remains unchanged. Hence

and from Eq.

3.1,

(3.2)

However, at saturation Pv

=

_{Ps(T c)' where Tc is the temperature at the onset} of condensation, and using Eq.

1.4

(B-f: ) Tc p :::: 10

v o

For the gener al case of a mixture of. gases, the following relation between the initial vapour pressure and the initial total pressure holds (using the

equatlons of state for the mixture components) mv Il.)'v w o = _ _ _ 0 _ _ = _ _ .--;o~ _ _ = or where Ct

=

IJ.v/lli' m. ~ o + mv o lliPi 0 + Il.)'v 0

apv

o p + (Ct-1)p o v o

(3.4 )

Inserting Eq.

3.4

into Eq.

3.3

and multiplying by the conversion factor

1.0325

x

10)

dynes/crrf3. yields

0: ... w

(0:-1)

o Po

=

w

-o A (B - - )

10

Tc

. 1.0325

x

105

dynes 2 cm

For a given initial condition in the driver section of the shock tube Po, To ' wo' and for a given

0:,

Eq.

3.5

gives the temperature Tc at which equilibrium condensation starts. The values of the other flow variables at this point can then be obtained from the isentropic relations (Ref.

25).

Instead of Po' To ' wo,the initial state mayalso be given by the quantities <Po, _{wo, To, where ceo,is the relative humidity and is defined as the} ratio of the vapour pressure to the saturation pressure at the same temperature,

PVo

<Po

=

ps(T o) From Eqs.

3.3

and

3.6

we obtain

A(.L .L)

cp 0 =

10

To Tc

(3.6)

Again, for given values of <Po, _{wo, To ' Eq.}

3.7

determines Tc. Values of <Po are plotted as a function of Tc/T o for two cases as indicated in Fig.

4.

If for the pure water vapour case .. CPo

=

0.857,

then from curve

1,

Fig.

4,

we obtain Tc/To

=

0.987

or Tc

=

324.1

o

K

for the temperature of the onset of equilibrium condensation

4.

Solution of Problem

4.1

Calculation of the Equilibrium Condensate Mass Fraction Differentiating the equation of state, Eq.

2.5

yields,

B....

Tdg + R ( 1 - Wo + Wo - g ) (dT

+:E

dp)

I-lv I-l_i I-lv p

=

-Then from the energy equation, Eq.

2.3,

the enthalpy relation Eq.

1.11

and Eq.

4.1,

R

C dT - Ldg + - Tdg

Po I-lv

R (

l~:

Wo + W

o~~

g) (

dT +

T~P

)

=

0

The integrated form of Eq.

2.4

can be written as

~v

p (w 0 - g)

T

= exp (

C -

RL

T)

I-lv

(4.1)

(4.2)

(see also Eq.

1.3),

where the left-hand side is the expression for the partial vapour pressure. Differentiating Eq.

4.3

and using again Eq.

4.3

results in,

"

[ ( L ) dT dp ] dg = (w - g) 1 - - - - +

-o

B....

_T T P

(4.4)

Ilv

Eliminating the density from Eqs.

4.2

and

4.4,

gives

l - W ) [ ( l - w w _{- g)}

J

_;T

( _ L + R o L 0 0

d

=

1 - - - +

c T _{c P Ili w - g} g c T Ilv Il. _Ilv

Po 0 , P 1

0 0

Introducing the new variable Cp T/L

=

À, the following differential equati~n for1fhe equilibrium condensate 0 mass fraction is obtained,

dg Wo - g À - (wo - g) - A

dA

=

'

A

t3

AA

+

(w _

g) . (

4

,

.5)

v 0

where, A

=

(1 '- wo) Ilv/Ili and t3_v

=

R/Cp Ilv

=

(llo/llv)(Yo -'1)/70' . Equation

4.5

can be integrated by using the substitu~ion

w -

g

o v =

---.---A

Equation

4.5

then becomes

t3 A)

+

...:!-v dÀ

The solution of Eq.

4.6

is,

~ t3 AÀl V C Y + W z~ny

=

z _w I ,C z~ nz

+!::....

_w (z _ 1) o o 0

(4.6)

(4.7)

where, y

=

_{(wo - ge)/w o' z}

= À/Àé

are nondimensional variables. When

z ~ 1 (onset of condensation), y ~ 1 and ge ~ O. When z ~ 0 (complete

expansion) y ~ 0 and ge ~ wo' For the particular case of pure water vapour, Wo

=

1. Eq.

4.7

reduces to 1 - g _e Cp T T T = _ 0 _ ~n

-2

+ - or L T T c cp oT T ( , T ) g = - - ~n - + 1 -e L T T c c

which is the value obtained by Buhler (Ref.

24,

see also Ref.

23).

4.2

Equilibrium Speed of Sound

(4.8)

Substituting Eq.

4.4

into the differerrtial form of the expression for errthalpy Eq. 1.11 we get

Here b

=

L!lv/R. dI'

=

( I '

But

From Eqs. 1.20 and 4.4

W o - g I' ) _P

d [

+ I' ~ W 0 - g ( 1

b)

J

dT l - g p - " i l l - g

-'T

I'

T

~v v

_H....

_(4.10) R I'

= -

p(w v ~ 0 v g)T

=

~

p(w _ g)T

~l

-

g)~

~ 0 1 - g)~ v

=

~

P(l -

g)T ~ W _ g

H....

_0:--_

=

~v 1 - g W - g ~ ~o _ _ I'

="il

l - g v Hence Eq. 4.10 becomes

dp [ ( b ) J d T

dI'

=

(I' - I' ) - + I' - I' 1 - - - .

v p v T T (4.11)

Eliminating the temperature from 'Eq.

4.9

of enthalpy

and 4.11 we obtain for the differential c T - L(w - g)

(1 _

È )

I' 0 T

dh=

0 ..

C

b)

I' - p_v. _' 1. - -_T .. (4.12)

Inserting Eq. 4.12 into the energy equation, Eq. 2.3, and af ter performing the necessary transforma:tions, we finally obtain

T 0 ~ + L ~v

(w

)

[ l - W 2

J

G_p 1 _ g

jl:-

~ 0 - g ~

=

0 l. R

(1 _

g) T

(4.13 )

dp CPoT -

~

(1 - g)T

+ (

1 -

LR~V)

(wo -

g{

~

-

L)

~

But dp/dp for constant entropy (the flow being isentropic) is equa1 to the square of the equilibrium sound speed ae. Hence

2 R

a

=

r -

(1 -

g)T

e e ~

where, reis the equil:ib rium isentropic index, and is defined by

r

_e

=

1 -

w

W - g 1 ~ _ _ o ~ + _0---.,.:-1 - g

~i ~v"e2 ;

W - g 1 -

~.

(1 - W ) +

°

2

~2

(1 -

2~

I\

,

'z)

l.

°

A 1\ " v C I-'v

C

z (4.14) (4.15)

where ~.

=

R/c.... ~.. At the onset of equilibrium condensation, from Eq. 4.14 l. !l0 l.

where IJ. w

(1 _

w ) ~

+

0 o IJ. -₁ A >. 2 I-' 1\,: v e ree

=

---w~~---and Partieu1ar Cases: 1 -

~-(1 -

w ) + 0 2 (1 -

2~

" ) 1 0 A " . ve I-'v ~ ,

r

IJ. 0 - ~ . (1 - g) -- z - re IJ. e (4.17) (4.18)

(a) On1y the inert: gas is present; then Wo

=

0, and g

=

°

and from Eqs. 4.14 and 4.15 ~T

~T

2 IJ.-1 IJ. i _R a_

₌

₌

,_ -- T 1 1 - ~i

,-

-

1 _{1 lJ.} i 1 - 1

,-1

(b) Pure water vapour: _Wo

₌

1, ~v

=

('v - l)'v· From Eqs. 4.14

(1 _ g) __ R T

2 IJ. v

a =

---~---~--~-e 1 _ 2" .(

'v

-

1 ) + _'_v_-_1 ,,2 IV

'v

1 - g

whieh is the va1ue given by Buh1er (Ref. 24, see a1so Ref. 23).

4.3 The Equilibrium Isentrope in the P, T-plane Inserting Eq. 4.4 into Eq. 4.2 results in

- (

~

-

~v

) (wo

[( * -

~v

) (wo

- g)

+

~-(1

1 -

w )

0 +

~

V

(w

0

-

g)J

dp P

or af ter some sirnp1ifieations

dp = 8(z) dz

P

z

and 4.15

where

w _

g

+

0

2(w -

g) o

ti

z c ~ ,,2Z2 -8( z)

= _ _ _ _ _ _

--:-:-v_c~ _ _ _ _ _ W - g ~.(l-w) ]. 0 + 0

1\

z c

Integrating,Eq. 4.19 y-ields the equilibrium isentrope in the P,T p1ane.

[J

Z 8(z)

~z

J

- P P

= -

= exp Pc 1 Particular Cases: (4.20) (4.21)

(a) Only the inert gas is present·w

=

0, L

=

0, g

=

0.

and 4.20 ' 0

Then from Eqs. 4.19 1i

-

1 1 _~i 1 ":' dp=

-

_{- =} dT 1i _-dT ₌ 1 dT P _~i T _1i

-

1 T _{' i}

-

1 T and (b) Pure vapour:

w

=

1.

o From Eqs. 4.19 and 4.20 we obtain

1 _+~1\2 1 - g

_-

2(1 -

_i\

g)

elf>

= ___

v_-=--_ _ _ _ _

P

y

~

=

(1 1

_g +

~

_~"

-

i )

d"

Us:i,ng Eq. 4.8 and integrating Eq. 4.22

P 1

- =

=...

-z2(1 _ " tnz)

c

which agrees with the value given in Ref. 22.

v

4.4 ;Relation Betwee!]. the Tatal Pressure and Nondimensional Temperature From the equation of state, Eq. 2.5, we have

p

=

~

(1 - g)PT and p

=

~

P T

c I.l. c C

(4.22)

Hence

:P

=

L

= Ilo (1 _ g)

E-Pe Il Pc

. z

(4.24)

where, the value of pjpc is given by Eq.

4

'

.2l.

water vapour we have from Eqs.

4.23

and

4.8

In the particular case of pure

(4.25)

which agrees with the value given in Ref.

22.

4.5

Determination of the Nonstationary Partiele Velocity

We now turn tothe ful1 gasdynamic,:eq(ll.a'tliimS given in Section 2

to derive the expression for the nonstationary partiele velocity u. Along a particle path

a 2

=

~

or a 2 dp -

~

=

0

e dp e dt dt

From Eqs.

2.1

and

2.2

we have

2 dp + 2

ou

a -_{e dt} a P"'Lx=O _{e ox}

du

Op

pa - + a ",,==0 e dt e ax Combining Eqs.

4.26

and

4.27

we get

Adding and subtracting Éqs.

4.28

and

4.29

give

Hence along the lines

the following relations hold

dx

-

=

dt dp ± P a du = 0 e

(4.26)

(4.27)

(4.28)

(4.29)

(4.30)

(4.31)

Equations 4.30 and 4.31 are the characteristic re1ations for the one-dimensiona1 nonstationary flow where equilibrium condensation occurs in the rarefaction wave in a shock tube.

For the Q-centered rarefaction wave (Fig. 3) we have (Ref. 25),

dx_

dt - u - ae dp + pa du

=

0

e

for the entire disturbance

From the second equation of Eq. 4.32 and Eq. 4.26 we::"deri ve

du·= - a -_e dp _p

(4.32)

(1+

.33) Substituting in Eq. 4.33 the values of

4 .19 and integrating, we obtain dP/P and of ae given by Eqs. 4.14 and

u - u

J

Z [

r

I-l J1/2 dz

_~c

=

Ü

= _

~. (1 _

g)

..2

Z

8(z)

-ae re I-l Z

c 1 c

(4.34)

The quantity Ü is a re1ative velocity with respect to the condensation front, which has a constant speed (a1ong one of the characteristics, see Fig. 3). Consequent1y, when U_c = 0 the water ,vapour is initia11y at the saturation temperature and pressure in the driver section of the shock tube and in this

case the characteristic c (Fig. 3) is at the head of the rarefaction wave. If the re1ative humidity of the vapour is initially 1ess than one, the characteristic c may still be regarded as the head of the rarefaction wave where equilibrium condensation occurs and propagates into a gas mixture at temperature Tc, pressure pc,density Pc in which the particles move to the right with a constant velocity uc' ' _{The va1ue of uc in function of the}

temperature Tc is given by the usua1 re1ation for a Q-rarefaction wave in a perfect inert gas (Ref. 25),

(4.35)

4.6 Re1ation Between the Characteristics Slgpes and Nondimensiona1 Te~erature

This re1ation is given by the first Eq. 4.32,

1 dx x -

-N = - .- - - = - = U - a

ae dt ae t . e

c c

(4.36) where the va1ues of Ü and a~ as functions of the nondimensiona1 temperature z are given by Egs. 4.18 and 4.34. We may then estab1ish simi1ar re1ations between N and the nondimensiona1 variables ge,

ä

_{e ,}

iS,

:P,

and Ü.

For a given inert gas/water vapour mixture the solution depends on the nondimensional parameter Àç = Cp T c/L, as is readily seen from Eqs. 4

.7,

4.18, 4.21, 4.24, 4.34 and 4.36. 0

Tt should be noted that the ratio of thr equilibrium speed of sound a e to the frozen speed of sound af

=

(7 0 RT/llo)"2 becomes discontinuous from thg condensation front onward. c

5. Strength of the Tra.nsmitted Shock Wave

Does equilibrium condensation affect the strength of the 'tra.nsmitted shock wave in a shock tube, and if so to what extent? This question is of particular i1I!Porta.nce when operating shock tubes in which condensation of moisture or other gas constituents in the driver might occur.

At the contact surface (see Fig. 3) the pressure a.nd velocity are continuous and therefore,

(5.1) For a forward facing shock wave we have for the velocity u2 (Ref. 25)

(5.2)

where, the subscript 1 refers to the channel conditions of the shock tube and P21 = P2/Pl' is the pressure ratio across the shock wave. The solution for the flow variables in the region 3 was obtained in Section 4.

From Eq. 5.2 we obtain

71(7

1 + 1) U;1

{ r ( 7

1 + 1)

U

21 J2}1/2

P 21 = 1 +

4

+ 7₁ U₂₁ 1 +

L---r-4----..;;=

(5.3)

For given values of 70, To , 71' Tl and for 1 > z > 0 we can compute, using the conditions given in Eq. 5.1,

u u _{a e ae P3} 2 ₌

_-L

c

Ü

3 c P 3c U = - = -21 al _{a e c} al al Pc (5.4)

We may then find P21 from Eq. 5.3 and Pl P 1c P2 ₁ P

₃

1 P3c = _{= - x - - = - x - - = P} 21 Pc Pc _P21 Pc _P21

Thus for 1> z > 0 Eq. 5.5 gives the dependence P_lc = f(P 21).

6.

Discussion of Results

Variations of the equilibrium flow variables with the nondimensional slope of characteristics we re co~uted on the basis of Eqs 0

4.7, 4.18, 4.21,

4.24, 4.34, 4.36

and

5.5

derived in the preceding sections. Two cases were

considered: nitrogen/water vapour mixture with initial conditions in the driver section characterized by the values Wo

=

0.01776

Ca mass ratio of steam to nitrogen of

1.776%),

~o

=

0.973

(a relative humidity of

97.3%),

To

=

295.3°K,

and pure water vapour (steam) with initial conditions given

by Wo =

1,

CPo =

0.857,

_To =

32804

oK.

All quantities are made nondimensional with respect to their equilibrium values at the onset of condensation (Fig.

3).

Flow variables prior to condensation can be computed from the usual expressions derived for inert gases (see, for example, R~f.

25)

with the appropriate values of I for the mixture of gases considered. The results are plotted on Figs.

5

to

20.

I t is worth noting from Fig.

5,

that for a given initial mass ratio

wo' relative humidity CPo and te~erature _{To ' the position of the equilibrium} condensa'tion front c is determined by CPo only. On the other hand, it is Wo

that determines 'the subsequent flow quantities through the amount of heat addi tion. For exa.n;ple, in Fig.

5,

i f Wo were increased for the same T 0 and CPo, then the curve would be raised and the interval (6N) in the approach to the as~tote would enlarge. The asymptote is ideally reached only for a complete wave (Eqso

4.7

and

408)

~nd then g

=

wo' In an isentropic expansion for an incomplete wave only a finite amount of vapour is condensed. However, all quantities are plotted from the condensation front c, which is assumed t o

take place at the head of the rarefaction wave for convenience (i.e., N

=

-1, as everything is _{made nondimensional by aec ' Eq.}

4.36).

Also shown on the figures are the frozen flow curves computed under the assumption that no condensation takes place during the expansion. In this case the variables are made nondimensional with respect to their

frozen values at the onset of condensation, and N is made nondimensional by af , this time.

c

~n examination of all the plots shows that some of the flow variables for the mixture and those for pure vapour follow different trends, due primarily to the amount of water vapour present which governs the heat addition to the flow.

Figure

5

shows that in the case of a mixture the vapour is nearly completely condensed in the characteristics interval -1 :::: N ;S

0.2,

af ter which ge approaches as~totically to wo' as the flow expands to vacuum at N ~

6.

This interval increases with increasing Wo and for Wo

=

1 (pure vapour), the production of the liquid phase is as shown on Fig.

13.

It is seen that, in this case, ge asymptotically tends to unity as the expansion proceeds to a vacuum at N ~

140.

Temperature plots (Figs.

6

and

14)

follow similar trends for the two cases. That is, for a gi ven N the equilibrium curve has the highest temperature owing to the heat addition resulting from condensation, and the frozen curve has the lowest ternperature. However, it is seen that the te~erature difference between the equilibrium and frozen flow states is very large in the case of a pure vapour. For frozen flows the te~erature is

zero at N ~ 5 for the nitrogen/water vapour mixture, and at N

=

6 for the pure vapour, while computations show that the corresponding values of N for equilibrium condensation are respectively N '" 6 and N '" 140. Also shown on the equilibrium curve are the corresponding values of the condensate mass fraction ge'

Variations of equilibrium and frozen sound speeds show different trends in the two cases (Figs. 7 and 15). Il1 the case of pure vapour the variation of a/ac is similar to that of temperature . Tt is almost linear for small values of N; for higher values of N it becOlnes strongly nonlinear and tends asymptotically to zero as the flow expands to vacuum. For nitrogen/ water vapour mixtures the variation of the nondimensional equilibrium speed of sound is shown on Fig. 7. Af ter all the vapour is condensed it decreases linearly to zero with a slope almost equal to that of the frozen flow curve. The equilibrium and frozen values of the particle velocity differ by small amounts at smaller values of N (Figs. 8 and 16). Again, in the mixture case, af ter the entire water vapour is practically condensed the velocity rises linearly with a slope almost identical to that of the frozen curve. However, in the case of pure vapour, the continuous heat addition also continuously increases the kinetic energy so that at greater values of

N, the equilibrium particle velocity becomes increasingly greater than that for the frozen flow. The limiting values for the full expansion are

N '"

6 and N "'140 for the mixture and pure vapour cases, respec ti vely. The Mach numbers (Figs. 9 and 17) which involve both the velocity and temperature , therefore become extremely sensitive to the process of heat addition.

For the two cases considered the frozen and equilibrium curves of pressure and densi'ty variations lie close together (Figs. 10 and 18 and 11 and 19). The pressure curves do not exhibit a cross-over point, while those of density do. The differences in these flow variables do become significant only at large values of

N.

The differences in density are more appreciable in the case of mixture than for pure vapour, while the opposite is true for the pressure . Tt is worthwhile noting that the plots of flow variables for equilibrium condensation obtained are similar in trends to those presented in Ref. 26 for the flowthrough a nons·tationary expansion including effects of equilibrium gas imperfections in air.

Equilibrium condensation of water vapour in a shock tube does not affect appreciably the strength of the shock into air for fini te values of the modified diaphragm pressure ratio Pl c (P41 = _{P4c • Plc ) (Figs. 12 and 20).} However, this effect becomes appreciable when Plc ~ 0 (P41 ~ 00) for pure water vapour condensation. In this case

lim P ~O lc whereas, P 21 _f ~ 100 P 21 ~ 20000 e I

These limits we re determined from the numerical results. For the mixturi, however, the limits are close to that for air/air, i.e., P21 ~44 (Ref. 25). It should be noted that the limiting equilibrium value P 21 is more of

theoretical than perhaps of practical interest, due to the breakdown of assumptions made in the formulation of the problem (equation of state, condensate velocity and temperature , e te ..• ) . Nevertheless, i t does point to the possibility of using a near liquid driver for the production of strong shock waves (Ref. 26).

PART 11. NONEQUILIBRIUM CONDENSATION

7.

Nucleation and Droplet-Growth Equations

In Part I of the present investigation i t. was tacitly assumed that condensation takes place on certain "surfaces" , or "foreign nuclei". Such nuclei are small cen'tres for condensation, and act as catalysts for

'the agglomeration of vapour into liquid or solid form. These may be ions, dust from any source, salts, the surfaces of a container, or condensate particles formed by prior condensation of another vapour. Also, in the equations derived in Part I we have neglected the surface energy of the liquid drops, which is very small compared to the latent heat if the droplets are large. Under 'these conditions condensation takes place ne ar thermodynamic equilibrium at supersaturations close to unity.

However, in a chemically and physically pure gas, i.e., in the absence of foreign condensation nuclei of any kind, centres of condensation must be created in the gas itself. In this situation the condensation is said to take place through homogeneous nucleation and i t may become much delayed with respect to the equilibrium state,;. The theory of the formation of nuclei of the liquid phase in pure supersaturated vapour has been developed by a number of authors (Refs. 1 to 5). A detailed presentation of the theory with references to original papers may be found in Frenkel' s book (Ref.

4).

In this section only the main ideas of the theory will be presented.

Nucleation

Surface phenomena play an important role in processes connected with phase changes. The new phase is generated in ·the form of "nuclei", which are usually considered very small vapour bubbles, liquid droplets , or crystalline bodies . Therefore, their smallness implie s an abnormally large ratio of their surface to 'their volume, compared with the case of usual

macroscopie bodies . Their surface energy or surface tension must accordingly constitute an important part of the total change of energy or the free energy of the whole system involved in the process of the formation of a nucleus. If the surface energy of ·the droplet with respect to the vapour is taken into account, the notion of thermodynamic equilibrium between the two phases can be exte~ded in such a way that the droplet of a given size and shape would remain in equilibrium with the vapour, despite the fact that the vapour is not in equilibrium with respect to a fully developed liquid, separated from it by a plane-boundary surface.

This extension of the notion of equilibrium between two phases has been applied by Thomson (Ref. 27) to the problem of condensation of a supersaturated vapour. He has shown that the pressure of a vapour in equilibrium with a droplet of liquid at a gi ven temperature is larger, t:re

smaller the radius of the droplet . saturated in the usual sense, i.e., with infinitely large radius (plane respect to small droplets .

Therefore, a vapour which is super-with respect to a "drop" of liquid boundary), can be nonsaturated with

The functional relationship between the vapour pressure Pv' and the radius r, of the droplet in equilibrium with the vapour in which i t is suspended may be obtained by considering a spherical drop having the same temperature T as the vapour. The condition of mechani.cal equilibrium between the droplet and 'the vapour is expressed by the equation,

(7.1)

where, Pd is the internal pressure of the spherical droplet , cr the surface tension corresponding to r ~ 00. Thermodynamic equilibrium of the system

under consideration is determined by the condition ,

o

(7.2)

Here, <I> is the thermodynamic potentialof the whole system, consisting of

the vapour and the liquid droplet and is expressed by the relation,

the 'term 47TT2cr being the surface free energy of the drop (provided it is not too small), Nv, Nd the number of particles constituting the vapour and

J,.iquid phases respectively, and CPv' CPd' their "chemical potentials" referred to one particle.

Equation 7.2 combined with the condition Nv + Nd == const, leads to the relation

. 2

+

4w

d(r )

=

0

CPd - CPv dN

d

Denoting the volume of a liquid par'ticle by vd we have vd == 47TT3/3Nd and, therefore,

2cr

cp _{d -} cp _v + - v _{r d}

o

(7.4)

In the limiting case of r ~ 00, this equation is reduced to the ordinary

equilibrium condition CPv == CPd. Differentiating, Eq. 7.4 for a fixed value of T and noting that in this case dCPv == vvelJlv and dCPd == vddpv' Vv being the volume occupied by one molecule in the gas phase, we get

If vd is neglected compared to Vv and if the vapour is treated as a perfect gas for which Vv == kT/pv' this equation can be written in the form

or

r = kT]n(p lp )

v

s

(7.5)

where, Ps is the vapour pressure corresponding to r ~ 00, or the saturation pressure of the vapour phase.

It should be noted that reannot be smaller than the average distanee between neighbouring molecules in the liquid phase. In this limiting case the enibryonic drop consists of a small mmlber (2-8) of

molecules. Tt is clear tha:t the preceding macroscopie theory cannot be applied to such a smaJ.l drop. Le't us estimate the limiting degree of supersaturation, i.e., the maximum value of Pv/Ps corresponding to the smallest possible value of N. Putting N ~

8

and vd :::::::

4

r

3

we get from Eq.

7.5,

Pv

1

:::::::

8crr2

Ps max kT

If we assume cr = 100 erg/cm2 , r = 4 x 10-8cm and T = 300oK, then

,e

n

I

p v

I

=

4

or . Ps max

p

I

p v

I :::::::

100

. smax

For any degree of supersaturation, the equilibrium between the drop of radius r, and the vapour pressure

Pv,

given by Eq.

7.5

is an unstable one. Indeed, if only one molecule of vapour condenses on the surface of the drop the radius of the drop increases, the pressure of the surrounding vapour being unchanged. According to Eq.

7.5

a lesser equilibrium vapour pressure should correspond to the enlarged drop. Hence the vapour pressure is higher than the new equilibrium pressure and condensation of the vapour on the surface of the drop will continue. If, on the other hand, only one molecule

evaporates from the surface of the drop, the radius of the drop decreases, and the new equilibrium pressure will be higher than the actual vapour pressure and evaporation will continue.

The instability of equilibrium between a supersaturated vapour and a liquid drop of radius r, wi th respect to a variation of the lat ter is expressed formally by the fact that the thermodynamic potential ~ of the system vapour-drop has for r = r* not a minimum value, as is the case for ordinary stable equilibrium but a maximum value. In fact, if ~o is the thermodynamic potential in the absence of the drop, according to Eq.

7.3,

then,

i.e. ,

2

= -

(~v

-

~d)Nd +

4rrr

cr