Mixing shocks and their influence on the design of liquid-gas ejectors

MIXING SHOCKS AND THEIR

INFLUENCE ON THE DESIGN

OF LIQUID-GAS EJECTORS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP AAN DE TECHNISCHE HOGESCHOOL TE DELFT OP GEZAG VAN DE RECTOR MAG-NIFICUS IR. H. j. DE. WIJS, HOOGLERAAR IN DE AFDELING DER MIJNBOUWKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP VRIJDAG 21 DECEM-BER 1962, DES NA MIDDAGS TE 2 UUR,

DOOR

JOHANNES HENDRICUS WITTE

werktuigkundig ingenieur

geboren te Breda

. /

&t

3/

Iy

.!.l~

UITGEVERIJ WALTMAN - DELFT.

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR PROF. IR. J. O. HINZE

CONTENTS

Introduction

page 9 CHAPTER I Definitions, assumptions and dimensional analysis

of the flow process in the mixing tube

§ I Definitions . . . Il 13 13 14 IS 15 16 17 § 2 Description of the shape of the flow channel

§ 2.1 On the shape of the nozzle

§ 2.2 On the shape of the flow channel . . § 3 Assumptions on the flow in the flow channel

§ 3.1 On the flow in every cross-section of the flow channel . § 3.2 On the flow in special cross-sections . . . . § 4 Dimensional analysis of the flow process in the mixing tube .

CHAPTER 11 The isotherm al model of the flow in the mixing tube

§ 1 The equations of continuity, momentum and state . . . 21 § 2 The bundie of mixing shock parabolas . . . 24 § 3 Notes on the conditions downstream a stationary mixing shock 25 § 4 Mixing shock relations . . . 28

CHAPTER 111 The adiabatic model of the flow in the mixing tube § I The equation of energy and the temperature change across the jet flow

and the mixing shock . . . 29 § 2 Entropy increase across the jet flow and the mixing shock as a function

of ë and f). . . . . . . . . 32 § 3 The physical impossibility of supersonic flow alter a compression

mixing shock . . . 37 § 4 On the possibility of an expansion mixing shock 39 § 5 The cavitation shock . . . 43

CHAPTER IV Flow of a homogeneous mixture in the diffusor 44

CHAPTER V Experimental equipment and procedure

§ 1 The flow circuit . . . 51 § 2 Measurement of mass-flow rates, pressures and temperatures . . . . 54 § 3 Measurement of the volumetric hold-up factor cl in a horizontal tube

through which a two-phase mixture flow> 58 § 3.1 Principle of the method . . . 58 § 3.2 The calibration and alignment procedure . . . 61 § 3.3 Errors in the calibration diagram and systematic errors of the

page CHAPTER VI Measurements and their results

§ I Relative importance of the various dimensionless numbers governing the gas entrainment process . . . 64 § 2 Relative errors of the dimensionless numbers calculated from the

measurements . . . 64

§ 3 Measurements on the mixing process . . . 66 § 3.1 The inftuence of the surface tension . . . 66 § 3.2 The inftuence ofthe nozzle configuration and the distance to the

entrance of the mixing tube . . . 69 § 3.3 The inftuence of the Reynolds number 72 § 3.4 The inftuence of the area ratio cp . • . 73 § 3.5 Final remarks . . . 75 § 4 Measurements on the pressure increase in the diffusor 77 § 5 The compression ratio across the cavitation shock. . 80 § 6 Measurements of the volumetric hold-up factor in a horizontal tube 80

CHAPTER VII Efficiency and design of a liquid-gas ejector

§ I Theoretical and measured efficiencies 82 § 2 The design propos al . . . 87

Tables of measured data of the diagrams 94

List of subscripts, symbols and quantities 105

References 107

INTRODUCTION

The liquid-gas ejector with the liquid as a driving medium has not received a great deal of attention lately. This is not surprising if we consider, that its effi-ciency is low compared with the efficiency of a typical displacement pump.

On the other hand there are several technical merits:

an unsurpassed simplicity on account of the absence of moving parts; low purchase and maintenance costs;

no sealing problems ;

suitable for the entrainment oflarge gas volumes at low pressure.

The good sealing possibilities suggest applications in the pumping of

poi-sonous and radio-active gases in the chemical industry and in nuclear reactors. Owing to the large contact surface between liquid and gas in the ejector, it seems possible to allow a chemical reaction or a physical transport process to

take place between the liquid and the gas. This opens the possibility to combine the functions of a pump and a reactor. Another field of application is the

evacuation of large spaces, for instance windtunnels.

On a small scale, we see, that waterjet pumps have been used for a long time in the evacuation of receivers etc. The list of investigators who published data

about liquid-gas ejectors or two-phase mixtures being of importance to the present thesis is rather short.

PFLEIDERER [1] points out, that a liquid-gas ejector operates as a displace-ment pump with an infinitely small dead volume. This makes the entrained .

gas volume per unit time independent of the backpressure downstream the ejector.

VON PAWELL RAMMINGEN [2], was the first to observe, th at in the mixing tube of the ejector a rather peculiar flow phenomenon takes place. He

de-sc ribes this phenomenon as a sudden change of jet flow to froth flow

accom-panied by energy dissipation and pressure build-up. It has not been found possible to compare the theory expl<iined in the present thesis with his

experi-mental results. This is caused by the fact th at he does not give a formula describing the contour of his mixing tube of non-uniform cross-section and the reader has to guess, where the mixing region in his mixing tube is situated

with each test run.

FOLSOM [3] gives a formula concerning the pressure increase in a cylindrical

CAMPBELL and PITSCHER [4] apply the laws ofconservation ofmass, momen-turn and energy, to the propagation of plane shock waves in a gas-liquid mixture and derive shock wave relations. Their theoretical discussion shows, that these relations assume a simple form, when the temperature rise across the shock, which is shown to be quite small, is neglected. They illustrate this with good experimental confirmation with the aid of a smallliquid-gas

shock-tube. Their way of coping with the problems involved, although having no direct concern with liquid-gas ejectors has been most inspiring to the writer in deriying his theory.

DIN and PLESSET [5, 6] treat the propagation of sound in a liquid-gas mixture with a homogeneous and isotropic distribution of gas bubbles. The bubbles

are assumed to be small and numerous, so that the mixture may be considered to be a uniform medium. They include the effect ofheat conduction and show,

that for the range of volume ratios of general interest, the accoustic compres-sions and rarefactions are essentially isothermal.

CHAPTER I

DEFINITIONS, ASSUMPTIONS AND DIMENSIONAL ANAL YSIS

OF THE FLOW PROCESS IN THE MIXING TUBE

§ 1 Definitions

FIG. 1 displays a diagrammetic picture of a liquid-gas ejector with the liquid as a driving medium. The liquid spouts through a nozzle a. centrically into the mixing tube. The gas flows through feed pipe d. to the suction chamber b. and is entrained in the mixing zone e. The back pressure p. is adjusted in su eh a way, th at the end of the pressure build-up in the mixing zone coin-cid es with the end ofthe mixing tube. This adjustment has been maintained in all test runs and the theory and the assumptions have been evaluated in accordance with this configuration. It is possible to connect a diffusor to the mixing tube, where part of the kinetic energy of the mixture may be converted into pressure. At the above-mentioned adjustmeht, the flow phenomenon described by VON PAWELL RAMMINGEN takes place. In the present thesis this phenomenon is called mixing shock.

Definition of a mixing shock. A mixing shock is a sudden change of jet flow into froth flow accompanied by a pressure increase and energy dissipation.

Definition of a

jet

flow. By a jet flow is meant a flow consisting of a core of fast moving liquid jets and droplets, surrounded by a mixture of gas and vapour and a liquid film developing on the wall of the mixing tube. In general, a velo city difference or slip will exist between the gaseous and liquid phases. Definition of afrothflow. A froth flow is a gas-vapour-liquid flow in which the mixture of gas and vapour is dispersed in the form of bubbles in the liquid.

The diameters of these bubbles are small compared with the tube diameter; the slip between the bubbles and the surrounding liquid is negligibly small. Definition of the mixing shock thickness. The mixing shock thickness is defined by the distance between the commencement and the end of the pressure in-crease in the mixing zone.

Experiments show, that at constant liquid mass-flow rate, the mixing shock thickness depends on the entrained gas mass-flow rate; an increasing gas mass-flow rate giving an increasing thickness. The order of magnitude of the mixing shock thickness could be measured to lie between 0.3 and 0.03 m in our experiments. Probably the mixing shock thickness also depends on the density

SECTION ° SECTION • SECTION .. SECTION di

~

f-'

Zo MIXINGTUBE _Lmt DIFFU OR d b a

l

GAS

:

1-

l

e p t

~.oooo

o

~ooo~

_LlQUID ... _{_Qn Dmt} 0 ° 0 0 0 0 0 o 0 0 0 oe,d~ o. 0

-

o 0 10 10 010 f 0 0 0 0° 00

l

/

(

t

I

MIXING--.- - -JET FLOW

SHOCK I-FROTH

FLOW- r-P_dim P~=P" ₉ + P _y P

_p[

_{= P~} o °

_p[

_{= Pg+Py} Pl = Pg UO U· 9 9

\-UO U· l l

-U u'l= U·~ Udim Tl . / / ' ATis _ATjf T Tl T~ = T~

I

I

FIG. I. Qualitative change of p, U and T along the mixing tube and the diffusor of a

liquid-gas ejector.

and the viscosity of the liquid and of the gas-vapour mixture. In the present thesis we shall not go into this subject, however. The mixing shock shows some similarity and also some dissimilarity with the plane shock wave in gasdy-namics. The following characteristics are similar :

1. Downstream the mixing shock, the pressure is higher and the velocities are lower than upstream the shock.

2. This effect is accompanied by dissipation of kinetic energy.

homoge-neous with a special equation of state, it may be proved (see CHAPTER 111,

§ 3), that the flow downstream the shock is always subsonic. The most important differences are:

1. The temperature changes across such a shock are very small. The flow process may be treated with good accuracy as being isothermal.

2. Before the mixing shock a velocity difference or slip between the liquid and the gaseous part of the jet flow will generally exist. Thus we cannot speak of a homogeneous mixture, neither can we speak of su personic mixture flow before the shock. Before the mixing shock the gas velocity may be higher thlin the liquid velocity.

3. The mixing shock thickness is much larger than the thickness of a shock wave in gasdynamics.

Definition

of a

"pure" mixing shock. A "pure" mixing shock is a mixing shock with only gas and liquid but no vapour present. This flow phenomenon may be approached, when a driving liquid with a very small saturared vapour pressure is used.

Definition of the cavitation shock. A cavitation shock is a mixing shock with only vapour and liquid but no gas present in thejet flow upstream the shock. Down-stream this shock only liquid is left.

This phenomenon may be created by simply shutting off the gas stream to the ejector. By adjusting the back pressure, this particular kind of mixing shock may be obtained. Before this shock we have ajet flow composed ofa liquid film on the wall of the mixing tube and a core of liquid jets and droplets moving in vapour. As in the case of the mixing shock, bubbles are formed, but these implode spontaneously and the vapour contained in the bubbles condenses. This is caused by the fact, that the pressure behind the shock is higher than the saturated vapour pressure of the liquid.

So af ter this phenomenon only liquid is left. For obvious reasons we called this shock cavitation shock.

§ 2 Description of the shape of the How channel

§ 2.1 On the shape

of

the nozzle

In the experiments nozzles with one or more holes are used. It is necessary to describe how these holes are distributed relative to the nozzle centre. As a starting point we will assume the following. The centres of the nozzle holes are arranged on the angular points and on the sides of several similar regu1ar polygons, situated inside each other. These polygons have a common centre. See FIG. 2. Of each nozzle, the radial pitch between the angu1ar points of these polygons Prn and the pitch of the centres of the holes on the sides P sn

is the same everywhere. The central hole and the mixing tube have a common axis; the axis of the other holes are parallel to the mixing tu be axis. The holes have all the same diameter dn . If the total number of holes is given by N, the

apparent nozzlediameter Dn is defined by:

Dn = dnVN . . . . . (1.1 )

Let D kn be the diameter of the circumscribing circle of the outer polygon.

FIG. 2. Distribution ofnozzle holes with respect to the cross-section ofthe mixing tube.

With Dn according to (1.1), it is always possible to construct the exit surface

of an arbitrary nozzle of regular polygonic form, when we know:

Dn, Dkn , N, n.

§ 2.2 On the shape of the flow channel

See FIG. 1. The central nozzle hole, the suction chamber and the mixing tube have a circular cross-section and the same horizontal axis. A perfectly smooth

surface on the inside of the mixing tube is assumed. The distance between the nozzle exit and the mixing tube entrance is denoted by Zn; the length and

the diameter of the mixing tube by Lmt and Dmt• The assumptions concerning

§ 3 AssulIlptions on the flow in the flow channel § 3.1 On the flow in every cross-section of the flow channel

The flow of the gas-vapour mixture and the liquid upstream and downstream

the mixing shock is assumed to be continuous. This holds good absolutely when the number of droplets or gas bubbles passing a control surface perpendicular to the flow direction per unit time is infinitely large and their dimensions in-finitely small. In practice, this number appears to be so large, th at the entrained

gas mass-flow rate is not subject to measurable fluctuations.

The flow is assumed to be one-dimensional. This assumption means, that differences in the properties of the flow perpendicular to the flow direction are small compared with differences in the direction of the flow.

In a cross-section we assume, that Tl = Tg

=

Tv. This assumption can be made plausible by pointing out, that the contact surface between the gas-vapour mixture and the liquid part of the flow is very large.

The temperature differences in the direction of the flow are very small. In CHAPTER lIl, these differences are calculated for our experiments to be of the order of 0.1 oK. Thus the evaporation heat rand the saturated vapour pressure

p

v

of the driving liquid in the flow channel may be assumed to be constant.

The gas and the vapour are assumed to form a perfect mixture; hence

everywhere in the flow channel" it applies :

Ug = Uv

Wall friction is neglected. The pressure increase across the mIxmg shock

amounted to 3 X 105 _{- 8 X 10}5 _N

f

m

2 _{or 3 - 8 atmospheres in our experiments;} compared to this the friction pressure loss across thejet flow and the mixing shock was found to be small and to amount to 102 _{- 10}3 _N

f

m

2 _{or 0.01- 0.1 atmospheres.} It is further assumed, that the 1iquid is incompressib1e and that the gas and the vapour follow the perfect gas law. The influence of the vapour on the flow phenomena is not so great, that a VAN DER WAALS correction on the equation of state of the vapour is required.

We will neglect the excess pressure caused by the surface tension in the bu b bles in the mixture af ter the shock; the wi thdra wal of surface tension

energy from the total energy of the flow will also be neg1ected.

The influence of the surface tension and of the viscosity of the~ gas, the vapour and the 1iquid on the gas entrainment mechanism in the mixing shock will not be neg1ected. The influence of the gravitation force on the flow pro-cess will be neg1ected. Before the jet flow hits the mixing zone, there will a1ways be some downward flow in an ejector with a horizontal axis, as the greater part of the jet flow consists of jets' and droplets describing ballistic trajectories. Calculation shows, that this vertical displacement amounts to 0.5 mm at a

r

typical nozzle exit velocity of 50 mjsec and a distance between the end of the nozzle and the commencement of the mixing zone of ,....",0.6 m. This

down-ward displacement of 0.5 mm is small compared with the mixing tube dia-meter of 30 mmo

§ 3.2 On the flow in special cross-sections

Cross-section _ 0

See FIG. 1. It is assumed, that dry gàs is supplied and that the gas temperature

is equal to the liquid temperature or Tlo

=

T/. Further we have Plo =

p/.

Other important quantities are:

AIo = An and A/ = Amt-An.

Cross-section - '

At this section the gas is saturated with vapour. This may be verified

experi-mentally. According to DALTON'S law, we may write: PI' = pg' +pv. Gas pres-sure differences across the j et flow are neglected; hence:

pg'

=

pg ° eg' =

e/

and also Agv' = Amt - Az'

Cross-section - "

A homogeneous mixture of liquid with dispersed gas-vapour bubbles is

created. Considering the small dimensions of these bubbles and their associated

high drag coefficients, it is assumed, th at no slip occurs between the bubbles

and the surrounding liquid hence:

Uz"

=

Ug" = Uv"

=

Um" and further Agv"

=

Amt- Az".

The quantities which are ofimportance for the flow in cross-sections _0, - ',

- " are shown in the following tabie.

TABLE 1

Cross-section _ ° Cross-section - ' Cross-section _"

T U p

e

A T_z O U_z O g T_g O U_g O A_z ' g

eo'

=

eg

° v _l

_I

_g

_I

I Tz

"+

_Tg"

~

T_v ' Uv'= Ug' U~If= Ugli = Pv = Pz'-Pr/

e

v

Other quantities, important in the description of the flow process are:

cz' (cv) g' r, Ro, Rv, az, 1]z' 1]0' 1]v' Dkm Dm N, n, Zm Dmt' Lmt·

§ 4 DiDlensional analysis of the flow process in the Dlixing tub,e

In CHAPTER 111, it will be shown, that the influence of the temperature on

the mechanical behaviour of the flow process in the mixing tube is negligibly

smaU. This means, that T may be dropped as an independent parameter and

that the flow process is assumed to be isothermal. In that case the thermal quantities Cz, (cp), r, Rg, Rv may be dropped. The length and diameter of the mixing tube and the quantities describing the nozzle configuration must be taken along in the considerations because they influence the dispersion of the jet flow in the mixing zone, which will influence the gas entrainment

mech-anism in the mixing shock. The following 18 independent quantities are con

-sidered to be of importance in the treatment of the flow process in the mixing tube:

the velocities

the pressures the densities the surface tension the viscosities the lengths

the nozzle configuration

UZO, U/; v

Pg

O

, pv; "

ez, e/, ev; v az;

'Yjz, 'Yjg, 'Yjv;

Zn, Dmt, Lmt ; Dn, Dien, N, n.

Dimensional analysis of th is group leams, th at the 18 quantities can be ar-ranged in the following group of 15 dimensionless numbers based on the quantities Ut,

pg

0, Dmt :

N Dien Lmt Zn Dn UgO ev egO }

,n'Dm/Dm/ Dm/Dmt' Uzo'ez'~'

. . . (1.2)

pv 'YjZ 'Yjv ez U/ 2 ezUzODmt, ezUz02Dmt.

pg 0' 'Yjg' 'Yjz'

---;;;0-'

'Yjz az

However, these dimensionless numbers are not very suitable to work with. Therefore, others obtained by combinations of these dimensionless numbers

wiU be used instead.

Thus the following dimensionless numbers will be introduced. The ratio

gas to liquid volume-flow ra te :

Q/

e

=

Qt .

.

. . . .

. . .

. .

.

.

.

.

. .

. . . . .

.

(1.3) The Euler numbers: ezU*2 a = - - 0 - .

pg

(1.4) eg °U*2 b= - · - .

p/

(1.5) 17

d =

e

VU*2

Pv

where U* is defined by:

M

O

U*

= __

1_ •

elAmt

the superficia11iquid velocity in the mixing tube.

(1.6)

(I. 7)

The area ratio between nozzle and mixing tube section is denoted by: An

cp = - . . . (1.8) Amt

The Reyno1ds number and the Weber number, both based on the apparent

nozzle diameter Dn: and elUlO Dn Re= -'fJl elUl02 Dn We= - - -al (1.9) . . (1.10)

Substitution of (1.3)-(1.10) in 7 dimension1ess numbers of (1.2), resu1ts in:

(1.11 )

Observation of (1.11) 1eams, that group (1.2) mayalso be written as follows: Dkn Lmt Zn 'fJg 'fJv ev

(J, a, b, d, Re, We, cp, N, n, D- ' -D ' D- ' - , - , -mt mt mt 'fJl 'fJl el

(1.12)

With these 15 (independent) dimension1ess numbers, the gas entrainment

mechanism in the mixing tuoe can be described with the unknown function:

( Dkn L

mt Zn 'fJg 'fJv ev)

(J

=

Fl a, b, d, Re, We, cp, N, n, D- ' - , - , - , - ,

-mt Dmt Dmt 'fJl 'fJl el a, b, and d give a ratio between inertia and pressure forces.

(1.13)

Re and We give a ratio between inertia and viscous or surface tension forces respectively.

Dkn Lmt Zn .

cp, N, n, - -, - -, - -, determme the shape of the flow channel.

Dmt Dmt Dmt

'YJg 'YJv ev 1 . . 1 .

- , - , - , on y contmn maten a propertles.

'YJ1 'YJ1 el

The following dimensionless numbers relating to quantities being important in cross-section _ " and quantities in cross-section _ 0 are introduced. The

pressure ratio of the gas across the mixing tube, which according to the as-sumptions made in § 3.2 is equal to the pressure ratio of the gas across the mixing shock:

(1.14 )

.

.

.

em"

Um "

Other Important quantltles are - - and U 0 •

el 1

The theory given in CHAPTER U will show, that using T = constant and the equations of continuity, momentum and state with some neglections, it is possible to derive formulae of the form:

c

=

F2((), a, cp), see (U.45) with (11.31); (1.15 ) (1.16)

em"

-

=

F4((), a, cp), see (U.50). (1.17)

[2l

Thus these ratios may be expressed in terms of the dimensionless numbers of group (1.12).

(1.15) relates the compression ratio c to the gas to liquid volume-flow ra te (); (1.15) will give an idea of the gas compression process in the mixing tube. In

CHAPTER U, (1.15) wiU be written in the form:

() = F5(c, a, cp), see (U.27) .. . . . (1.18)

It will be demonstrated, that (1.18) is the result of a macroscopie approach in which nothing is said about what happens between two chosen control planes;

(1.13) describes the detailed process in the region between these control planes.

In conclusion, the ratio between the mass-flow rates Mg and MLo is given by:

Mg

The absolute humidity in cross-section _ , is denoted by:

X' =

e

v, ; • • • • • . . . . • . . . " (1.20)

eg

eg'

=

eg

O

according to the assumption given in § 3.2 of this chapter.

ft and x' may be written in terms of some of the numbers of group as follows: ft = b

(~)

. . . . and X' =

(e

v) (::) ,el b (1.12), (1.21 ) (1.22)

CHAPTER 11

THE ISOTHERMAL MODEL OF THE FLOW IN THE MIXING TUBE

§ 1 The equations of continuity, Dl.OInentuDl. and state

In this chapter a theory is discussed based on the assumptions in CHAPTER I

§ 3 and on the following additional assumptions :

1 The temperature differenees in the mixing tube may be neglected,

T lo

=

Tl'

=

Tl".

2 The mass-flow rates of the vapour and the gas may be negleeted eompared

with the mass-ftow rate of the liquid.

3 It wiU be shown, that the inftuenee of the saturated-vapour pressure

pv

may be neglected in the momentum equation.

Negleeting the vapour mass-flow rate eompared with the liquid mass-flow rate ean be made plausible when it is eonsidered, that in an arbitrary eross-seetion of the mixing tube we have:

(11.1 )

It is 'known that Ug is of the same order as UI or Ug ,...", UI and Agv = Amt- Al ; this yields in (II.l):

Mv ,...", ev (Amt _

1).

MI el Al

In our experiments with water at room temperature as a driving 1iquid,

ev Amt

- = 2.10-5 _{and 0}

<

- -

-

₁

<

_{4, hence we no te roughly:}

el An

Mv

0

< - <

10-4

MI (11.2)

The range of gas mass-flow rates compared with the 1iquid mass-ftow rates

amounted to:

Mg

0

< - <

3.10-3

MI (11.3)

(11.2) and (11.3) demonstrate, th at the error invo1ved by neg1ecting Mv and Mg eompared with Mz is very small indeed.

The equation of continuity in every cross-section of the mixing tube and with the above-mentioned additional assumptions reads:

MI

=

(lIUIA I

=

constant. . . (I1.4) The equations of state are given by:

(ll = constant of the liquid;

pg = _RgTg

(lg

pv

- = RvTv

(lv

of the gas with Tg

=

Tl

=

constant;

of the vapour with Tv

=

Tl

=

constant

(I1.S) (11.6)

(I1.7) The integral momentum balance applied between cross-sections _ 0 and - ' with the approximations given above and the assumptions of CHAPTER I § 3, see TABLE I, reads:

pgoAmt+(lIUl02An = p/A~t'+PvAmt+(lIUI'2Az' . . . (I1.S) Hence:

p/Amt+(lIUl02An

=

I

+

pvAmt

pg °_Amt+_{(lIUz' 2A I'} pg °_Amt+_(lIUI,2AI' (I1.9)

The same procedure between cross-sections ~o and _ " results in:

(I1.10) or:

p/Amt+(lIUl02An _ I pvAmt

p"A _{g mt (ll}

+

U ,,2A " _{I I} -

+

P _{g mt (ll} /IA

+

U /l2A" . _{I I} . . . . . (I1.II) The last terms in (I1.9) and (I1.II) are always smaller than 10-3 _{in our}

exper-iments and are neglected compared with l. In th at case, we obtain from (I1.9): (lIUl02An = _{(lIUI,2A I'} . . . .

ahd from (I1.II) :

pgOAmt+(lIUl02An = _pg"Amt+_(lIUI,,2A/'

Substitution ofthe continuity equation (I1.4) in (I1.12) yields:

U1o= Uz';. . . . . . .

which in its turn gives substituted in (I1.12) :

An = Az' . . . . .

Substitution of (I1.4) in (I1.13) results in:

(I1.12)

(I1.13)

(I1.14)

(I1.IS)

pgoAmt+MIU1o = pg"Amt+MIUz" . . . . . . . (I1.16)

the influence of the vapour phase is dropped. This appears to be permissible

in those considerations in which only mechanical quantities play a part.

However, since the latent heat content of the vapour phase is substantial, we cannot neglect the vapour influence in calculations containing thermal quan-tities; this will be shown in CHAPTER lIl.

From (I1.16) we derive:

P "

_{_g _ _ 1}

₌

_{_}

M

_{_}

1_ (Ut- UI") .

pg

0

pg

° Amt . . . . (I1.17)

The liquid holdup factor a" in section _ f t is by definition:

A"

a" = _ I . . . . . . . (I1.18)

A mt

and since according to (H.4),

Ml

o

=

MI,

U*

may be written as:

U*

= MI . . . . . . . . . (I1.19)

elAmt

p/' An

With (11.18), (I1.19), s = - 0 and cp = - , (11.17) may be formulated as

follows:

pg

Amt

s- l

=a(2

_

~)

. cp a . . . (11.20)

The equation of state of the mixture af ter the shock, can be derived from: Agv" Ug" eg"

A/'U/'el . . . (I1.21)

Af ter the shock we assumed UI"

=

Ug", see TABLE 1, and with (11.18), (I1.21)

yields:

( M

g

)

(e/')

=

1

,~"

. .

MI

el a . . . (I1.22)

Since T = constant, BOYLE'S gas law holds good:

pg"

eg"

s = -

=

-

. .

. . .

.

.

.

. . . .

.

. . .

.

. .

(I1.23)

p/

e/

With fJ =

(Z-:)

C:l

o

)

substitution of (I1.23) in (11.22) results in:

, , _ ê

a - - - . . . .

s+fJ . . (I1.24)

Between em", el, and a" the relation exists:

" " +(1 ")"

Considering, th at in the experiments 0.7

<

a"

<

1, (!g"

<

0.01 (!l, holds good, we may write with an accuracy better than 0.3%:

"

a" = (!m . . . (II.26)

(!l

(II.24) gives the relation between the reduced mixture density a" according

to (II.26) and the reduced pressure t: and mayalso be seen as an equation of state of the mixture written in dimensionless symbols.

Substitution of (II.24) in (II.20) yields the expression:

. . (II.27) According to the assumptions given in § 3 CHAPTER I, see TABLE 1,

p/

=

pg'

and

(!/

=

(!g'-; because Mg

=

constant, Qg °

=

Qg'. It is further known that

ULo

=

Ut' and An

=

At' which results in Qlo

=

Ql'. The dimensionless symbols

e,

a and rp can be given as:

e

=

~:

:

=

~::

. .

.

.

.

.

.

.

. . . .

.

.

(II.28) (II.29) (!IU*2 (!IU*2 . a= - - = - -

_pg

_o

_{pg' ,}

wüh An Al' rp= - = -' . . . . A_mt A_{mt '} . . . . (lI.30) and we knowalready: pg" pg" t:= =

-pg

o

pg'

(I.14 )

Since (1I.27) only contains

e,

a, rp and t:, it is demonstrated with (lI.28), (II.29),

(II.30), (I. 14 ), that (II.27) can be used in considerations between cross-sections - ' and _ " as weU as between cross-sections _ 0 and -".

§ 2 The bundIe of IDÎxing shock parabolas

In the foUowing section and also in CHAPTER lIl, in many formulae the term

1

- - 1 will be found.

rp

For simplicity we write:

1

1=-

-

1

rp

and (II.27) then reads:

1

() =

-{-e2+(l+af)e} . . . (11.32) a

The slip factor in cross-section _ 0 is defined by:

°

U/

(J

s = Ulo

f

(11.33)

Since according to (11.14) Ut = UI' and because

Q/

=

Qg'

and Agv' =

= Agvo

=

Amt-An; U/

=

Ug'.

Hence:

SO

=

s' . . . (11.34)

It is noted, that when q; and hence

f

is a constant, SO is proportional to (J. When SO

=

1, (J = fand (11.32) has the following solutions:

el = af e2 = 1. . . . . . (11.35)

In a (J-e-diagram (11.32) represents a bundie of parabolas with the Euler number a as parameter and a given value of

f.

The axis of these parabolas have two points in common:

e

=

0, (J = 0 and e

=

I, ()

=

f.

The top of each para bola is given by the coordinates:

1 +af . _ (1 +af)2

e

=

- 2- ' () - ~- . . . (11.36)

The geometrica1 locus of these tops is found by the elimination of parameter a

from the coordinates (11.36) yielding: fe2 0= -2e-1 In the domain: e> 1 0 < e

<

1 . . . (11.37) () > fe

î

°

<

()

<fe

f

..

....

.

.

...

(11.38)

for a certain pOSltIve value of

f

(11.32) gives negative values of the Euler

number a. Since negative Euler numbers are impossible, we cannot enter

domain (11.38) experimentally. The course of the mixing shock parabolas is

illustrated in diagrams land 11, page 36 and 37.

§ 3 Notes on the conditions downstream a stationary mixing shock

If a mixing shock para bola is sketched in a ()-e-diagram for a certain value of the parameters a and 1, two solutions of e for each () value are found. The physical significance of these solutions may be clarified, when the isotherm al

sound velocity in the mixture af ter the shock is calculated. The fact, that the

isothermal instead of the adiabatic sound velocity in such a two-phase mixture

must be used, was rigorously treated by DIN and PLESSET [5, 6] and was

experimentally confirmed by CAMPBELL and PITSCHER [4].

(11.26), being used, the isothermal sound velocity in the mixture is given by:

Cm"

=

l/(oPm::)

=

U*

l

/~ (~)

. . . (11.39)

V

oem T

V

a oa T From (11.24), we derive: ()a" e=- -1 - a " . . . (11.40)

o

e

Af ter evaluating oa" from (11.40) and substitution in (1I.39), we obtain:

C "= U*

1

/~(_

1

)

m Val- a"

Elimination of a" with (1I.24) results in the expression :

Cm" = U*

(e~)

. . . .

. .

...

(11.41) We remark that lim Cm"

=

+

=

8-.+0

This is caused by the fact, that the liquid is assumed to be incompressible giving an infinitely large sound velo city in the liquid alone. For finite

e,

Cm"

is small compared with the actual sound velocities in the liquid and the gas

separately. This may be shown by the following current values:

U*

=

15 m/sec, e

=

6, a

=

3,

e

= 3 yielding Cm" = 45 m/sec.

The mixture velocity af ter the shock is found with:

Um " =

~~,

which with the aid of (11.24) results in:

a

Um "

=

U*

C

:

e

)

. . .

(11.42)

The Mach number defined by the ratio between the local mixture velocity and

the local sound velocity, calculated with (11.41) and (11.42) becomes:

" Um" Vae Mam

=

-

c

, , = - . . . m e (11.43) Wh en (11.32) is solved, we obtain: 1 +af±V(1 +af)2-4ae e1,2 =

2

. . . (11.44)

Subscript 1 indicates the

+

solution and subscript 2 the - solution.

. (1 +af)2

See DIAGRAMS I and II. Smce êl

>

ê2 when 0

<

()

<

,

the

4a

+

solutions are situated on the right hand and the - solutions on the left hand branches of the mixing shock parabolas.

Algebraic rearrangement of (lI,44) yields:

êl.2 =

Va8

{I +af

±

V

'

(1 + af)2 - l}. . . (II.45)

2

Va{) 4a{)

Substitution of (II,43) in (lI.45) results in the Mach numbers corresponding

with êl and ê2: 1 (Mam"h 2 = . . . (lI.46)

.

f

l

1

+af

±

1

/

(1 +af)2 _

I}

2

~

V

4a{)

The () ordinate of the top of every parabola will be called {)top. From (lI.36), it is known that:

(1 + af)2

{)top = . . . (lI.4 7)

4a

Substitution of(II,47) in (lI.45) and (lI.46) gives:

- {)top

-

V

()top êl.2 = Va{)

(V

e

±

8

-

1)

.

. . . (I1.48) and 1 (Ma m"h.2 = . . . (I1.49) 1 / () top

±

1 / () top _ 1

V ()

V ()

The expr~ssions (I1.48) and (11.49) clearly illustrate, that on the tops of the parabolas,

êl.2

=

Va{) and (Mam"h.2

=

l.

When ()

<

()top. two possibilities exist:

l. For the

+

solutions, situated on the right hand branches of the parabolas

as shown above, the following holds good:

êl > Va{), (Mam"h

<

l.

2. For the - solutions situated on the left hand branches of the parabolas, we have:

Hence the right hand branches of the parabolas correspond to subsonic flow

after the mixing shock and the left hand branches of the parabolas correspond

to supersonic flow af ter the mixing shock. lt is further observed th at (II.37),

being the geometrical locus of the tops of the parabolas, is a line of constant

Mach nu mb er Mam" = 1.

Observation of the bundie of mixing shock parabolas in DIAGRAMS 1 and II

leads to two important problems, th at will be discussed in CHAPTER IIl:

1 When by a compression mixing shock is meant a mixing shock satisfying ê

>

1, the question is whether supersonic mixture flow af ter such a shock

is possible; only subsonic flow is found experimentally.

2 When by an expansion mixing shock is meant a mixing shock satisfying

o <

ê

<

1, the question is whether such a shock is physically possible.

§ 4 Mixing shock relations

(II.45) and (11.46) give ê and Mam" as a function of the entry conditions a,

e

andf (or

cp),

From (II.24) and (II.26), we derive the ratio mixture-density to liquid density:

em" ê

. . . (II.50)

el

ê+e

with ê according to (II.45).

Since

eg'

=~,

and since a and bare the Euler numbers given and

eg'

=

er/,

el

a

we obtain for the ratio mixture density to gas density before the shock:

em" a ( ê )

eg'

=

b

ê

+

e . .

.

.

.

. .

. . . .

.

.

. .

.

.

.

.

(II.51) with ê according to (II.45).

Because Um" = U*

(ê~e),

ULo = Ul', ULo =

~*,

the ratio mixture veloc-ity to liquid velocity before the shock reads:

~:'

=

cp(ê~

e)

. .

.

. . . .

.

.

. . .

(II.52)

with ê according to (II.45).

The ratio mixture velocity to gas velocity before the shock may be evaluated with (II.52), Ug '

=

Ugo and

r:;

:

=

e

(1

cp )

.

The result is: l

cp

. . . (II.53) with ê according to (II.45).

CHAPTER III

THE ADIABATIC MODEL OF THE FLOW IN THE MIXING TUBE

§ 1 The equation of energy and the telDperature change across the jet flow and the IDÎxÎng shock

In this chapter, the flow process in the mixing tube win be assumed to be adiabatic. It will be seen, that without heat exchange with the environment, the temperature changes in the jet flow as a result of evaporation and in the mixing shock caused by dissipation of kinetic energy and vapour condensation are very small. With the aid of expressions giving these temperature changes explicitly, we may evaluate the entropy change of the flow process in the mixing tube. Application of the second law of thermodynamics then yields possible and impossible domains in the s-B-diagram. This approach is needed in order to answer the questions raised at the end of CHAPTER II § 2. The theory will be based on the assumptions of CHAPTER I § 3; further assumptions win not be made tin a clear view is obtained of the relative importance of the various terms in the energy equation.

See TABLE I. We win use:

The integral energy balance applied between cross-sections _ 0 and -", with H denoting the total or reservoir enthalpy, then reads:

MloHlo+MgOHgO

=

Mt"Ht"+Mg"Hg" +Mv"Hv" where:

P

o

U

02

P

"

U

"2 H_I o

₌

_-

I

+

_{- -}

I

+

_Cl T_{I ,} O H" _I

₌

_-

I

+

_-

I

_-

+

_Cl T" _{I ,}

el

2

el

2 (lIl. 1 ) U 02 H O - g

+

(cp)"T_IO g -

-

2

-

"

. U "2 H " -g -

-

2

I

-

+ ( ) Cp g T" I , U "2 R" _v

₌

_{- 2}I _-

+

_Cl T"+ _{I r}

The external work done per kg vapour is included in the evaporation heat r. Because M g and Mv are not neglected compared with MI initially, we obtain the following continuity equations:

M lo= MI+Mv = constant, }

Mg = constant

Substitution of (111.2) in the first 1ine of (111.1) yields:

(111.3)

Hence:

Th e a b 1 so ute uml lty h 'd' x " at sectlOn . - ,," lS glven y x b " = -ev f t •

eg

Since Mv"

=

x" Mg

=

flX" Mlo and Mg

=

flMlo, (111.4) may be written as:

(Hlo-Hz" )+fl(H/- Hg" )+flX"(Hz" - Hv")

=

0 . . . (111.5) The differences between the tota1 enthalpies are given by:

po p" U 02 U ,,2

HIO_HI" = I - I

+

I - I

+

CI( _{T l}o-Tz" ) (111.6)

el 2

i j 02 _ U,,2

Hg O-H"= g g

2

I +(C) (Pg TI O-T"I ) • • • • • • • (1117• ) and for Hl" -Hv" we may write:

" I f

(P "

l U,,2 l Til ) (U"2 l " ) p" l

Hl - H_v = - + - - + C I I - - -+cITI +r =--r~-r

el 2 2 el (111.8)

We may justify the last approximation when it is considereq, th at in our

experiments the following ho1ds good:

When

pz"

r = 2.45.106

J

_{jkg, -}

<

₁₀3

J

jkg. el D.. _{T js} = TI" - T lo, . . . (111.9) substitution of (111.6), (111.7) and (111.8) in (111.5) yields the temperature change across the jet flow and the mixing shock, as follows:

!:l.T

js

=

I - I

+

I - I

+

fl g - I - flx"r (111.10)

1

{P

°

p"

U 02 U ,,2

(U.

02 U "2') }

CI+fl(Cp)g el 2 2 .

Ifit is considered, that Cl is ofthe same order as (cp)g, Ulo is ofthe same order as U/ and fl

«

1, we can neg1ect fl(cp)g compared with Cl and

(

U_g 02 _ Uz"2) . UI02_ Uz',2

fl 2 compared wlth 2

flX" r cannot be neg1ected because the evaporation heat r is very large. Hence:

1

(P

°

P"

U 02 U ,,2 )

D..Tj8

=

_

I - I

+

I ~ I _ flx"r . . . (111.11)

WithPz°

=

pg

o

andpl"

=

pg"

+

p

v

,

(lIl. 1 1) may be written as:

~T

iS

=

~(P/-Pg"

_

p

v

+

UI

02

-UI_{,,2 _ flX"r)}

Cl el el 2 . . . (III.12)

We introduce the dimensionless temperature change:

~Tis . . 0 U* U* Vjs

=

- -

and wlth UI

=

-

,

U!,'

=

-,_

,

,

n

°

~

a

p

/

~

Pg"

=

U*2

C

a

ë),

x"

e

v

eg"

. . . . (III.I3) and ft

=

(~]

8,

the equation (III.I2) becomes:

U*2 [2(I-ë) 1 1

2

_P

_v

}

ev

r

(8)

Vjs

=

2Cl TlO

1

a

+

~2

- a"2 - e1_U*2 - elCI

T;

ö

~

(lIl. 14) From (I1.20), it is known, that the combination of the continuity and the momentum equations applied between cross-sections _ 0 and - " and written

in dimensionless symbols results in:

ë-

I

=

aG

-

a~')

. .

.

.

. .

. . .

. . .

_(I1.20) Substitution of ë-l a ë 1 - - ---,;, a" = - ,

f= -

-

1 . . . (III.I5) ~ a ë+8 ~

in (lIl. 14 ) yields the following expression of Vis as a function of ë and 8: U*2 {(

2

Pv )

(8)21

ev

r

(8)

Vis

=

- - 0 f2 - - -2 - -

f

-

- - 0 - • • • • • (lIl. 16)

2c l T I elU

*

ë elCITI ë

In our experiments the quantity -2U

P:

₂ was very small compared withf2. The el

minimum value off2 amounted to 2.25 and the maximum value of the term with

p

v

amounted to 0.026 with

p

v =

2650 N/m2, U*

=

14 m/sec and el

=

1000 kg/m3; th us the term containing

p

v

is neglected.

Expression (I1l.I6) may then also be written as follows: Vis = Vdiss

+

Vevap , where

) (HI.l7)

U*2

{(8)2}

evr

(8)

Vdiss = - - 0

1

2 - - and Vevap

=

-

- - - 0

The first term Vdiss describes the temperature increase as a result of dissipation

of kinetic energy in the shock. The last term Vevap describes a decrease in tem-perature. This is caused by the fact, that not all the vapour that is created in the jet flow condens es in the shock; part of it is taken away in the gas-vapour bubbles leaving the shock.

The temperature decrease in the jet flow between cross-sections _ 0 and

- ' may be given as:

Vj[

= -

(~)

8 . . . . . . . (lIl. 18)

(!lclTI

The contribution to the temperature increase between cross-sections - ' and

_ " as a result of condensation Qf the vapour in the shock is given by:

lIcond

=

-Vj[

+

lI~vap

=

~

(8-1)

(~)

(!lclTI 8

(IIl.19)

ln the case of the "pure" mixing shock without vapour influence, we obtain: U*2 {

1

8)2}

Vsp

=

Vdiss

=

- - - 0

f2

-

(

-

and Vj[

=

Vcond

=

Vevap

=

0 (Ill.20)

2clTI 8

In order to give a view ofthe temperature changes given by (Ill.17), (IIl.18) and (lIl. 19) , a numerical examp1e is given valid for a water-air-vapour flow at room tempera tu re with the following q uantities: (!l = 1000 kg/m3, Cl = 4200 J/kg, Tlo

=

295 oK, (!v

=

0.020 kg/m3, r

=

2.45 X 106 J/kg, U*

=

15 m/sec.

TABLE 2

q; = 0.25 0= 3 e= 5

diss

_I

eond evap js jf

v. lOs 78.8 9.5 - 2.37 76.5 - 11.8

LlToK 0.232 0.028 - 0.007 0.225 - 0.035

q; = 0.25 0= 3 e = 50

diss eond evap js jf

v·lO" 82.0 11.6 - 0.24 81.8 - 11.8

LlToK 0.242 0.034 - 0.001 0.241 - 0.035

§ 2 Entropy increase across the jet flow and the ntixing shock as a

function of 8 and 8

In the preceding section, the equations of continuity, momenturn, energy and state, have been used for an adiabatic process, the energy equation being in fact the first law of thermodynamics. We now want to make use of the

second law of thermodynamics, stating, that the entropy of an adiabatic process always increases or at least remains constant. In symbols, this law is given by: 6.S ;;;. 0 . . . . . (IlI.21) Expressions describing the entropy increase across the jet flow and the mixing shock 6.Sjs and across a "pure" mixing shock 6.Ssp will now be derived. The entropy increase per kg mixture is given by:

AS 6.Slv+ ft6.Sg h

Ll = W ere

1+ ft

6.Slv = the entropy change of 1 kg liquid

+

saturated vapour

6.Sg

=

the entropy change of 1 kg gas

} . . (III.22)

The entropy increase across the jet flow and the mixing shock of 1 kg liquid

+

saturated vapour, considering th at :

6. Tjs V'S

=

- -

""'

1 0-3 _{- 1 0-}4 J T IO , is given by: T IO+6.Tjs

(6.SIV);s

=

Cl In Tlo C::'. CIVjs . . . . . . (IlI.23)

The entropy increase across the jet flow and the mixing shock of 1 kg gas is found to be:

TIO

+

6.

Tjs pg"

(6.Sg);s = (cp)g In T O - Rg In - 0 C::'. (cp)gvjs- Rg In e (lIl.24)

I

pg

Substitution of(lIl.23) and (lIl.24) in (IlI.22) results in:

(

CI+ft

(~

P

)g

)

ftRg In e

6.SjS = 1 + ft Vjs - 1 +ft

orsinceft(cp)g«cl and ft « 1:

. . . (lIl.25)

Similarly in the case of the "pure" mixing shock:

6.Ssp = Clvsp-ftRg In e . . . . . (IlI.26)

6.SjS and 6.Ssp will be written as functions of e and (J. The dimensionless tem-peratures Vjs and vsp are given by (lIl.17) and (lIl.20) respectively. The quantity

f

is a constant.

Acccrding to (1.21):

. . . . . . (1.21)

From (Il.32), we derive: 8(8- 1)

. . . (IlI.27)

a=

-f8-(}

Substitution of (IIl.27) in (1.21) results in:

ft = b

(~)

(1

8- (}) . • . . . .

8 8-1 . . . (IIl.28)

Substitution of (IIl.28) and (IlI.17) in (IlI.25) and a1so of (IIl.28) and

(IlI.20) in (IlI.26), af ter some rearrangement yields the expression of /::"Sjs and

/::,.Ssp in terms of () and 8: /::"SjS = bRg("P(}2_~js(}+1/2f2) where ) In 8 1

f

In 8 v (!vr "P = 8(8-1) - 282'

~jS

= 8-1

+~,

v = (!lU*2 . . . (IlI.29) and fIn 8

1

/::"SSP = bRg("P(}2_~sp(}+1/2f2) where ~SP = -8-1 . . . (IlI.30) and"P as in (IlI.29).

It is noted, th at "P, ~jS and ~sP are functions of 8 only;

f

and vare constants.

If /::"SjS = 0, /::"Ssp = 0 b =J::. 0, two quadratic equations in () are obtained:

"P(}2_~js(}+1/2f2 = 0, (IlI.31)

"P(}2_~sp(}+1/2f2 = O. . . . . . (IIl.32)

with "P, ~jS and Çsp according to (IIl.29) and (IlI.30).

The discriminants of these equations may be given in the form:

Dijs =

f2

Cn

~

-

~r

+

v

Gr

+

2ift

(

~n~lJ

. (IIl.33)

(In 8 1)2

Disp =

f2

- -

-

-

..

. . . .

. . .

.

. .

8-1 8 (IIl.34)

We remark, that in the region of interest 8> 0, Dijs > 0 and Disp > O. The solutions of (IlI.32) may be written without square root; they are:

((}SP)1

=

f

8, the geometricallocus of the + solutions (IIl.35 )

and

f8(8-1) .

((}SP)2

=

,

the same of the - solutlOns . . . (IIl.36) 28 In 8-8+1

Because in a (j-e-diagram in a certain interval ca

<

e < eb, holds good

Dijs

>

0, Disp

>

0, (III.31) and (III.32) each yield a pair of real lines cor-responding with the geometrical locus of the

+

and - solutions of these equations in the given interval. In the case of (III.32), these curves are given by (III.35) and (III.36); they border on regions conflicting and not con-flicting with the second law. The answer to the question which side of such a line is physically possible can be given. Evaluation of (III.29) and (III.30)

~ IMPOSSIBLE DOMAlN FIG. 3a and 3b. e ~>o e ~<o (ejS~ (ejs), (ejs)2

(%

'///"-:;

-

.,'?'?/

I---r/'

FIG. 4a and 4b. L ~" ~ éb ~>o ~<o

learns, that these functions in a ~S-{j-diagram represent two parabolas for a

given value of e and constant values of bRg,

f

and v. These parabolas have a

common second derivative 2bRg1p. As bRg is a known constant, 1p fixes the sign of this derivative on which it depends whether these curves have a maximum or a minimum value. With this knowledge and the requirement ~ S ;;:;. 0, the following domains in ~S-{j-diagram for a given e, are defined, see FIG. 3a and 3b and TABLE 3.

TABLE 3

() < (()is). (()is). < () < (()iS) 1 () > (()is)l

() < (()sp). (()sv). < () < (()sv) 1 () > (()SP)l

1p> O possible impossible possible mln

1p < O impossible possible impossible max

The same regions can be discerned in a (j-e-diagram, when pairs of lines are sketched according to the geometrical locus of the solutions of (III.31) and

0 < s

<

0.287

s>

0.287 See FIGS 4a and 4b.

VJ

<

0,

VJ > O with lim VJ =

1

/

2

e--+ I

1

J

. . .

. .

(IIL37) We will now define the (0}S)1.2 and the (OSP)1.2 isentropes. These lines re p-resent the geometrical locus of the

+

or - solutions of the corresponding !:l.S = 0 equation in a O-s-diagram. The

+

solutions are indicated by sub-script 1, the - solutions by subscript 2.

The (0}S)1 and (e}S)2 isentropes are solved from (IIL3l), the (eSp) 1 and (eSP)2

isentropes are given by (IIL35) and (IIL36). In order to give the reader a view of the path of the curves defined up till now, the DIAGRAMS land II have

been composed to be characteristic for air-water experiments at room te m-perature.

DIAGRAM 1.

9=~ _2(.-1

~.1

DIAGRAMS land II are based on the following data:

~j~ WA aIo I . 2.11 lY. 0.202 M_z

=

11.1 kg/sec, Amt

=

7.05 X 10-4 m2, [Iv

=

0.020 kg/m3, ez

=

1000 kg/m3, cp = 0.322, RgTzO = 0.79x 105

1/

kg.

With these data we compute:

f

= 2.11, U*

=

15.6 m/sec, v

=

0.202.

The path of the (ejs)!' (e}S)2' (OSP)1' (OSP)2 isentropes has been computed with the aid ofthe LB.M. 650 computer ofthe Mathematical Centre at Amsterdam. The diagrams also display, the Mam"

=

11ine (II.37), the lines s

=

1,

e

=fs and the mixing shock parabolas for:

a = 0.10,

=

1

/

j

= 0.472, 2.00, 3.00, 3.85, 4.70, 5.80, 7.51, 9.90, 13.4, 17.1, 20.8, 23,5.

A theory will be introduced now concerning the possibility of supersonic flow

af ter a compression mixing shock (§ 3). The possibility of an expansion

mixing shock will als 0 be discussed (§ 4). We repeat, that by a compression

mixing shock is meant, a shock satisfying e

>

1, with an expansion mixing shock, a shock satisfying 0

<

e

<

1.

POSSIBLE ,0.472

DIAGRAM Il.

DOMAlN

~ IISjs(O ~"" Q(O

f.2.11

\Y,0.202

§ 3 The physical Ïlnpossibility of supersonic flow af ter a

compression mixing shock

See DIAGRAM 1. We have shown, see (11.38), that since the Euler number a

cannot be negative, for e

>

1 the forbidden domain is given by:

e>

1, ()

>

je

or ()

>

(esp)!, . . . (II.38)

In order to see, whether a supersonic solution is not conflicting with the second

law of thermodynamics, the regions bounded by the (())tS1.2 isentropes and by

the (()SP)1.2 isentropes must be superimposed on the bundie of mixing shock parabolas given by (II.32) and also on the forbidden domain (II.38). We then

have to check, whether the left hand branches of these parabolas cross possible regions. As is shown in CHAPTER 11 § 3, these left hand branches correspond

with supersonic flow af ter the mixing shock. When (111.37) is considered, it is

seen that when e

>

0.287, '!jJ

>

O. In th at case, the regions situated between the (OjS)1.2 isentropes or the (OSP)1.2 isentropes, are physically impossible. This is illustrated in TABLE 3.

In the case of the "pure" compression mixing shock, the impossible region is given according to (111.35), (111.36) and TABLE 3:

e> 1,

1

f e(e- l)

(OSP)2

<

0

<

(OSP)l or 2e In e-e+ 1

<

0 < f e

. . . . (111.38)

It is observed, th at (OSP)1

>

(OSP)2 in this region. From (11.38), we see, that in this case the a

<

0 domain bounds entirely on the f1Ssp

<

0 domain given

in (111.38). Thus in the case of the "pure" compression mixing shock, the only

domain which may be entered experimentally is given by:

. . . . (111. 39)

It can be shown further, that for all values of e

>

1, the (OSP)2 isentrope is

situated under the Mam" = Iline given by (11.37), or when: fe(e- l) fe2

e> 1,

<

---

. .

. . . .

..

..

..

(111.40)

2eln e-e+ 1 2e-1

This implies, th at the (Osp) 2 isentrope crosses only the right hand branches of

the parabolas yielding subsonic solutions. Hence in the only possible domain

(111.39) no supersonic solutions may be found.

In the case ofthe actualflow process in the mixing tube, it is concluded from (111.29)

and (111.30), that since e

>

0 and v

>

0:

~jS

>

~sP; . . . . . . . . (111.41) this being the case and because '!jJ

>

0 when e

>

1, we have:

~js+ V ~js2-2f2'!jJ ~8P+ V~sp2-2f2'!jJ

---=----

>

or (OjS)1> (OSP)1' . . (111.42) 2'!jJ 2'!jJ ~jS-V~js2-2f2'!jJ ~SP-V~8p2_2f2'!jJ

- -

- - - <

or (OjS)2

<

(OSP)2 . . (111.43) 2'!jJ 2'!jJ

The last inequality may be easily proved by assuming .~jS

=

~sp+f1~;

It is seen from (III.42) and (III.43), that the ((jjS)l isentrope is situated

above the ((jSP)l isentrope; the ((jSP)2 isentrope runs under the ((jjS)2 isentrope.

Hence, the impossible

6.s

js < 0 domain ((jjS)l < (j

<

((jjS)2 overlaps the e-qually impossible a

<

0 domain given in (II.38) and the only region which may be entered experimentally is found with

. . . (III.44)

Because it is shown by (III.43), that the ((jjS) 2 isentrope runs under the ((jSP)2 isentrope, and the ((j8P)2 isentrope runs in its turn below the Mam"

=

lline, which is demonstrated in (III.40), there are no supersonic solutions to be found in region (III.44). So it is expected that we will not observe supersonic mixture flow af ter a compression mixing shock.

§ 4 On the possibility of an expansion mixing shock

See DIAGRAM II. Here for 0 < 8

<

1 the forbidden domain as given by (II.38)

results in:

0 < 8

<

1, 0 < (j <f8 or 0

<

(j < ((jS'IJ)l. . . (II.38)

In order to investigate whether a flow process in the mixing tube with an

expansion mixing shock is not conflicting with the second law, the regions

bounded by the ((jjs)l,2 and ((jSP)1.2 isentropes must be superimposed on the forbidden domain (II.38).

In the case of the "pure" expansion mixing shock, in the interval 0.287 < 8

<

1 we have '!fJ

>

0 and according to (III.35), (III.36) and table 3, for a certain value of 8 situated within the interval, the impossible region is given by:

0.287

<

c

<

1, )

f8(8-1) . . . . (III.45)

((;lsP)l

<

(j

<

((jSP)2 or

J8

<

(j

<

-2dn8- 8+l

It is observed, that ((jSP)l < ((jSp) 2 in this interval.

From (II.38) it is evident, that in this case, the impossible a < 0 domain borders entirely on the impossible t1Ssp

<

0 domain given in (III.45). Thus the only region that should be attainable experimentally is situated above the

(8SP )2 isentrope or:

0.287

<

c

<

1,