Droplet size determination in evaporator tubes

Citation for published version (APA):Geld, van der, C. W. M. (1986). Droplet size determination in evaporator tubes. (Report WOP-WET; Vol.86.004). Eindhoven: Technische Universiteit Eindhoven.

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DROPLET SIZE DETERMINATION IN EVAPORATOR TUBES

C.W.M. van der Geld

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8606191 . -"-----------1

T,H.E!!\·JOHOVEN

Report WOP-WET 86.004

Eindhoven University of Technology

May 1986

... - '-.... - ...... --., .... ~ .... ee' __

TABLE OF CONTENTS

ABSTRACT

NOMENCLATURE

List of symbols

Acronyms

Subscripts

LIST OF FIGURES

1 Introduction and scope

1 .1 Some history of dry-out investigations at EUT

1 .2 Some history of droplet size detection

1.3 On droplet impingement studies

2 Thermo void probe measuring strategy

2.1 Determination of droplet size

2.1.1 First estimates of droplet size

2.1.1.1 First estimates; r th ,1

2.1.1.2 Correction parameters; r th ,2

2.1.2 Theoretical analysis of heat transfer during evaporation

2.1.2.1 Newton's cooling of a cylinder; 1 D case

2.1.2.2 Instantaneous spot cooling of a cylinder;

rotatoric symmetry

2.1.2.3 Uniform cooling; radial temperature drop

2.1.2.4 Non-uniform, bounded heat flows

2.1.2.5 Non-uniform, non-local cooling and reheating

2.1.2.6 Some typical evaporation curves

2.1.3 Measuring stategy and computation method

2.1.4 Experimental verifications

2.1.4.1 Thermo void probe measuring device

2.1.4.2 Experimental set ups

2.1.4.3 Large diameter thermocouple measurements

2.1.4.4 TVP verification measurements

2.1.4.5 On void fraction estimation

2.1.4.6 Varying measurement conditions

2.2 On droplet velocity measurements

2.2.1 Experimental calibration

2.2.2 A simple friction loss correction model

3 A computation model for estimation of droplet size at dry-out

3.1 Modeling assumptions and semiempirical equations

3.2 Additional governing equations

3.3 Solution procedure with measured wall temperatures

3.4 Discussion of results

4 On droplet impingement

4.1 Experimental results

4.2 Governing equations

4.3 Solution procedure

4.3.1 Collocation method

4.3.2 Dynamic contact angle algorithm

4.4 On the results

5 Suggestions for further work

5.1 Some design and conditioning improvements

5.2 Droplet velocity measurements

5.3 Other applications

5.4 Theoretical droplet impingement studies

6 Conclusions

REFERENCES

ACKNOWLEDGEMENTS

6

ABSTRACT

DROPLET SIZE DETERMINATION IN EVAPORATOR TUBES

Two ways of measuring droplet size are presented :

- a detection device, the "thermo void probe", to measure the amount of

cooling caused by evaporation of a droplet on a thermocouple at several

moments of time in order to allow for comparison with theoretically

calculated values;

- a rather indirect, computational method that requires the measurement of

system parameters and wall temperature after dry out has occurred in a

vertical test section.

Results are compared with those obtained with conventional methods.

Droplet velocities were measured with a thermo void probe with time-of­

flight method.

The dynamic spreading of a droplet on a surface was numerically studied with

a collocation method.

7

NOMENCLATURE

List of symbols

Roman letters

a

B

c

thermal diffusivity (m 2 /s) 2aa(1 + I( dk/2) ) I dkA

temperature drop at 0,4 t (see figure 12) x

temperature drop at 0,3 t (see figure 12) x

temperature drop at 1,25 t (see figure 12) x

constant proportional to the augmentation rate of a heat current

(equation 2.26)

Cd friction coefficient of droplets (equation 3.5)

Ck thermocouple constant defined by equation 2.3.b

C heat capacity (J!kg.K) p C heat capacity of vapour (J!kg.K) pv d droplet diameter (m)

dk = 2Rk o Effective diameter of rod representing a thermocouple (m)

d upper bound of droplet diameter according to computations max (section 3.3)

dv droplet velocity in axial direction relative to vapour phase (m/s) d diameter of thermocouple wire (m)

o tube diameter (m)

Df = 2Rf o ~iameter of evaporating liquid film

E magnitude of electric potential (V)

G total mass flux (kg/m2s)

Gd parameter that accounts for temperature dependance of (H P )

Gm parameter that accounts for temperature dependance of ( A PCp)

Gk parameter that accounts for temperature dependance of (Rk )

G mass flux of vapour (kg/m 2s) v H specific heat of evaporation (J!kg)

I R ~2~2 I ~ ~. Relative importance of second to first

derivative of temperature (-)

I(dk!2) approximately equals 1

1 mean radius of thermocouple welding

b f (m-3) n num er 0 droplets per unit of volume v p pressure (Pa)

9

r

r th ,1

r th ,2 r w Rf

Rk Sea)

t

t x

T a T avgk T 'inlet T

o

Tsat T 'tc T

v T

tLJ

heat flux towards hot junction of a thermocouple (W/m2)

radiative heat flux between droplets and tube wall (W/mZ)

total heat flux to droplets (W/m2 )

convective heat flux between vapour and droplets (W/m2)

heat flux between tube wall and impinging droplets (W/m2)

amount of heat (J)

= r th ,1 • First estimate of droplet radius (m) (Figure 14)

radial coordinate (m)

first estimate of droplet radius {m} (section 2.1.1.1)

second estimate of droplet radius (m) (section 2.1.1.2)

= r th ,2 = r / 4> w ( m ) = Df /2. Radius of evaporating liquid film (m)

= dk/2. Effective radius of rod representing a thermocouple

measurement error of quantity a

time (s)

time measured between droplet impingement and maximum temperature

drop (s) (see figure 12)

ambient or environmental temperature (oc) measured mean temperature level (oe) temperature at the inlet of a test section (oe) initial temperature of thermocouple (oe) saturation temperature (oc) temperature of thermocouple (oc) vapour temperature (oC)

inner wall temperature of a tube (oC)

mean vapour velocity (m/s) (section 2.1.3)

droplet velocity (m/s)

liquid velocity (m/s)

x = x • Distance of centre of evaporating liquid film to thermocou-o

pIe welding

x = G / G. Actual steam quality (-) a v

x equilibrium steam quality (-) (equation 3.11) e x = x. Distance of centre of evaporating liquid film to thermouple o

welding (m)

z axial coordinate (m)

10

Greek letters

a convective heat transfer coefficient (W/m2K)

awv

convective heat transfer coefficient between wall and vapour

B 6

£ 1 £ v e e tx

JJ v

P d

P 1

p v 0"

T

(W/m2K) (equation 3.1)

= Of / r = 2Rf / r w w (-) Dirac's delta function

void fraction (-)

absorption coefficient of liquid (equation 3.3.b)

absorption coefficient of vapour (equation 3.3.c)

T - T • Temperature relative to environment (ae) a maximum temperature drop measured during evaporation of a

droplet on a thermocouple

heat conductivity (W/mK)

dynamic viscosity of vapour (kg/m.s)

mass density of droplet (kg/m 3)

mass density of liquid (kg/m 3 )

mass density of vapour (kg/m 3 )

surface tension (N/m) (section 1.3)

time of evaporation of a droplet

normalized temperature (equation 2.23)

normalized temperature for non-local heat extraction

(section 2.1.2.5)

~ " normalized temperature for non-uniform heat currents

(section 2.1.2.4)

~ "' normalized temperature for non-uniform, non-local heat extraction

(section 2.1.2.5)

~ 00 correction parameter for subcooling and superheating

(section 2.1.1.2)

~ 01 correction parameter for spheroidal effect (section 2.1.1.2)

~ 10 correction parameter (sections 2.1.1.2 and 2.1.2)

w = x / I 4a t x

11

Acronyms

Bi Biot number (equation 2.7.a)

D1F measure of differences between vapour temperature gradients (eq. 3.19)

EUT Eindhoven University of Technology

Fo Fourier number (equation 2.7.b)

LH5 left hand side

MK5 international system of standard units

Nu Nusselt number (section 2.1.2.6)

O(a)

Pr

Pr v Re

Red Re

v RH5

order of magnitude of a

Prandtl number (section 2.1.2.6)

- U c I A • Vapor Prandtl number - v pv v Reynolds number (section 2.1.2.6)

= p d (v - vl ) I u • Droplet Reynolds number v v v = G x Diu • Vapor Reynolds number

a v right hand side

51 international system of standard units

TC thermocouple

TVP thermo void probe

We Weber number (sections 1.3 and 3.4)

12

Subscripts

S,r = r. Partial derivative of s.

This notation is only used if total as well as partial derivatives

are used in a chapter.

Other subscripts : see list of symbols.

13

LIST OF FIGURES

1 3-D Heat transfer topography

2 Burn out fluxes at natural circulation (1966)

3 Burn out heat flux versus inlet subcooling (1970)

Influence of pressure and surface roughness

4 Burn out flux and pressure drop versus subcooling (1970)

5 Burn out quality versus mass flux (1976)

6 Dry-out wal temperatures versus steam quality in the presence

of a cooling spot (1977)

7 Droplet impingement and evaporation history on a capillary tube

8 Spreading factor versus impact energy

Influence of static contact angle

9 Normalized temperature curves; influence of B

10 Normalized temperature curves; influence of c

11 Normalized temperature curves; influence of Xo and Of

12 Cooling curve schematics and measuring parameters

13 Estimation of evaporation location from measured parameters

14 Flow chart for the calculation of droplet size

15 Thermo void probe measuring device (collage)

16 Thermo void probe electric conditioner

17 Specimen of temperature history; 10 mm thermocouple

18 Specimen of temperature history; 0,1 mm thermocouple

19 Specimen of temperature history; plateau reheating

20 Droplet velocity measurement with time-of-flight method

21 Droplet hitting a thermocouple in downflow

22 Flow chart of calculation procedure of mean droplet diameter at dry-out

23 Computational results at point of dry-out for various start conditions

24 Computational results; steam temperature and quality

25 Computational results; heat fluxes at dry-out

26 Adiabatic droplet impingements on a stainless steel bar

27 Droplet impingement and evaporation history on a thermocouple

28 Coordinate system, collocation angles (N=9) and dynamic contact angle

29 Dynamic contact angle versus interfacial velocity parameter

30 Schematics of signal conditioning with a compensation

31 Schematic of a solid fuel combustion chamber

32 Thermocouple readings at various locations along a fuel grain

15

1 INTRODUCTION AND SCOPE

It is common experience, that accurate measurements of droplet sizes and

velocities in superheated steam are difficult to achieve (see, for example,

Nijhawan et al., 1980, Azzopardi, 1979 and Oelhaye and Cognet, 1984).

Optical methods, favorable since they do not affect the flowfield of study

(see Hirleman, 1983, Drallmeier and Peters, 1986, Jones, 1977 and McGreath

and Beer, 1976) are often difficult to apply at elevated pressures (see Van

der Geld, 1985). Two measuring strategies for the determination of droplet

sizes in superheated steam were developed and studied. nne method is

t X ::J rl <0-..., ro OJ

:t:

oNB departure from nucleate boiling I ,

I , , I I

, I I I

I , 1 ,

, 1 critical heat flux

I~ ,,",$o, / ...... , ',::" I I , /-0°, ,

\ , I , ~ I ~ I ,

~/ \~~~ \ /;..", " O),l

/~~~ \ ~!oo. ... :~/ '~\';:[:': ~Qj ,/ / ~ljl \ ~ \ ,'. "~:: 6' I I lo~ \ '" \ I .::? 0 I l;-

I /I~\~ \ , ::.;. O)Qj ,/~ I /' \ \ / . (1;<" l,f

I I \ I .4< I ,.;. I \ I 0'" I....,

"

\ , I 'h~ ~ \ I 1 ~

I ;~, \ I <' I I \ '_.v " ,f I I \ \ ,,(Ji

I I \ \ I '" I I \ \ I I I \ \ I , I \ \ ' I I 0.\ I

I \'\ \ , , I ~ \ , ,

, ~ \ \ " I ('I' \ \ ,

, \ 1 I St \ \ I &~ \\ I

9v<l.l' ,I " J.ty ,\ I

"'" " 1

Heat transfer topography (after a drawing by G.L. Shires)

3-D Heat transfer topography

17

electro-mechanical and intrusive, and the other method is based on a semi­

empirical physical model.

1.1 Some history of critical heat flux investigations at EUT

At the European Two Phase Flow Group Meeting 1981 in Eindhoven, G.l. Shires

presented heat transfer topography in a three dimensional schematical

drawing. Figure 1 is based on his drawing.

It clearly shows that critical heat flux may occur at low steam qualities,

when it is called departure from nucleate boiling (DN8), but equally well at

high steam qualities, when it is usually called dry out.

The research presented in this paper finds application in experimental and

theoretical studies of dry out and the transition from annular flow to

dispersed droplet flow, called point of dry out.

Natural circulation

t T = 200 DC sat

........ 160 ft

E u "- Burn out ~ x ::l

.--! 140 '" <0-..., ...

" ('() ... , , !Il ...... ", :r

....... " 120 1r........ -,' ---

Instability threshold

100

o 10 20

Inlet subcooling (OC) ~

Figure 2

Burn out fluxes at natural circulation (1966)

During the last two decades, critical heat flux research at the Eindhoven

University of Technology (EUT) gradually shifted from the low quality region

in the sixties to the moderately high quality region in the seventies and

18

and the very high quality region in the early eighties.

In 1963 the influence of tube geometry and unequal heating on burn out was

investigated by Bowring and Spigt in a 7-rod bundle (see Spigt, 1963). One

of the findings was, that burn out heat flux seemed to decrease with

increasing test section length. Until ca. 1967 much research was performed

on a contract basis, e.g. in collaboration with Euratom (ISPRA). At that

time Germans, Frenchmen and Italians joined the research team in Eindhoven

(*). Early measurements are reported by Anonymous (1966) and Spigt (1966). They

deal with natural circulation at several pressure levels in a closed loop

with a vertical test tube heated by electrical current.

A typical result is shown in figure 2. The occurrence of an instability

threshold led to careful analyses of causes and effects of instabilities

(Spigt. 1966). Some notes on the possible occurrence of two different

mechanisms of heat transport, already clear from figure 1, will be given

later.

In collaboration with Westinghouse Electric Corporation (Atomic power

divisions), in Eindhoven the effects were studied of flow agitation and

special pipe configurations on critical heat flux (Tong et al., 1966). The

following conclusions were stated :

- The decrease of critical heat flux due to the proximity of unheated walls

at a constant local quality can be minimized by an additional mixing effect

generated by the roughness of the unheated wall. This benefit of roughness

is more significant at higher flow rates.

- The amount of reduction of the critical heat flux due to the line contact

with an unheated wall at a constant local quality is smaller at higher water

mass velocity.

In 1966 Spigt and Boot report some new progress made with burn out research

with a 7-rod cluster fuel element.

In 1970 a research program was started in collaboration with Interatom,

Gesellschaft fur Kernenergieverwertung in Schiffbau und Schiffart (GKSS),

Reactor Centre Netherlands (RCN). Stability characteristics and interchannel

mixing of "Otto Hahn" reactor cooling system were investigated (see

Anonymous, 1971).

(*) P.G.M.T. Boot, private communications

19

The effects of an additional unheated wall,

agitation were further studied by Vinke during his

The inlet velocity was carefully kept constant,

wall roughness and flow

Msc thesis work (1970).

among other things by

throttaling and smoothening the flow inlet, and measurements of temperature,

velocity and mixing rate were performed locally in a rectangular duct with

two transparent walls.

t ----~

E 275 u

" ~ ----~ ~

2~ 0

c ~ ~ n ~ 225 m x ~ ~ ~

~ 200 ro ru z

175

Figure 3

Additional rough plate P 4 bar

P 4 bar

30 40

Rectangular test section 2200 x 30 x 10 mm Heated surface 200 x 20 mm Inlet velocity 1 m/s

50 60

T - T (OC) sat inlet

Burn out heat flux versus inlet subcooling (1970)

Influence of pressure and surface roughness

Some typical and interesting results are shown in figure 3. It clearly shows

that heat transfer is improved if hydrodynamical mixing in the duct is

intensified or if system pressure is increased.

Figure 3 also exhibits the fact that two different mechanisms of heat

transfer may occur

differences between

at

bulk

each

and

system pressure. At

near wall temperatures

large subcoolings,

are large. Strong

oscillations in temperature, heat flux and pressure are observed. Bubbles

originating from the wall presumably enter into the fast core flow

stochastically, but more easily than at relatively low subcoolings, when

hardly any oscillations are found. In both regions of subcooling, heat

20

transfer is improved if subcooling is increased, since the latter

effectuates a better mixing rate and better supply of fresh water to the

wall.

t t~ 2 ~ '--'

---- 275 N c 0 0

~

" ~ ~ 1,62 u .... w

250 m

~ ~ ~ 0 m w c ~ ~ ~ ~ ~ 225 w

> ~ 1,47 0 m

~ x 0 ~ ~ ~

200 P 2,1 bar ~ ~ w ~ Inlet velocity = 1 m/s ~ m ~ W m

I 1,32 m w 175 ~

~

30 40 50 60 70

T - T (DC) sat inlet ~

Figure 4

Burn out flux and pressure drop versus subcooling (1970)

From these considerations, radial void distributions can be suggested as a

means to indicate the subcooling region present.

Figure 4 demonstrates how transition from one subcooling region into the

other is associated with a minimum pressure drop over the test section.

Critical heat fluxes at low steam qualities were found to depend on surface

roughness, surface contaminations, aging of test materials and other

parameters of the actual test configuration. Burn out at high steam

qualities, on the contrary, was found to depend mainly on flow parameters

such as averaged void fraction, steam velocity, etc ••

An example of this is given by figure 5, in which some results of Boat et

ale (1976) are shown.

As a follow up of an exercise of the European Two Phase Flaw Group (Rome,

1973), also the influence of a local heat flux disturbance was studied in

Eindhoven (Boot et al., 1977). Some typical results are shown in figure 6.

21

If the heat flux is decreased at some point were dry-out is already present,

alternate condensation and superheating induce a propagating perturbation of

flow and wall temperatures.

0,8

t >­~ ..... ~

~ 0,7

~ :J o C H

t5

0,6

0,5

~ __ - 40 W/cm'

50 W/cm' ---~

Nimonic 75 tube diameter 10/12 mm heated length 4,1 m

1000

Figure 5

1500

Mass flux (kg/m's)~

Burn out quality versus mass flux (1976)

2000

In 1982, Van der Geld et al. presented a simple method for calculating post­

dryout wall temperatures. Temperature values calculated with this model were

found to be very dependant on the droplet size at the point of dry-out. This

droplet radius was estimated from a maximum Weber number.

In subsequent years, the modeling equations were therefore

order to be able to predict droplet parameters at point of

measured values of the wall temperature. A computer program

Clevers, 19B4) to perform the computations.

in

dry-out from

was written (R.

In this paper the calculation model and computer program are presented.

To be able to verify computational results it was found most desirable to

halle some direct means of measuring droplet size. To this end, the thermo

22

void probe was developed. The next section and chapter 2 are devoted to this

droplet detection method.

t u o 450

400

350

300

60 W/cm'

50 W/cm'

0,5 0,6 0,7

Figure 6

Inlet quality 0,5 ~ass flux 1500 kg/m's Tube diameter 10/12 mm Arrows indicate locations of heat flux disturbance (length 100 mm)

0,8 0,9

Steam quality ~

Dry-out wall temperatures versus steam quality

in the presence of a cooling spot (1977)

In 1984 it was attempted to combine these two experimental methods of

droplet parameter determination in a 39 mm diameter tube in the large test

facility described by Van der Geld (1985). Unfortunately the power supply

was insufficient to create dry-out situations in the 8,23 m long test

section.

Further experiments, with smaller bore tubes, are planned.

23

1.2 Some history of droplet size detection

In 1974, C.A.A. van Paassen in Delft published his investigations of

atomization and evaporation processes using droplet detection thermocouples.

The detection method was based on the fast temperature fall if a droplet

evaporates on the hot junction of a thermocouple. The detection technique

was analyzed and applied to a wide range of test conditions, but especially

to spray coolers in attemperators at elevated pressures; droplet velocities

ranged up to 40 mls both in air and superheated steam. His results were very

satisfactory.

L.O.C. Heusdens, a MSc. student of Van Paassen, in 1976 made a numerical

study of heat flow and temperatures in a detection thermocouple,

disentangled the influences of some aspects of the cooling process, and

extended in this way the range of applicability of droplet detection

thermocouples.

In later years, experiments have succesfully been carried out at pressures

up to 100 bar (Van Lier and Van Paassen, 1980). A report is in preparation

and more experiments are contemplated (*).

The Delft investigations yielded starting points for the Eindhoven research

described in chapter 2 of this paper. This chapter describes, inter alia:

- a theoretical approach to the cooling process of a detection thermocou­

ple on which a droplet evaporates;

- a study of the interconnections and relative importance of correc­

tion parameters;

- a measuring strategy to facilitate droplet size calculations;

The theoretical results made it possible to give a practical form to

the measurement analysis;

- an introduction to droplet velocity measurements by means of time of

flight method.

Calculations were verified by measurements performed by van der Looy (1983),

Boonekamp (19B4), Boot, and Van Bommel (1986).

(*) C.A.A. van Paassen, private communications

24

1.3 On droplet impingement studies

It is clear, that a better knowledge of droplet behaviour on thermocouples

of various sizes would lead to improved prediction methods and hence would

contribute to a more accurate way of measuring droplet size and velocity by

thermocouple detection methods. In the course of the work on the thermo void

probe (see chapter 2), it was therefore decided to look into droplet

behaviour on surfaces in both experimental and theoretical manner.

If a droplet hits a surface that has a much higher temperature than the

droplet, an insulating vapour film is formed inbetween the surface and the

droplet. This Leidenfrost phenomenon (Leidenfrost, 1756) or spheroidal

effect (8outigny, 1850) was first mentioned by Boerhaave (1732).

In 1965, L.H.J. Wachters reported experimental impingement studies with

highly superheated surfaces. He found a breakup of droplets if the Weber

number

We = 2 P d v ~ r / 0

exceeded a value of ca. 80. In his thesis, Wachters shows and examines high

speed cinefilm recordings.

Recent studies of liquid drop behaviour on very hot surfaces are reported by

Adams and Clare (1983), Makino and Michiyoshi (1984), Mizomoto et al.

(1986), Zhang and Yang (1983).

Much less appears to have been published about the impact and spreading of

droplets on surfaces that are only slightly higher in temperature than

impinging droplets. In this case the Leidenfrost

important. Experimental investigations were reported

(1967), while Hoffman (1975) reported interesting

general nature.

phenomenon is not

by Ford and Furmidge

measurements of a more

In chapter four, some new measurements of isothermal droplet impingement on

curved surfaces are reported. Also, a numerical model for the calculation of

droplet spreading on flat surfaces is presented. Use was made of a

collocation method.

A collocation method has succesfully been applied to bubble growth by Zijl

(1977), and to bubble implosion by Sluyter and Van Stralen (1982). The

25

collocation method presented in chapter four for droplets (a kind of

"inversed bubble" case) was developed in collaboration with Mr. W. Sluyter

of the department of Physics (EUT).

In addition an algorithme was developed to account for the dynamic contact

angle where the liquid-vapour boundary touches the surface.

Although work is still in progress, this can be considered as a first step

towards improved prediction methods for the spreading of droplets on

slightly superheated, curved surfaces (see, for example, figure 7).

Figure 7 (overleaf)

Droplet impingement and evaporation history on a capillary tube

26

t

o 33,67

0,37 34,78

0,74 35,52

1,11 36,26

1,48 37,0

ms

Droplet impingements and evaporation history on capillary tube. (dmvn flow)

2 THERMO VOID PROBE MEASURING STRATEGY

An intrusive detection device, the "thermo void probe", is based on two

thermocouples (diameters are, for example, 0,026 and 0,10 mm) that penetrate

a dispersed ~oplet flow. The couples are heated by the superheated steam

and are cooled down slightly each time a droplet evaporates on it. Resulting

cooling and reheating curves are analyzed to infer droplet size and, if

possible, droplet velocity.

2.1 DETERMINATION OF DROPLET SIZE

A thermocouple, heated by superheated registers

fall if a droplet hits the hot junction and evaporates

a fast temperature

there (see also

section 1.2). Resulting cooling and reheating temperature curves are

analyzed in this section.

2.1.1 First estimates of droplet size

2.1.1.1 First estimate; r th ,1 If a droplet at saturation temperature, Tsat'

evaporates completely on a thermocouple that has temperature Ttc higher than

T a total heat 'sat'

(2.1)

is extracted from the environment of the droplet. Here H denotes the

specific heat of evaporation, and r the mean droplet radius. This radius

will be estimated from the time required for ,evaporation, T, and the

maximum temperature drop of the thermocouple junction during evaporation,

denoted with e tx. It is assumed that

- evaporation heat is only extracted from the thermocouple;

- vapour is not superheated by heat from the thermocouple;

- temperature drops in radial direction are neglected;

the heat flux towards the hot junction, q, is constant in time;

- the droplet impinges on the thermocouple welding, and flattens to

a circular liquid film with radius Rf

•

~ction 2.1.2.3 the following expression will be derived for spot cooling

1 28

during time of a circular cylinder with radius Rk = 0,5 dk

Here the material constant I ( A pC) is obtained by averaging over the p

corresponding values for chrome 1 and alumel.

Elimination of Qv from (2.1) and (2.2) yields

Let t denote the time required to reach maximum temperature drop after x droplet impingement. If T is estimated by t , equation (2.3) yields a first

x estimate of the actual droplet radius.

Equation (2.3) has been derived, in slightly different way, by Van Paassen

(1974).

2.1.1.2 Correction parameters; r th ,2 To improve the accuracy of the above

droplet radius estimation, the correction parameters <I> 00' <I> 01' <I> 10' are

introduced in the following way

(2.4)

The parameter <I> compensates for subcooling of the liquid and superheating 00

of the vapour produced

(2.5)

where T b is the liquid temperature at the moment the droplet collides with su the hot junction, and p C (T - Tsat) represents the vapour enthalpy vap p,vap sup yielded by the thermocouple.

This subcooling or superheating enhances the

measured, whence,j, ~ 1. 'I' 00

temperature difference

If heat is also extracted from the surrounding vapour at temperature T , vall this can be accounted for by the time averaged value of lhe parameter

29

(2.6)

Here Tvap denotes the vapour temperature, Tfilm droplet after spreading on the hot junction,

temperature of the

the thermocouple

temperature, and (l and (l tc corresponding heat transfer coefficients. vap Usually the (l / (l t ratio is much less than 1, and the value of <Il 01 is vap c close to 1. Only if a vapour film occurs between the liquid film and the

thermocouple, (l tc is reduced and <Il 01 may be different from 1. This

phenomenon is called the spheroidal effect, see section 1.3.

Quantification of this effect is often cumbersome, but experiments (see

section 2.1.4) showed that it may be neglected if (Ttc- Tsat) is in the

order of 30 K or less.

Expressions (2.5) and (2.6) have been derived before by Van Paassen (1974).

A droplet may evaporate partially if its speed at collision is high, or if

its diameter is large compared to the thermocouple size. Experiments showed

that good results can be obtained if Rtc is about 8 times as large as r.

Section 2.1.4 will deal with thermocouple measurements, while more details

on droplet collision phenomena are given in chapter 4.

The parameter <Il10

accounts for :

- heat exchange between thermocouple and surrounding vapour;

heat fluxes in the thermocouple wires that are not constant in time;

If the droplet temperature is lower than T t' initial heat fluxes sa are highest;

- droplet impingement at some distance from the hot junction;

If a droplet impinges at a greater distance from the thermocouple

junction, time of evaporation is larger and e tx is smaller.

- the fact that in the evaporation process the liquid film has a finite

extent.

In section 2.1.3 a measuring strategy and a computation method for <Il10

will

be presented.

30

2.1.2 Theoretical analysis of heat transfer during evaporation

In this section the evaporation of a droplet on a thermocouple is considered

in more detail than in section 2.1.1. The interconnections and relative

importance is evaluated of the various effects that should be accounted for

by the correction parameter ~10 that was introduced in section 2.1.1.

The analysis starts with a simplified cooling problem, from which some

interesting conclusions can be drawn. This case is also of practical

importance, as will be demonstrated in section 2.1.4.7.

More complicated cooling situations are subsequently studied with the aid of

some assumptions that are based on the conclusions of the simplified cooling

problem.

Typical theoretical cooling curves are calculated and compared with the aid

of a computer. Main features and dependencies of the correction parameter

~ 10 are deduced in this way.

=2~.~1~.=2~.~1~N~e~w~t~0~n~'~s~~c~0~0~l~i~n~g __ =0~f __ a=-~c~y~l~i~n~d~e~r~;~~1~D~~c=a=s=e~. Let a thermocouple be

represented by an infinitely long cylinder with radius R = D/2. The

consequences of this simplification will be accounted for in the measuring

strategy of section 2.1.3 and in chapter 5, where thermo void probe design

adaptations are discussed.

At initial moment t = 0 the cylinder has a uniform temperature T and is o

placed in a medium with temperature T , that is constant a

than T. The radial temperature profile in the cylinder o Newton's cooling from the outside, and is determined by

dimensionless parameters

(2.7.a) Bi = R a / A (Siot number)

(2.7.b) Fo = a t / R2 (Fourier number)

in which a denotes the thermal diffusivity A / pC. P

in time and less

is affected by

the following two

The ranges of SI values that are typical for Thermo Void Probe (TVP)

application are listed below :

31

(2.8.a) (2.8.b) (2.8.c) (2.8.d)

o E (26.10-6, 5.10-4) m

a ~ 5,35.10-6 m2 /s

A ~ 26 W/mK

a E (100, 104 ) W/m2 K

These values will be further discussed in section 2.1.2.6.

Let e ~ T - T • The governing heat equation in cilindrical coordinates is: a

(2.9)

The last term on the RHS of equation (2.9) can be neglected in the present

case.

It is noted that there was no need to introduce a partial derivative

notation like

a e or or e ,r

in this chapter since no total derivates are involved.

The following boundary condition is obtained from Fourier's conduction law

and Newton's law at the surface:

(2.10) d T (R,t) + JL (T - T(R,t) 0 - cr:r A a

( ) ( -4 -2) . From 2.B Biot numbers are calculated in the range 10 ,10 ,whlle Fo

approximately equals BOOO t. For these small Biot numbers the exact series

solution of the present cooling problem, which can be found in Carslaw and

Jaeger (1959) for example, can be truncated to yield

(2.11) (T(r,t) - T ) I (T - T ) ~ 1 - J (-Rr 12 Bi) exp(-2 Bi Fa) o a a 0

Let T T(R,t) and let surf

B 2 Bi F 0 / t ::;: 2 a a I (A. R)

and since .; 2 Bi < 0,15 ,the temperature drop inside the

32

cylinder is small as compared to (T - T f). The cooling process depends a sur merely on heat transfer between the surrounding medium and the surface of

the cylinder. From (2.11) a relaxation time equal to 1/8 is deduced.

Let I denote the relative importance of the first term on the RHS of (2.9)

with respect to the second term on the RHS of that equation, i.e.

At the surface, I(R) = 1, as can easily be demonstrated with the aid of the

following equations:

L J = - J d Z 0 1

2.1.2.2 Instantaneous spot cooling of a cylinder; rotatoric symmetry. If a

spherical droplet impinges on a thermocouple that has a diameter, 0, larger

than the droplet diameter, it will spread out quickly (see chapter 4). Since

cooling mainly uniform and external, and since relevant 8iot numbers are

very small, it is now worthwhile to look into some elementary cases in which

rotatoric symmetry is assumed. Finite droplet size will be accounted for in

section 2.1.2.5.

Consider again a cylinder at an initially uniform temperature T • Striving a towards solutions of more general problems, a cooling explosion at time t = o and axial location x = 0 is now studied.

Again the governing heat equation is given by (2.9), in which the last term

on the RHS is now important during the entire cooling process. As soon as

axial temperature gradients become small, the cooling problem for each cross

section has some bearings to the one discussed in the previous section

(2.1.2.1). It is therefore expected, that the first term on the RHS of (2.9)

contributes to wall cooling curves characteristics in about the same way as

in the uniform cooling case of section 2.1.2.1. This assumption can be

formally phrased as follows.

In a region close to the wall, where e (r) ~ O. a local heat transfer

33

coefficient a (r) can be defined by

(2.12) a(r) q(r) / e (r) _ A d e / e (r) d r

Differentiation of this defining equation yields

Note that a (R) can be replaced by a • From this relation and the defining

equation of I (see section 2.1.2.1), one easily establishes

I(R) Bi + a R) / (~) r R

In the case of uniform cooling I(R) equals 1, as was seen in section

2.1.2.1. For TVP applications, the Biot number has values in the range (10- 4

, 10-2 ), and I(R) can therefore be approximated as follows;

(2.13)

In analogy with the case of uniform cooling it is now postulated that

(2.14) ~ (R) « 1 d x e e (R)

x and ~ (R) « 1 d e (R) d ted t

The consistancy of this approach can of course be checked by putting a

solution into equation (2.13) to evaluate the terms in (2.14).

With the definition

(2.15) B (1 + I(R) )

the following equation is now derived from equations (2.9) and (2.10). It

describes the temperature profile along the surface:

(2.16) d 8 (R) d t (R)

Let 6(x) be Dirac's delta-function and Q be the total heat extracted from

the thermocouple during the cooling process. Let ~ be equal to

34

Since Q = 1T P Cp It _oo/''dx (Tsurf(t=O) - To)' the axial surface temperature profile imposed by an explosive spot cooling at t=o can be

written as

Equation (2.16) with boundary condition (2.17) can be solved in the usual

way, by splitting of variables and by a Fourier transformation, to yield:

(2.18) x2 8 f(x,t) = (0/ (2pC 1TR~/1Ta t» exp( ---- B t) sur p ~ 4a t

2.1.2.3 Uniform cooling; radial temperature drop. Now suppose that heat q is

extracted uniformly during the time T • If t < T then :

(2.19) t 2

a surf(x,t) = 0 f dt' q exp(- 4~t'

Using an adapted Laplace transformation the integral in

primitivated to obtain:

(2.19) was

(2.20) e f(x,t) ={q / 4pC T 1(a8)} .{exp(/(Bx2/a».(-1 + erf( sur p

+ l(x 2/4a t) ) ) + exp(- I(Bx 2 /a».(1 + erf(1'I31t - l(x2/4a t) ) ) }

Ifx=O:

(2.21 ) a f(Ott) = q erf( 1l3f) / (2p C T 1T R~ sur p ~ )

If B = 0, t = T and x = 0 equation (2.19) yields

(2.22) 8 surfeD, T) = q / ('IT It I( 1T A p Cp T) )

The suffix "surf" will now be dropped. Strictly speaking only surface

temperatures will be calculated in the following. but in view of the results

of section 2.1.2.1 and because of the fact, that Biot numbers are less than

0,01 , temperature drops inside the cylinder are small as compared to the

35

temperature difference T - T • a surf With this prerequisite the weighted thermocouple radius, Bk, can be

substituted by Rk• Equation (2.22) with this substitution was already employed

in section 2.1.1.

2.1.2.4 Non-uniform, bounded heat fluxes. Define, for t ~ T:

(2.23) ~ (x,t, T) = e (x,t) / e (0, T)

where e (x, t) and e (0, follow from equations (2.20) and (2.21 )

respectively. Let an amount of heat, q, be extracted by a heat current that is

uniformly rising in the time interval (0, 1):

(2.24) q(t) = 2q t / 12

Note that q = IT dt q(t) and that Q(t) = (2/ T ) o

t f dt' q / 1 , if t ::ii 1 •

o The latter integral implies that during each time interval dt' a spot heat

sink of strength q/1 has become active. The heat current q is therefore a

superposition of heat currents of the type discussed in section 2.1.2.3.

For the heat current defined by equation (2.24) the following normalized

temperature profile holds:

(2.25) t

~1!(x,t,1,2) = (2/1) J dt'f.(x,t-t',1) o

The index 2 will soon become clear.

A more general heat current is defined by

(2.26) q(t) = c q t / 12 + d q / 1

with d = 1 - c/2. The second term on the RHS of equation (2.26) was already

treated in section 2.1.2.3, while the first term on the RHS of (2.26) contains

the parameter c that was equal to 2 in equation (2.25). If t::iii 1 :

(2.27) t; "(x,t, T ,c) = (c/2) ~ II(X,t, T ,2) + (1 - c/2) f. (x,t, 1 )

If t> T then

(2.28) ~ "(x,t, 1 ,2) = (2/ T ) tdt't;(x,t-t',1) o

36

wi th for t > T :

(2.29) ~ (x.t, T) = (e (x,t) - e (x,t-T) ) / e (0, T)

in which two continual, uniform heat currents with equal strength but

different starting point were combined.

2.1.2.5 Non-uniform, non-local cooling and reheating. If a droplet impinges on

a surface it will spread out to form a thin circular liquid film with

approximately constant radius Rf (see section 1.3 and its references). Let its

centre be at distance x from the thermocouple welding, and let y measure the a

axial distance from its centre. It is now assumed that heat extracted in the

volume between y and y + dy is proportional to the area covered by the film,

which equals 2RfdY 1(1 - (Y/Rf )2). If Q denotes the instantaneous heat release caused by the liquid film, then

yields the following expressions for the normalized temperature distributions

due to a circular liquid film with radius Rf

:

(2.31 )

(2.32)

Of course, if t exceeds T , then ~" , ( t)

Since ~ = e (x, t) / 8 (0, T), the correction parameter 4>10 (see section

2.1.1.2) is given by

(2.33) 4>~O= E;,~x • I(tx I T)

where ~ '" denotes the maximum value of the normalized temperature curve max that is represented by E;,"'.

37

2.1.2.6 Some typical evaporation curves. The notation for c, d, T and xo ' Rf is adopted from the previous sections 2.1.2.

Equations (2.8) and (2.15) show that typical B values lie in the range 0 -

300, if I(R) ~ 1. Values of the product B.T are therefore less than about 20.

t 8 en

~ 0 ..., u ro .... en 4 c

-rl TI ro OJ H D.

U1

0

glass

~ beeswax

cellulose acetate

o 2 4 6 -6

Impact energy (10 Nm) ----t ......

After Ford and Furmidge (1967) data for water on various substances

Figure 8

Spreading factor versus impact energy

Influence of static contact angle

It is noted that for chromel-alumel, values of the thermal diffusivity, a, may

vary from 5,3.10-6 m2 /s at 380 K up to 6,3.10-6 m2 /s at 800 K.

The convective heat transfer coefficient, ~ , depends on thermocouple geometry

and flow parameters, but can be derived from values of :

Nu = ~ 0 / A­

Pr = p C / A p Re = V 0 / v

(Nusselt number)

(Prandtl number)

(Reynolds number)

with the aid of a correlation of the type

38

Nu = a + b (Ttc I

Values of the correlation parameters a, b, c, d and e are given by Grober et

ale (1961), and depend on thermocouple geometry and Re. Typical values for a

cylindrical geometry are a=O; b=0,8; c=0,25; d=O,39; e=0,31.

Typical values of a for TVP application range from 100 up to 104 W/m2 K, as

already noted in expression (2.8).

The spreading of a droplet is measured by the radius of the droplet film after

collision, Rf , or by the ratio 8 = 2Rf/r. Spreading phenomena are considered

in more detail in chapter 4. If the impact energy, the sum of kinetic and

f ° -6 Q ° 4 d sur ace energles of a droplet, less than 6.10 Nm, ~ varles between an

8 (see figure 8, adapted from Ford and Furmidge, 1967). For impact energies -9 less than 10 Nm the spreading is determined by the static contact ,¢

(270 for water on nickel) in the following way

(2.34) s 3 = 32 sin 3 ¢I {1 - COS¢)2 (2 + cos¢)

yielding a value of 4,4 for the water-nickel combination.

A nominal value of 8 for TVP application is 6. A typical droplet diameter is -5 -5 5.10 m, and a typical value of Rf is therefore 8.10 m.

o ......... <D ....... <D

c. a '"' "0

Q) k ::J ..., '" k Q)

g. Q) ...,

-0 Q) N

0r1 ...; ro

~ ~

o~----~~ ____ ~~~ ________________ -,

0,2

0,4

0,6

0,8

-----

x :: 0 o

o '" 0 f

B :: 0

1,oJ-----~------~----~----_T------r_----~-----0,05 0,10 0,15 0,20 0,25 0,30

Time (5) __

Figure 9

Normalized temperature curves; influence of B

39

It will be demonstrated in section 2.1.4.5 that an uncertainty in the value of

of about 70 % is still acceptable for droplet size estimation.

~

r>

0 <D "-([)

.. a. 0 1-1 -0

III 1-1 ::J ..., co 1-1 III a. E III ..., -0 III N

.,..;

.-! co E

t 1-1 0

Z

0 __ --------------------------------------------,

0,4

0,8

1,2

0,05 0,10 0,15 0,20

Figure 10

x ,. 0 o

o ,. 0 f

C ,. 0

0,25 0,30

Time (s) _

Normalized temperature curves; influence of c

Typical cooling curves were calculated from equations (2.31) and (2.32). It

turned out that if x ~ , cooling curves with distinct features were o obtained that could be analyzed in a relatively easy way.

~ a r>

0 ([)

"-([)

0,2

a. 0 0,4 1-1

-0

III 1-1 ::J ..., ro 0,6 H III a. E III ...,

-0 0,8 Q) N

.,..; ,....; co E t 1,0 H 0

Z 0,05 0,10

Figure 11

o ,. 10-4 C = 2/3 B,. 25 f

-4 Xo ,. 3.10 Of = 0 C,. 2/3 B,. 0

0,15 0,20 0,25 0,30

Time (5)_

Normalized temperature curves; influence of Xo and Of

40

The following conclusions concerning cooling curve characteristics were drawn:

the product B T generally has great influence (see figure g);

if a heat flux is not constant in time, results are notably affected

only if Icl > 2/3 (see figure 10);

the spreading of a droplet has hardly any influence (see, e.g., figure

11) ;

if a droplet impinges besides the thermocouple junction, i.e. if IXol >

0, cooling curves are strongly altered, although the possible

influence of c is strongly reduced (see, for example, figure 11).

41

2.1.3 Measuring strategy and computation method

In view of the conclusions reached in section 2.1.2.6, the accuracy of the

correction parameter ¢10' and hence of the droplet size determination, is

mainly dependent on the knowledge of the evaporation time, T, and the

evaporation location, x • An estimate of T is offered by t (see section o x

2.1.1), but there is no way of estimating x directly. Since calculations o

showed (see also section 2.1.2.6) that the initial slope of a cooling curve

is strongly influenced by variations in xo

' this slope was quantified by the

temperature ratio 81/82 (see figure 12) and used for estimating x • o

t

1.2Stx 0.4 tx

"--~-+c

0.3 tX--l* .....

Figure 12

T

Time_

Tavg k

Cooling curve schematics and measuring parameters

For given values of the parameter c and the droplet film width Of = 2Rf

,

equations (2.31) and (2.32) allow for the computation of T and 81/82 for

each set of given values of x and t • This was done by solving the equation o x

by varying T while keeping other parameters constant. The results for c = 0

and Of = 0 were used to construct figure 13 that yields

42

first estimates x o

if not only tx but also B1/B2 is measured from a cooling curve.

Since figure 13 contains many such curves, each corresponding to some value

of B, it was named "calibration field", rather than "estimation curve", in

the corresponding box of figure 14.

Ul5

1,8

1,75

1,15

t 1.65

N 1,6

CD

........ 1.55

CD 1,5

US

t.4

1.35

1.3

1.25

1.2

1.15

1.1

UlS

[E D,=O

.04 .08 .12 .18 ,20 .24 ,28 ,32 .36.4 ,44 .46 ,52 .56 .6

Figure 13

W=X/~­x

Estimation of evaporization location from

measurement parameters

This figure 14 summarizes the whole computation method of droplet size, and

will now be further discussed.

Apart from parameters defined in figure 12 or listed in the nomeclature, in

figure 14 the following notations are used:

43

a thermal diffusivity (m 2 /s)

B 2an (1 + I(dk/2) ) / dk A

dk

::;;: 2Rk• Effective diameter of the rod that repre­

sents a thermocouple (m)

d <p diameter of thermocouple wire (m)

Gd parameter that accounts for temperature dependance

of (H P )

G parameter that accounts for temperature dependance m

of (A p e ) p

Gk parameter that accounts for temperature dependance

1

r

I' W

T avgk T sat v vapor

of (Rk

)

radius of thermocouple welding

::;;: r th ,1 ' the first estimate of droplet radius (m)

::;;: I' th, 2 ::;;: I' / ¢ w (m) measured mean temperature level (ae)

saturation temperature (ae)

mean vapor velocity as estimated from mesured

superficial velocities (m/s)

vd droplet velocity, either measured directly (section 2.2)

or estimated from v , I' and correlations (m/s) vapor

x ::;;: xo ' the location of evaporation as measured from

the thermocouple welding (m)

S ::;;: Of / rw ::;;: 2Rf / rw (-)

w ::;;: x / 14a t x

It is noted that I(dk/2) approximately equals 1.

In calculation block 1, a value for r r th ,1 is determined from equation

(2.3) with the aid of the parameter ek that depends on Gd, Gm and Gk•

In calculation block 2, ¢10 is iteratively determined by minimizing

(2.35) II (8./8. 1 - 8. /8. 1 ) 2

i=1,2,3 1 1+ calculated 1 1+ measured

where 84 81 (cyclic permutation). Each individual quadratic term W9S given

a lower bound to minimize these three factors together.

The three input parameters 81/82, 82/83 and 81/83

allow for the computation

44

BEGIN

CALCULATION BLOCK 2

v vapor ...

B

... tx ... ~/B2

W

tx INPUT

a

... CAUBRATION Tav9k CURVE

B,/B2 } B,/B3 B2/B3

Of

t. Ot x

Gk

dk CALIBRATION CURVE

Gm

... CALIBRATION

Tavgk CURVE YES

Gd

CALIBRATION Tsal CURVE

NO

Figure 14

Flow chart for the calculation of droplet size

45

of the three unknowns x, 1 , and c (see also equation (2.32)). The value of

B was set equal to 4a il I ( A dk ) •

Since the droplet film diameter, Of' was found to have only a minor effect

on the calculated results (see also section 2.1.2.6) and since only rough

estimates for S can be given, this spreading factor S was kept constant

each iteration of block 2. The value of 8 can be adjusted once

(see figure 14).

The varying of Of by ± 70 % resulted in a spread of ± 10 % in the calculated

values of ¢ 10'

Only if distinct, ' , thermocouple readings are being analyzed, i.e.

only if x :;:; 14a or w ;;; 0,62, and if the drople is found to evaporate on

the thermocouple welding, i.e. if x < 0,95 1 , the effective diameter, dk ,

is adapted to some value larger than d¢ and close to 2· 1, and calculation

block 2 is entered again (see figure 14).

If the thermocouple junction has about the same diameter as the thermocouple

wires, this procedure is unnecessary since dk is equal to d¢ at every

location. The manufacturing of small cylindrical thermocouples is possible

(see chapter 5).

46

2.1.4 Experimental verifications

2.1.4.1 Thermo void probe measuring device. The intrusive measuring device

primarily consists of two thermocouples inserted in a tube of highly

degassed ceramic that is reinforced by a stainless steel capillary tube (see

figure 15).

The outer diameter of the ceramic tube is 1,2 mm, while wire diameters used

are in the range 0,026 - 0,1 mm. Although thermocouple junctions shown in

figure 15 are large as compared to the wire diameter, it is possible to

manufacture thermocouples in almost a cylindrical shape (see chapter 5).

Teflon shieldings and a lava sealant in a special mounting unit (see figure

15) electrically insulate the thermocouples from the stainless steel tube

wall that can be heated with Joule's heat, and allow for a pressure drop of

more than 200 bar in the device.

The mounting unit is readily installed in a tube (see figure 15) with use of

O-rings. Therefore rapid replacement and easy repair are possible. An

electric connector at the end of the unit also facilitates quick replacement

(see figure 15).

Figure 15 (overleaf)

Thermo void probe measuring device

47

THERMAL VOID PROBE

thermocouples in a ceramic tube

mounting unit

unit mounted in a test tube

electrical connection

A TVP unit encompasses two thermocouples. Each thermocouple is

electronically conditioned in the way depicted in figure 16. The optional

thermocouple (T/C2) is at constant temperature level close to the mean

temperature of a TVP thermocouple, and is used to increase the sensitivity

of the conditioning and hence the measurements accuracy if the TVP is

operated at high system pressures.

I sntht'll!'1d1 plafle Offset

ric C===

Htdlt-in

Figure 16

Thermo void probe electric conditioner

The conditioner has an automatic ice point compensator.

Cer,>:';,al power suer>,,· ..

:-.. ~ tch .. 1:>-' ~r '

A TVP unit can be mounted on locations wher.e the electric potential is

different from the earth (common) ground potential. The signals are

registred and monitored in the way described by Van der Geld (1985).

2.1.4.2 Experimental set ups. Measurements were performed in two

experimental set ups:

Type I (Enlarged TVP simulation set up)

Droplets were produced with the aid of adapted capillary tubes;

superheated steam heated up a chromel-alumel thermocouple made of

bars with a diameter of 10 mm. The system pressure at measurement

49

Type

location was about 1 bar.

Inlet temperature of droplets was measured with another thermocouple

just before they were released from a capillary tube.

Droplet sizes and velocities were measured with high speed

fotography, and weight of droplets was measured with the aid of a

precision balance after gathering a number of droplets on ice so as

to prevent evaporation.

:IT (actual TVP set up)

Droplet detection thermocouples of a thermo void probe (see section

2.1.4.1) catched droplets produced by liquid atomizers in

superheated steam. System pressures up to 10 bar.

Droplet sizes and velocities were measured with high speed

cinematography (frame speed 4000 pictures per second) and with a

laser doppler velocitometer.

Thermocouple readings were analyzed with the aid of a Hewlett Packard

frequency analyzer (HP 5420A).

2.1.4.3 Large diameter thermocouple measurements. A typical temperature

measured with the 10 mm diameter thermocouple bar (type I

measurement. see section 2.1.4.2) is shown in figure 17.

I measurements allowed for repetition of droplet impingement and

keeping constant of droplet size, and hence for reduction of the error in

the mean value of the calculated droplet radius. Relative reading errors for

input parameters amounted to

3 % for S(B. / S.) / (B. / B.) 1 J 1 J

4 % for S( e t) / e tx 1 % for Set ) / t x x

% for S(Tavg ) / T , k avgk

Typical relative errors for calculated parameters amounted to

7 % for S( Cl ) / Cl

2 % for droplet size determination by measuring weight loss

9 % for S ( ¢ 10) / ¢ 10

50

The result for a typical test, after averaging over a number of droplets was

r th ,2 = 1,45 ± 0,03 mm

while weight loss measurement yielded

rweight = 1,46 ± 0,02 mm

t u o

154

20

Figure 17

60 100 140

Time (5) _

Specimen of temperature history; 10 mm thermocouple

In general the agreement between values of r th ,2 and rweight was very good.

2.1.4.4 TVP verification measurements. A typical temperature history ~~~~--~--~~~~~~--~~~~~~~

measured with a TVP is shown in figure 18; the diameter of the thermocouple

amounted to 0,10 mm.

The best results were obtained with droplets with diameters in the range

0,04 - 0,07 mm and with thermocouple diameters of 0,0265 mm and 0,1 mm.

From 18 droplet size measurements during a single test run a mean droplet

51

radius of 0,04 ± 0,003 mm was calculated. For error estimates see section

2.1.4.3. The temperature drop e tx varied between 20 K and 71 K for the

9nal1er couple, and between 3 K and 9 K for the larger couple. 20 to 40 % of

the total evaporation took place at the smaller thermocouple.

t 120

u 0

QJ '-' :J

-'-' ro '-' (l) II 117 E ill t-

114

f'~ / I

1 \

I I I ,

0,5

Figure 18

, I

I j

J I

1,5 2

Time (5) ..

Specimen of temperature history; 0,1 mm thermocouple

This experiment was repeated without the smallest, leading thermocouple.

The mean droplet radius was determined under the same conditions with the

aid of only the 0,1 mm thermocouple. The result was 0,036 ± 0,004 mm, which

is in good agreement with the former result.

No other means of comparison was available, since high speed cine recordings

for these specific experiments were found to be systematically unsuccesful.

52

2.1.4.5 On void fraction estimation. Figure 18 clearly shows how the vapor

void fraction, £ , can be estimated from

(2.36) 1 - £ = (1/A ). N~ T) _1 4 1T r3 / 3 t . vd avg

with At = effective scattering cross sectional area (1T d2 /4);

N(T) / T = number of hits per second, which can be determined

directly from a thermocouple reading (see figure 18;

vd = mean droplet velocity;

r = mean droplet radius. avg At high steam qualities, the droplet velocity can be estimated from the

superficial steam velocity and some correlation for the relative droplet

velocity as a function of the droplet size (see chapter 3). Another way is

to measure droplet velocities directly, and will be discussed in section

2.2.

No accurate comparison data were available for the calculated values of the

vapor void fraction.

2.1.4.6 Varying measurement conditions. In annular-mist flow a water film is

flowing adjacent to the tube wall. If such a water film can reach the top of

a superheated thermo void probe, the reheating of thermocouples after

evaporation of the initial cooling water may appear to happen non-uniformly.

Special precautions have to be taken to prevent this from happening.

Figure 19 clearly shows that "isothermal plateaus" occur during the heating

up of a thermocouple. High speed fotography showed that in downflow these

plateau's were caused by some water, that was gathered underneath the

capillary tube of a TVP, moving towards the thermocouple and evaporating

there. Not until all this water was evaporated the isothermal plateaus had

disappeared.

It is noted that temperature rises inbetween the isothermal plateaus are

described by equation (2.11) of section 2.1.2.1.

An effective way to avoid plateau-heating at annular-mist flow is the

mounting of some water barrier near the end of the capillary tube of a TVP

unit. A small piece of stainless steel that surrounds the capillary tube

proved to be efficient.

The following practical recommendations are added to the measuring strategy

53

- 1 -

- 2 -

- 3 -

in section 2.1.3

The smallest thermocouple wire radius preferably should be about six

times as small as a typical droplet radius, although under certain

conditions measurements succeeded with radii that were about three

times as small as typical droplet radii;

t 140

90

40

o 50

Time (5) ..

Figure 19

Specimen of temperature history; plateau reheating

The largest thermocouple wire radius prefarably should be about

eight times as large as a typical droplet radius; often several test

runs are required with differently sized thermocouple wires to

establish a typical droplet radius and hence the appropriate wire

diameter;

Skip test runs at which the above mentioned "plateau reheating" has

occurred.

54

2.2 ON DROPLET VELOCITY MEASUREMENTS

2.2.1 Experimental calibration

Under certain test conditions droplet velocities can be measured with a

thermo void probe by a time-of-flight method. The two thermo void probe

signals shown in figure 20 exhibit the time delay caused by the adherence to

the leading thermocouple and by the time of travelling towards the

following thermocouple. The experimental set up used is described in section

2.1.4.2. The adherence of a droplet to the leading thermocouple will be

subject of section 2.2.2.

120 J1f 0.0265 mm

t °C

70

0 4 B

Time (ms) -

122

t [40.1 mm

°C

116

0 4 B

Time(ms) -

Figure 20

Droplet velocity measurement with time-of-flight method

Time-of-flight measurements turned out to be impossible if the thin

thermocouple had about the same size as a droplet. Droplets of that size are

likely to stick onto the leading thermocouple (see section 2.1).

55

For comparison it was attempted to measure droplet velocities with a

different technique. A 35 mW He-Ne backscatter laser doppler velocitometer

was applied. The type of analyzing equipment used demanded a regular,

continuous stream of nearly equally sized droplets at more or less the same

velocity. Unfortunately such a stream can not be produced by commercially

available liquid atomizers at the test conditions. At that time,

more appropriate electronic equipment (with the so-called "tracker" option)

could not be applied, whence no useful comparison data were obtained (see

Van Looy, 1983; Boonekamp, 1984; Van Bommel, 1986).

Only recently relatively large droplets could be traced by means of high

speed cine films (see figure 21). No comparison with thermo void probe

has yet been made.

Droplets smaller tha~ a thermocouple were found to hit the welding and then

either to bounce away or to stick to it (see figure 21). On this phenomenon

of catching of droplets the working principle of droplet detection

thermocouples is based (see section 2.1). The observation shows that the

leading thermocouple has to be smaller than droplets to be measured (see

also section 2.1.4.7).

Figure 21 (overleaf)

Drop et hitting a thermocouple in downflow

2.2.2 A sim Ie frict on loss correction model

A that slide$ over a smaller sized thermocouple loses energy due to

friction:

(2.37) Efric = - 0 IT fw . v dt

It is now supposedlthat frictional deceleration is proportional to the

velocity squared. Th~ integrating of the deceleration yields

(2.38) v. ;: v. 1 - dt q. ( v. 1 + v.)2 1 1- • 1 1- 1

Here v. denotes theldroplet velocity after i time intervals each with 1 '

56

t

o

0,59

1,18

1,77

2,36

ms

Droplet hitting a thermocouple. (down flow)

duration dt. The constant q. is approximated by 1

with A the effective contact area between liquid and thermocouple, md the

time-dependent amount of mass of the evaporating droplet, and A. a 1

resistance coefficient that is appropriate for a flat plate and Reynolds

numbers less than 3.105 (see, for example, Hinze, 1959):

(2.40) A • 1

1 ,238 / ( P d 2 r vi / II d) 0

,5

Major uncertainty in q. as determined by espressions (2.39) and (2.40) is 1

caused by the product 1,238.A. which is some kind of effective resistance

factor. This factor was therefore given values between 0,6 1f R; and 1,8 1f R;

in order to obtain some boundary limits for the frictional loss of energy.

It was intended to adjust this effective resistance factor with the aid of

experimental 'calibration' results (see section 2.2.1), since this seems to

be the most effective way to increase accuracy.

Solving equation (2.38) for v. yields 1

(2.41 ) Vl'_

1 = (1/2q. ).(1 - 2q.v. - (1 - 8q.v. )0,5 )

1 1 1 1 1

This equation allows for the step-by-step computation of the advancing

interface. The initial condition is the droplet velocity at the time it

detaches from the thermocouple. This velocity can be approximated by the

velocity, measured with the time-of-flight method, of the droplet as it

travels from one thermocouple to the other.

The displacement step, x., can be calculated from 1

(2.42) x. = v. dt - 0,5 dt2 (q./dt) (v. 1 + v.)2 1 1 1 1- 1

The total displacement is the sum of all displacement steps. If it exceeds a

certain value, the droplet is assumed to detach from the thermocouple and

the step-by-step computation is stopped.

The integral of equation (2.37) was numerically solved with the aid of the 1 3 - Simpson rule.

The following results were calculated for a droplet with radius 0,02 mm and

58

velocity 2 m/s at time of detachment from a cylindrical thermocouple of the

same radius. Material properties were evaluated at a pressure of 2 bar and

at T t = 120 oc: sa

A A / 1T R; Ef . r~c

Velocity loss

10-7 mJ cm/s

0,6 0,8 14

1,2 2,6 31

1,8 5,2 52

The calculated force was about 1000 times as large as the gravity force on

the droplet.

The radius Rf

was estimated by 0,028 mm. Actual thermocouples employed yield

smaller values of Rf , and consequently the actual velocity loss is less than

the values indicated in the above table.

It is concluded, that the velocity loss of a droplet during sliding over the

leading thermocouple of a thermo void probe is in the order of 10 cm/s. The

effective resistance factor AAi has to be determined experimentally in

order to achieve accurate calculation of this velocity loss.

59

3 A COMPUTATION MODEL FOR ESTIMATION OF DROPLET SIZE AT DRY-OUT

In high-quality, vertical flows a way to deduce droplet size is the

measuring of outer wall temperatures in axial direction and the subsequent

application of a semiempirical physical model that evaluates flow and heat

transfer after dry-out has occurred. This computation model takes into

account these measured temperatures since temperature profiles downstream

the wetted part of a test section usually show a steep increase at dry-out

and hence are characteristic and indicative of flow and heat transfer

processes.

The model described in this chapter employs several well-established

correlations, and infers a cross-sectional average of droplet size if steam

quality is known.

3.1 MODELING ASSUMPTIONS AND SEMIEMPIRICAL EQUATIONS

The calculation model aims at calculating post-dryout heat transfer and flow

development in vertical evaporator tubes, and at determination of droplet

size by minimizing differences between measured and calculated wall

temperatures.

The location where annular-mist flow alters into mist flow is called point

of dry-out.

The following assumptions are made

-1- flow and heat transfer are stationary;

-2- there is rotatoric symmetry around the tube axis;

-3- liquid and vapour mixtures are flowing vertically upward;

-4- mist flow with highly dispersed droplets occurs directly downstream

of the locus of dry-out;

-5- fluid is at saturation temperature;

-6- calculations at every axial location can be performed with droplets

distributed uniformly over a cross-section, and with a mean droplet

60

-7-

-8-

(3.1)

radius;

radiative heat transfer between wall and vapour can be neglected;

convective heat transfer between wall and vapour is adequately

described by the following correlation (Moose and Ganic, 1982):

= (A /0) 0,023 ReO,S Pr1/ 3 (/ )0,14 a wv v v v W v 1.1 v, w

Rev G xa 0 /\..I v ; Red = pvd(vv - v1)/\Jv Prv - Wv Cp/ AV

o represents the inner diameter of the tube, \..I denotes dynamic v,w vapour viscosity close to the wall and d the droplet diameter;

-9- direct contact heat transfer between tube wall and droplets is

adequately described by (Filonenko, Petukhov Popov; see Webb, 1971):

(3.2.b) f = (1,58 In Re - 3,28)-2 v

f exp(1 - (T /T t)2 ) W sa

G denotes the total mass flux (kg/m2s) and T the wall temperature. w

The exponential

efficiency;

factor in equation (3.2.a) is the evaporation

-10- radiative heat transfer between wall and droplets is adequately

described by the following correlation (Deruaz and Petitpain, 1976):

(3.3.a) qr CJ (T 4 - T \) E E /( €: 1 + E (1 - E l) ) w sa w w

(3.3.b) 0 1 + (v/vl )(P/Pl)(1 - x )/x -1

E 1 = 1 - exp(-2,22 d 1 - a a

(3.3.c) E 0,66 w

-11- heat transfer between droplets and vapour is adequately described by

the following correlation (see, far example, Grober et al., 1961):

(3.4)

61

-12- the friction coefficient for droplets can be obtained from (White,

1974) :

(3.5) 24 Cd = Re + 6/(1 + IRe) + 0,4

-13- vapour velocity increases only gradually in axial direction, whence

droplet accelerations relative to the vapour phase need not be

considered:

(3.6)

(3.7)

-14-

(3.8a) (3.8b) (3.8c)

dv is positive and follows from

first order Eulerian integration is suffices

x (z + dz) = x (z) + dz. ddXa (z) a a dd z

d(z + dz) = d(z) + dZ.--d (z) dz z

vi = dt

62

3.2 ADDITIONAL GOVERNING EQUATIONS

The vapour mass density, p , vapour dynamic viscosity close to the wall, v

W ,vapour heat capacity, C , vapour heat conductivity, A and surface vw pv v tension, a , are all dependent on temperature such, that a dry-out model has

to take these dependencies into account. The essentially one-dimensional

flow problem was therefore solved numerically. Equations that are needed for

the gradients in axial direction are derived in this section.

Almost by definition of the actual steam quaE ty, Xa ' tIle following

expression for the void fraction, £ , holds:

(3.9)

while the slip factor is determined by

(x !(1-x )) ~.2.-1 a a £ P

v

This equation together with equations (3.5) through to (3.7) determines the

velocities Vv and VI"

The equilibrium steam quality, xe ' is the quality that would be present if

all vapour superheating energy would have been used evaporating droplets :

It is immediately clear that

dx (3.12) ~ = 4qt I (0 G H)

with the total heat flux given by qt = qwv+qwd+qr. Differentiation of

equation (3.11) yields

(3.13) dT d?' (~ddX - (H + C (T - T )~» z pv v sat dz

from which Tv can be determined if ~ is known.

Let n denote the number of droplets per unit of volume. Almost by v

definition :

63

(3.14.a) n v

G (1 -

Differentiation of (3.14b) yields

In first approximation, only gradual changes of In(n vI PI) are considered.

The gradient of x is then approximated by a

dx (3.15.b) If ::: - (1 - x ) a 3 dd d dz

In second approximation, equation (3.15.a) is calculated. The second term

on the RHS of (3.15.a) in almost all cases investigated proved to be very

small as compared to the first term.

In the same way axial changes of the droplet diameter are treated.

Difference between in- and out flux of liquid in a cross-section of the tube

are due to evaporation

(3.16)

In first approximation follows

(3.17) dd dz

With the aid of equations (3.1) through to (3.17), the boundary condition

T (z 0)::: T t' given values of D, T qt' G and dz, and with start v sa 'sat' values of x ::: x(z 0) and d ::: d(z ::: 0), the vapour temperatures and heat o 0

fluxes can now be calculated if the wall temperatures downstream of the

point of dry-out are also known. An appropriate calculation procedure is

presented in the next section (3.3).

It is noted that it is sufficient to measure wall temperatures on the

outside of a test tube at dry-out. The inner wall temperatures, that are

needed to perform the computations, can be calculated if system pressure and

mass fluxes are known (see Van der Geld, 1982).

Of practical importance is also the fact, that droplet size is calculated at

64

Flow chart for the cak:ulatton 0' the droplet diameter at dry~out from measurements wIth thcfmocooPlfl mounted e)!ternally at the wall

INPUT PRESSURE

CALCUlATION OF A REGRESSIVE CURVE FOR THE WALL TEMPERATURE

Z A:X IAl COQROINA TE FRDM DRYOUT'

Z 0

MASS FLUX DENSITY WALL I-fEAT FLUX DENSITY MEASURED WALL TEMPERATURES STARTVALUE MASS QUALITY DRYOllT, XO STARTVALUE DROPLET DIAMETER DRVOUT, 00

PROPERTIES OF THE VAPOR PHASE AS A FUNCTION OF PRESSURE AND VAPOR TEMPERATURE VELOCITIES OF VAPOR ANO LIQUID PHASE HEAT TRANSPORT BY DIRECT CONTACT, WALL·DROPLETS HEAT TRANSPORT BY RADIATION. WALL-DROPLETS HEAT TRANSFER COEFFICIENT BY CONVECTION, WAlLNAPOR

COMPARISON OF THE INCREASE OF THE VAPOR TEMPERATURE WITH THE CALCULATED GRADIENT BASED ON A HEAT BAlANCE. DURING 10 INTERVALS

OUTPUT: ACTUAL MASS QUALITY EQUILIBRIUM MASS DUALITY VOID FRACTION

Figure 22

DROPLETS DIAMETER NUMBER OF DROPLETS VAPOR VELOCITY DROPLETS VELOCITY VAPOR TEMPERATURE HEAT TAANSFER FLUX DENSITIES - CONVECTION WALL-VAPOR - CONVECTION VAPOR-DROPLETS - DIRECT CONTACT WALL -DROPLETS - RADIATION WALL~OROPlETS

Flow chart of calculation procedure

of mean droplet diameter at dry-out

65

every axial location downstream of the point of dry-out. This allows a

comparison with measurements performed with a thermo void probe (see chapter

2), that has a fixed position on a tube while the locus of dry-out is

unknown in advance.

3.3 SOLUTION PROCEDURE WITH MEASURED WALL TEMPERATURES

The numerical calculation of mean droplet diameter at point of dry-out is

now elucidated with the aid of the flow chart shown in figure 22.

Pressure, mass flux and total heat flux are measured and serve as input

parameters.

Measured wall temperatures are interpolated by means of a regression curve

in order to allow for temperature calculation at every axial location

downstram of the point of dry-out.

Some start values are selected for vapour quality and droplet diameter.

After these and other initializations, the main block for

several vapour temperatures is entered (see figure 22).

encompasses the following.

calculating

This block

Start value of T (z v 0) is T sat'

is T (I.dz). v

For each natural number I, start

value of T ((I+1).dz) v Velocities are calculated with the aid of equations (3.5), (3.6), (3.7) and (3.10). Vapor temperatures are subsequently calculated from

a follows from equation wv from (3.3).

(3.1); q d follows from (3.2), and q w r

If the calculated value of T is not in accordance with the start v

value of T , this vapour temperature is iteratively determined. v

Equations (3.4), (3.15) and (3.17) now yield values of x a and d at

the next axial location, according to (3.8).

66

The main calculation block is entered again, until 10 values of T, v corresponding to 10 axial locations, are calculated.

In the next phase of the calculation procedure, the d ' t dT (1) 1.'S gra len "(iZ"l

calculated from these 10 vapour temperature values.

Another value for the vapour temperature gradient is obtained from (3.13)

with the aid of (3.12), and denoted with ~ (2).

The value of

(3.19) OIF 10 dT dT i ~ 1 { "(iZ"l (1) - "(iZ"l ( 2)} 2

now minimized by varying x and d and starting all former calculations a 0

allover again. This is done by selecting, at each iteration step, eight new

sets of (x ; d ) values: o 0

(0,98

(0,98

x • 0'

x • 0'

o x ~

>.

;:! 0,7 ...... co :::J rr E OJ Ql ....., tn

0,5

0,3

0,98 d ), (0,98 o

1,02 d ), etc., o

0,1 0,3

Figure 23

x • 0'

d ), a

0,5

G 1000 kg!m's qt 400 kW!m' P " 70 bar dz 0,1 m

0,7

Droplet diameter, x (mm) IP o

Computational results at point of dry-out for various

start conditions

67

and calculating eight new values of DIF according to (3.19). The iteration

is terminated if the DIF value corresponding to (x ; d ) is lower than o 0 the DIF value corresponding to any of the other eight sets.

Typical results of this variation are gathered in figure 23. The

curve was obtained by varying the start values of (x f d ) while keeping 0 0

other conditions the same.

It is clear that figure 23 allows for the calculation of the droplet

diameter at point of dry out if

can be found, an upper bound of

Similar relationships between

x can be estimated. If no estimate of x o 0

this droplet diameter can be calculated.

droplet diameter and steam quality are

calculated at various axial locations, which allows for comparison with

thermo void probe results (see chapter 2).

With the aid of the upper

velocities, temperatures and

bound of do, dmax ' the program

heat fluxes until all liquid

68

calculates

has been

3.4 DISCUSSION OF RESULTS

Various wall temperature profiles, both measured and made up ones, were used

to calculate the steam quality, x , and the droplet size, d, at point of o 0

dry-out with the calculation procedure described in the previous section

(3.3). Some of the results are discussed in this section.

The cross-section averaged Weber number as determined from

at point of dry-out was found to be less than 1. Here d denotes the max maximum droplet diameter that can be calculated by varying input parameters

for the computational procedure of section 3.3 (see figure 23).

1 u o

500

300

G ~ 994 kg/m's qt = 502 kW/m' p = 70 bar o = 1 em

o 2 3

Distance from point of dry-out {m} ..

Wall temperatures after CtFAD et al. (1974)

Figure 24

Computational results; steam temperature and quality

A Weber number of, for example, 7 corresponds to droplets with a diameter of

about 2 mm, that are not likely to prevail in a tube with a nominal diameter

of 5 mm. In addition, We represents an average over a cloud of droplets aug

69

while also droplets are mainly created by turbulent shear acting on liquid

originating from or adherent to the wall. Hence We avg is hardly comparable

to the maximum Weber number that prevails for example for droplets in a free

stream (see Hinze, 1959).

Cumo et al. (1974) have measured wall temperatures at dry-out in a vertical

pipe at a system pressure of 70 bar and with a total heat flux, qt' of 502

kW/m 2• Using their results for a specific test case, a value of 0,21 mm for

d and a corresponding quality x 0,5 were calculated. In view of the max 0

preceding remarks these results are intelligible although verification seems

impossible.

With these values of x and d the heat fluxes (see figure 24), qualities a 0

and vapour temperatures (see figure 25) at various locations downstream of

the point of dry-out were calculated. Intervals of 0,1 m were employed.

i

~ ro w

£

D ill N

~

ro E H o

Z

0,6

0,4

0,2

o

It is clear from

G 994 kg/m's qt = 502 kW/m' p = 70 bar

D = 1 em dz = 0,1 m

o 2 3

Distance from point of dry-out (m) ~

Figure 25

Computational results; heat fluxes at dry-out

25 that radiative heat transfer to droplets is only

relatively important directly downstream of the point of dry-out if dry-out

wall temperatures are about 700 K. However, a much greater effect of

radiative heat transfer should be expected for wall temperatures higher than

70

ca. 800 K, since radiative heat transfer rates are roughly proportional to T4 , . Differences between actual and equilibrium vapour qualities are due to the

temperature rise of the vapour (see figure 24). This is the trend that was

expected. The vapour superheating also causes a relatively strong heat flux

from vapour to droplets (see figure 25).

It is concluded from these and other test findings that in general

calculation results are in agreement with expectations.

The present physical model is numerically practable, but primarily based on

experimental correlations. The methodology therefore needs a thorough

validation. A strong validation, however, is only possible with accurate

measurements, for example of droplet size and vapour superheating.

No accurate measurements of these parameters at relevant system conditions

were found in the literature. Chapter 2 of this paper attempts to contribute

to practical droplet size determination; relevant experiments were performed

while more experiments are planned (see the end of section 1.1). Nijhawan et

ale (1980) presented accurate vapour superheating measurements at relatively

low pressure levels, whence it is fair to suggest that strong validation of

computational results such as the ones presented in this section will be

made possible within the near future.

71

4 ON DROPLET IMPINGEMENT

Aiming at a higher accuracy of droplet detection methods and a better

understanding of droplet impingement, the dynamic spreading of droplets on

surfaces is experimentally investigated and numerically studied. The

governing equation are solved by means of a collocation method. Viscosity is

only accounted for via a semi-empirical dynamic contact angle algorithm.

4.1 EXPERIMENTAL RESULTS

With the aid of stainless steel capillary tubes, droplets at room

temperature WF!re generated. After freely falling 14,5 cm, the droplets

adiabaticall y impinged on a massive, stainless steel cylinder with a

diameter of 6 mm. Several droplets at a row were allowed to hit the bar; the

surface of the bar was therefore dry only for a leading droplet. It was

observed that the spreading of following droplets was in general facilitated

by liquid remnants on the surface, usually a very thin film or a monolayer.

The qualitative picture of droplet spreading was however found to be

unaffected by these liquid remnants.

High speed cinematography was used to study droplet spreading of three

different substances (see figure 26)

- demineralised water; ph = 7,6 specific ohmic resistance 850 k~

mean droplet diameter ± 2,5 mm;

- 99,9 % ethanol (C2H

5) mean droplet diameter ± 1,5 mm; maximum

film width after spreading 15 - 20 mm;

- turbine oil (DTE oil 105, Mobil) ; mass density 936 kg/m 3; dynamic

viscosity ± 360 cSt (mm 2 /s).

Consider the series of photographs corresponding to demineralised water in

figure 26. Notice that larger gaps between two pictures are indicative of

longer time intervals, and that time intervals are different for the three

substances investigated.

Figure 26 (overleaf)

Adiabatic droplet impingements on a stainless steel bar

72

t =

o

1,34

2,68

4,02

5,36

11,36

51,36

ms

deminera1ised water

oil

t t =

o o

0,67 2

1,34 4

2,01 6

2,68 8

4,02 982

ms ms

Cold liquid impingements on a stainless steel tube. (down flow)

t

o 1,25

0,25 1,50

0,50 1 ,75

0,75 2,00

1,00 2,25

ms

Droplet impingements and L>vaporation history on thermocouple. (do"n flow)

The time history shows that a water droplet

first milisecond. Afterwards the outer

spreads out quickly during the

rim that is formed near the

intersection of the liquid interface with the stainless steel grows thicker

and thicker until it has swallowed up almost all liquid and has adopted the

shape of a torus. Gravity and surface tension then cause contraction of this

torus in the direction of the lower part of the cylindrical bar. This

contraction phase if of less interest to the present investigation, since

under realistic conditions a droplet has evaporated by then (see figure 27).

In these and other pictures, surface waves with a relatively short

wavelength are observed.

After some time, 3 miliseconds typically, the free surface has lost the

star-shape property, that defined by the demand, that straight lines

through the liquid can be drawn from a given point to any point on the

interphase. It is clear from figure 26 that after this time the distance

function of the interface to the baryometric centre has become

valued. The analysis of section 4.2 will therefore be restricted

first two miliseconds of droplet impingement only.

multiple

to the

Similar time histories of droplet spreading are observed for ethanol and

turbine oil (see figure 26). Only the spread out times are different, while

also hardly any surface waves can be discerned at the oil interface.

These differences are attributed to differences in viscosity and surface

tension (see also figure 8 in chapter 2).

Figure 27 (see overleaf)

Droplet impingement and evaporation history on a thermocouple

74

4.2 GOVERNING EQUATIONS

In the present investigation, droplet spreading on a flat horizontal wall

after vertical collision is studied.

The normal velocity condition u.n=O is satisfied by introducing a "mirror

droplet" that makes the wall a of symmetry. The viscous liquid layer,

along which the liquid moves smoothly is assumed to be thin. The no-slip

boundary condition therefore plays no role. The dynamic contact angle is

dependent on the spreading velocity, which represents a boundary condition

that will be accounted for in a semi-empirical manner. This is the only

place where viscosity explicitly shows up in the equations.

Under the above mentioned restrictions, potential flow theory may be applied

in the liquid. To simplify the treatment, tangential velocities and

interfacial turbulence, induced by surface-tension gradients, are neglected.

Combination or the Bernoulli equation for the liquid pressure, that states

that for vortex-free potential flows the expression

(4.1) 0,5 }L2 + pi p + ¢, t - .9.-1.

is a constant througout the fluid, and the Laplace equation for the surface

tension results in the following boundary condition, which is written in

spherical coordinates (see figure 28):

(4.2) ( 2 -2 2) ( I) () ¢ ,t + 0,5 ¢ , r + R ¢ ,e + 1 - p v p 9 R cos e +

+ ( o/R p) W - Co = 0

on r R( e ,t). Here W denotes

Note that part of the normal in the point where the droplet first touches

the wall is a flowline for reasons of symmetry.

The constant c can conveniently be evaluated at the centre of mass. o

75

Directly after touch-down the centre of mass is still moving at

approximately the impact velocity vd since momentum exchange at the wall is

then still negligible. The expression (4.1) is therefore approximately equal

to

+ P / p + 0,5 g d a 0

at the centre of mass. It follows that the constant c can be estimated by: o

(4.3) c = 0,5 v 2 + 4 a /d o d 0

I I \ \ \

"­ "- ..... - 8 10

Figure 28

+ 0,5 g d o

Coordinate system, collocation angles (N=9) and

dynamic contact angle

Almost by definition

(4.4)

The kinematic surface condition puts equal flow velocity and displacement

rate of the droplet boundary, and is expressed by

(4.5) dR ,+- _ R-2 R dt = '¥ ,r ,8

The dynamic contact angle was determined by Hoffman (1975) for various

76

substances under varying conditions. He obtained the universal contact angle

curve that is depicted in figure 29. Application of this curve goes as

follows.

(I) ...... CJ\ C m ..., u t1l ..., C o u u

''-; E m C >. o

90

o

-4 10

After R.L. Hoffman(1974) data were taken at 24°C

Figure 29

100

Jl.)!.+F(6) .-y s

Dynamic contact angle versus interfacial velocity parameter

Take a certain wall/liquid combination. Let e be the static contact angle s for this combination. The curve of figure 29 relates the value of e to s some value Z on the horizontal abscissa. Now assume that the interface is

moving with velocity v .• The dynamic contact angle for this situation is 1

then the angle that corresponds to the value Z + v. n/o according to the 1

curve of figure 29.

For the present investigation the curve of figure 29 was analytically fitted

to obtain:

(4.6) 101og(G(8» -5,2957 + 0,14489* e - 0,00264* 8 2 + 2,7857 10-5 e 3+

- 1,494410-7 8 4 + 3,154710-1°8 5

If, for example, e s equals 27 degrees (water and glass), G( e ) equals s

77

0,0014628. If at some time a

interface velocity is equal to

dynamic contact angle of 8

( a / n ). (G (8 ) - 0,0014628) c

c attained, the

The boundary conditions (4.2), (4.5) and (4.6), combined with the Laplace

equation for the velocity potential represent a well-posed partial

differential problem if initial conditions are prescribed.

78

4.3 SOLUTION PROCEDURE

4.3.1 Collocation method

The solution of the potential equation with zero velocity at r=o (see figure

28), symmetric with respect to the plane e:: 1T /2 and non-singular at e =0

is:

(4.7) 00

<p(r,8,t):: E ak(t) r 2k+1 P2k(cos8)

k==o

Since the set of even Legendre polynomials is complete for axisymmetric

functions, the bubble boundary can be expanded in these polynomials as well:

(4.8) R ( e ,t):: I bk ( t) P 2k (cos e ) k=o

The expansion coefficients ak(t) and bk(t) can be determined by matching

(4.7) and (4.8) to the boundary conditions (4.2) and (4.5). For reasons of

computational efficiency this is done with the collocation method (see, for

example, Zijl, 1977).

The collocation method implies the disctretizing of a droplet cross section

through the normal at 0 (see figure 28) into a number of (N+1) so-called

collocation points. The series (4.7) and (4.8) are truncated after N+1 terms

and the boundary conditions are only applied in the collocation points. Now

the NxN matrix

{E • .} = {P2'(cOS eJ } 1J J 1

determines the droplet radius on the collocation angles e., and the matrix 1

{F .. } 1J

2' 1-1 == {R.J P2'(cOS e.) }

1 J 1

determines the velocity potential on the collocation points:

(4.9) R. (t) :: 1

(4.10) <p . (t) 1

N I E .. b.

j=o 1J J N I F •. a.(t)

j=o 1J J

79

Since the matrices E and F are not sparse, the number of collocation points

may not be too high and partial pivoting is recommended. If the collocation

cosines are taken as the zeros of a Chebyshev polynomial, the values of R. l.

and $. as determined by this method converge to the exact solution for N 1.

going to infinity (see, for example, Zijl, 1977).

4.3.2 Dynamic contact angle algorithm

At each timestep, the values of velocity and radius on the collocation

angles (see section 4.3.1) of the last iteration cycle are used as initial

conditions, and Eulerian integration of the velocities determines the new

positions of the droplet interface. The new position on the x-axis however

(see figure 28) is determined in the following way.

Through the collocation points at the liquid surface nearest to the wall a

quadratic curve is fitted. In figure 28 these points are denoted with L, L-1

and L-2. Let the coordinates of these collocation points be written as (XL'

YL)' (xL_1' YL-1)' etc •• Define P(1) and P(2) with:

P(1) = zL-2yE-1 xL-1yE_2 +

+ xL ( y E -2 - y E -1 )

Through the collocation point at the x-axis the tangent line to the

quadratic curve is calculated (see figure 28). This tangent line determines

the dynamic contact angle:

(4.11) e c = 1T /2 + arctan( P(1 )/P(2) )

With the aid of the universal curve represented by equation (4.6) the

velocity corresponding to 6 is now calculated. The new position of the c droplet interface on the wall is determined by Eulerian integration.

At the first timestep, when t=O, the distance, R , between the point 0 (see o

figure 28) and the point where the droplet interface intersects the wall

equals zero in reality. To initiate the advancing of the droplet interface,

R is given an arbitrary value, EPS, that is very small to suppres numerical o

80

dispersion.

One major advantage of the above algorithm to handle dynamic contact angles

is the fact that it is insensitve of the choise of EPS. If, for example, EPS

is given too large a value, the induced interface velocity is relatively

small, and the liquid interface is allowed to adjust itself to the value of

R • If, on the contrary, EPS is given too small a value, induced interface o velocities will be relatively high, causing a fast increase of Ro'

Although the introduction of EPS at first glance seems to be somewhat

artificial, it should be considered as the only non-quantummechanical way to

handle the breaking of the liquid interface when it touches the wall.

4.4 ON THE RESULTS

Results obtained sofar are only preliminary, since only a total number of

nine collocation points was exploited (see figure 28), and more tests have

to be carried out (see chapter 5).

The spreading and deformation of a droplet impinging on a flat horizontal

surface was found to occur:

- more or less independent of the start value EPS, as was expected

(see sectien 4.3.2);

faster if surface tension is increased;

- slower if mass density is increased;

- faster if impact velocity is increased.

The present author in coorporation with P. Sluyter will discuss final

results of their joint research in a separate paper.

81

5 SUGGESTIONS FOR FURTHER WORK

5.1 SOME DESIGN AND CONDITIONING IMPROVEMENTS

Very thin (0,015 mm) thermocouples can be manufactured in almost the shape

of a cylinder (see, for example, Nina and Pita, 1985). The effective

diameter, dk, always equals 2.Rk in the TVP measuring strategy applied to

the thin thermocouples (see section 2.1.3). Hence the use of these

thermocouples results in a reduction of computation time and in an increase

of measurement accuracy.

As opposed to the analytical approach of chapter 2, it is current practice

to take a mean time constant to compensate for the effects of thermal

inertia of thermocouples. This compensation is essentially done using two

methods: electronic or computational ones. An example of the latter method

T

dE T dt

Figure 30

I Time ~

Time ---I~ ....

E

Time --..,..

Time ---I ......

dE T- + E

dt

Schematics of signal conditioning with a compensation

is discussed by Cambray (1986). An example of the former method is now

discussed. For sake of clarity it is stressed that these compensation

82

methods are not relating to the measuring strategy of the thermo void probe,

since this strategy is essentially based on analyzing the consequences of

thermal inertia (see chapter 2).

M. Nina and G. Pita (1985) showed a way of how to compensate for the

response time of fine wire thermocouples. A special electronic circuit was

applied during processing of registred voltages.

Figure 30 illustrates the working principle of this compensation circuit.

Consider a cylindrical wire, representing a thermocouple, that is initially

at uniform temperature, and then is suddenly cooled according to Newton's

law of convective heat transfer. The wire experiences a cooling with a

relaxation time equal to t/(2 Bi Fa) (see section 2.1.2.1). If the

temperature drop inside the wire is neglected, T-T is approximately equal a to t~t (T-Ta ) during the cooling process. The compensation circuit is

therefore made to perform the following operation on the voltage E registred:

E = E + T ddt E new

where t can be continously adjusted between 0 and 30 ms. Figure 30 clearly

demonstrates how in case the intitial temperature drops stepwise, the step

is recovered by the above way of analyzing a thermocouple signal.

Nina and Pita (1985) measured values for T for several thermocouples. A Pt/Pt-13 % Rh wire with a diameter of 0,05 mm was found to have a relaxation

time of about 32 ms, and a 0,015 mm wire one of about 5 ms. Physical

frequencies up to 400 Hz could be measured thanks to the compensation

circuit.

83

5.2 DROPlET VELOCITY MEASUREMENTS

Droplet velocities can conveniently be measured with the aid of the

cylindrically shaped thermocouples described in section 5.1. Measurement

accuracy is increased by the application of these thermocouples, not only

because droplet size estimation is improved (section 5.1), but also because

knowledge obtained from simple laboratory experiments such as the ones

described in chapter 4 is made transferable to actual measurement

conditions. In addition, spurious signals caused by droplets bouncing from a

leading spherical thermocouple into all directions are avoided.

A strong laser doppler velocitometer with electronic "tracker" possibility

allows for instantaneous velocity measurements that can be used to sort of

calibrate the diagnostic procedure with the thermo void probe (see section

2.2). Further comparison is obtained from high speed cinematography, as was

already demonstrated in section 2.2.

The diagnostic procedure to account for friction losses (section 2.2) itself

can be improved by incorporating theoretical results of chapter 4 on droplet

impingement. The incorporation is not straightforward, however, since one

has to account for dependencies on impact velocity, temperature and system

impurities. Experimental verification is demanded whatever the degree of

sophistication of the model applied.

84

5.3 OTHER APPLICATIONS

At Delft university of technology (and Prins Maurits laboratory TNO), M.

Nina and the present author arranged temperature measurements during

combustion of polymethylmethacrylate with some sort of adapted thermo void

probe design (see Korting et al., 1986). The combustion chamber set up

consists of an injection chamber, a transparent fuel grain with a maximal

length of 300 mm and an aft mixing chamber (see figure 31).

diaphragm solid fuel (PMMA)

\ ~ air -

Figure 31

Schematic of a solid fuel combustion chamber

Flame stabilization after ignition was obtained with the aid of a rearward

facing step (diaphragm) located at the entrance of the fuel grain. Some 0,1

mm Pt!10 % Rh-Pt thermocouples intruded the flow at various distances from

the inlet (as indicated in figure 32). Some of the results are gathered in

figure 32. They show that satisfactory local and pseudo-instantaneous

temperature measurements can be obtained in hot combustion gases.

A totally different application of a thermo void probe is the studying of

bubbles in a subcooled liquid. Vapour bubbles formed at a tube wall are at

saturation temperature and therefore distinguishable from the surrounding

liquid. From the temperature curves registred by a thermo void probe, bubble

size can be estimated. This explains the paradigma "void" in the name thermo

void probe.

These measurements can conveniently be performed in the large test rig of

the Eindhoven University of Technology (see Van der Geld, 1985).

85

1750

t -::.! -

750

----------~-- -~----------,

initially flush with wall. l

200mm entrance . ""', ... ,.;' .... ~,~:\"'-'.' ,,_ .. 1,,""''''

" ..,y""y, ... r'

i~ ~-v I

I! • j~y.f'\~

II \J:/~WI / V ~ initially 3mm from wall.

265mm from entrance

PMMA/air mair=150g/s Pc = O,l.5 MPa L= 300mm dpo=L.Omm hid po = 0,3125

273 __ ~~~~~~~ __ ~ __ ~ __ ~ __ ~ __ ~ __ ~ __ ~ ___ ~ __ ~

t 1750

.-.. ::.! -Q) ... .P f: 1250

f ....

750

o 10 20 30 40 so 60

Time (5) --

Figure 32

Thermocouple readings at various locations along a grain

.~ I jrA~ initially 8mm \ :/ \ \ from wall

"

r I,\f,\i ~ : ~ ~ I initially 2mm , from wall

PMMA/air mair=150g/s Pc =O,45MPa L=300mm dpo =40mm h/dpo =O,3125 distance from

entrance 40mm

~3L-__ ~ ____ ~ ____ ~ ____ ~ __ ~ ____ ~ ____ ~ ____ ~ __ ~ o 10 20 30 40

Time {s} --

86

5.4 THEORETICAL DROPLET IMPINGEMENT STUDIES

The numerical model described in chapter 4 should be tested further by:

- varying the number of collocation angles;

- increasing the degree of the polynomium that approximates the

liquid surface in the neighbourhoud of the dynamic contact angle;

- increasing the time lapse of calculation;

- varying physical properties and impact velocity dependently from

each other.

Future work will be focussed at:

- allowing the surface on which a droplet spreads to be slightly

curved;

- allowing the surface on which a droplet spreads to be highly

curved;

- allowing the interface distance function to become multiple valued

at each collocation angle;

incorporating non-uniform

evaporation;

temperature distributions and

- studying boundary layer development and shock waves.

This research program is jointly being carried out by the present author and

Ir. P. Sluyter.

(111' \\'/i I"r/II'll~rviil; PMMA/air

~ 1/·1, 111\ mair = 150g/s

t 1750 nP1t.'ri "\f Pc =0,40 MPa t~I,~ ~ \~ij I

L= 300mm ~ I dpo =40mm '-"

initially 2 mm from wall III J..< h/dpo= 0.3125 ::J ..., 10

1250 center line distance from J..< III inlet 150mm i ~~ l-

i 750

~ 273

0 10 20 30 40 50 60

Time (5) -

87

6 CONCLUSIONS

Two measuring strategies for the determination of droplet size were

developed:

I a droplet detection method that can be applied in mist flow if the

nominal droplet diameter is in the order of one eigth of the diameter of the

thermocouple used, and if steam is slightly superheated (nominal values

ranging from 10 to 40 oC);

II a computational model based on semi-empirical equations that can be

applied to dry-out in vertical tubes if wall temperatures are measured.

For strategy I, the cooling and reheating process of a cylindrical

thermocouple on which a droplet evaporates in superheated steam is of vital

importance. This process was theoretically studied. A new device, the Thermo

Void Probe was designed that allowed for the performing of verification

measurements. These measurements showed good agreement with droplet sizes as

determined with other techniques.

Time of flight method can be applied to measure droplet velocities.

Frictional losses that a droplet experiences if it slides over a much

smaller thermocouple can be accounted for with the simple calculation model.

The computation model of strategy llgave satisfactory results when applied

to wall temperature data of Cumo et al. (1974). Further validation of the

model is necessary, and can be achieved, for example, with the Thermo Void

Probe.

To improve the accuracy of strategy I - results on droplet size and droplet

velocity, [nore understanding has to be gained of the spreading of a droplet

after it has hit a surface.

High speed cine film recordings revealed wavy structures on the interface of

a droplet after gravity driven impingement on a surface under adiabatic

conditions.

The droplet spreading was numerically simulated by means of a collocation

method. A special algorithm accounts for the dynamic contact angle at the

contact line of the liquid-vapour interface with the surface. Although first

results are promising, this work is still in progress.

89

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95

ACKNOWLEDGEMENTS

The MSc. students H. van Looy, H. Dietzenbacher, R. Clevers, J. Boonekamp

and D. van Bommel contributed enormously to the experimental and

computational part of chapters 2 and 3. Ir. C. van Paassen (Delft University

of Technology) gave encouraging advice and lend the 1 cm diameter

thermocouple. Ir. W. Sluyter took part in the numerical investigations on

droplet impingement (chapter 4). Prof. C. van Koppen suggested to

investigate droplet detection methods and stimulated development of the

instrumentation. Ing. P. Boot helped performing experiments and gathering

historical information for section 1.1. Mr. J. Verspagen helped with data

acquisition and with preparing figures.

The author is indebted to everyone.