On droplet parameter determination at dry-out

On droplet parameter determination at dry-out

Citation for published version (APA):

Geld, van der, C. W. M., & van Koppen, C. W. J. (1986). On droplet parameter determination at dry-out. (Report WOP-WET; Vol. 86.005), (European Two Phase Flow Group : meeting; Vol. 1986). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1986

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ON DROPLET PARAMETER DETERMINATION AT DRY-OUT

C.W.M. van der Geld .) and C.W.J. van Koppen II)

European Two-Phase Flow Group Meeting Milnchen, 1986

Eindhoven University of Technology Department of Mechanical Engineering

Report nr. WOP-WET 86.005, May 1986

I) Delft University of Technology Department of Aerospace Engineering

' I ) Eindhoven University of Technology Department of Mechanical Engineering

TABLE OF CONTENTS

1 Introduction and scope of the present investigations 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,l 2.1.1.2 Correction parameters; r

th,2

2.1.2 Theoretical analysis of the evaporation process 2.1.2.1 Newton's cooling of a cylinder; 1 D case 2.1.2.2 Spot cooling of a cylinder;

rotatoric symmetry

2.1.2.3 Uniform cooling; radial temperature drop 2.1.2.4 Non-uniform, bounded heat currents

2.1.2.5 Finite droplet size 2.1.2.6 Typical cooling curves

2.1.3 Measuring stategy and practical 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 Error estimates

2.1.4.4 Large diameter TC verification measurements 2.1.4.5 TVP verification measurements

2.1.4.6 Void fraction estimation 2.1.4.7 Measurement difficulties in

annular-mist flow pattern 2.2 On droplet velocity measurements

3 A computation model for estimation of droplet size at dry-out 3.1 Modeling assumptions and correlations

3.2 Modeling equations

3.3 Numerical calculation of d and x from

o a

3.4 Discussion of results

4 TVP design improvements; other applications

5 Conclusions

REFERENCES

ABSTRACT

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

velocities in superheated steam are difficult to achieve. An experimental

method to determine droplet sizes in superheated steam was developed and studied in recent years, and is presented in this paper.

The experimental detection device used is based on two thermocouples

(diameters are, for example, 0,026 and 0,10 mm) that intrude the droplet

flow. The couples are heated by the superheated steam and are cooled down

slightly each time a droplet evaporates on it. Cooling curves

characteristics are analyzed to obtain information concerning droplet size and, sometimes, droplet velocity.

In high-quality, vertical flows near dry-out, additional information

concerning droplet parameters can be inferred from a computation model that takes into account the measured outer wall temperature profiles in axial direction. These profiles show a steep increase downstream of the point of dry-out. The model employs several well-established correlations, and results are discussed.

1 INTRODUCTION AND SCOPE

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

presented a heat transfer topograph 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 (DNB), but equally well at high steam qualities, when it is usually called dry out.

This study will focus attention on dry out and the transition from annular flow to dispersed droplet flow, called point of dry out.

t

DNB

=

departure from nucleate boiling

critical heat flux

I I I I I I I I I I I I I I I / I

£t,

1,,1.:;'$0,

,/

I

,

\ I

/ -00'

~,' I I I

,ltr....

\

I.:::j I I I \~.. \ I «- I I I

/IJ~~-V'

_/

iiW·§::'"l0 /'

I

WI;

10.\

~

\ I .: .. ' 0 I L.., I

I

/I~\ ~ \ I :;.;.' &;Qi I ,Q; I II \ \ / . RJ'-, I ,fJ I I \ \ I

R-

,I

'-,'Ij I \ \ I "b'" I .",,"'" " \ \ I I n'!Y ~ \ I I ~ I ~"f'... \ I -<:' , I \ .... -~ / $' I I \ \ I .""Q; I I \ \ I &; I I \ \ I I I \ \ I I I \ \ I I I \ I I ~ \ \ I I ~ \ \ I I ' t , \ \ " I «' \ , I , \ \ I St \ \ I ~<J", \ \ I 91..!<JI· _-lty \ \ _{\\ I} " '-... \\ I J

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

Figure 1

1.1 Some history of critical heat flux investigations at EUT

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 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 on a contract basis, e.g. in collaboration with Euratom (ISPRA). time Germans, Frenchmen and Italians joined the research team in

(*).

performed At that 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.

Natural circulation

t

T '" 200 °C sat 160 Burn out 140 120 Instability threshold 100 o 10 20

Figure 2 Inlet subcooling (OC) ~

Burn out fluxes at natural circulation (1966)

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 I, 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 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).

t

~ " _E 275 u ... ::3 ...., :J 250 0 c '-' :J .n ...., ₂₂₅ ro x :J r l "-+' ₂₀₀ ro OJ :r: 175

Additional rough plate p = 4 bar

P 4 bar

30 40

Figure 3

Rectangular test section 2200 x 30 x 10 mm Heated surface 200 x 20 mm Inlet velocity 1 m/s T sat 50 T (0C) inlet 60

Burn out heat flux versus inlet subcooling (1970) Influence of pressure and surface roughness

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 throttling and smoothening of the flow inlet, and measurements of temperature. velocity and mixing rate were performed locally in a rectangular duct with two transparent walls.

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 at each system pressure. At large subcoolings,

differences between bulk and near wall temperatures are large. Strong observed

oscillations in temperature, heat flux and pressure are I measured. Bubbles originating from the wall presumably enter into the fast core flow stochastically, but more

hardly any oscillations transfer is improved i f

easily than at relatively low subcoolings, when are found. In both regions of subcooling, heat

subcooling is increased, since the latter

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

From these conSiderations, radial void distributions can be suggested as a means to indicate the subcooling region present.

t

t-..

275 t:: e a ..!:'. _1,62

....

.... ~ u

:x

+' 250

....

::l 0 (f) ro C

....

I-<

'"'

.B

₂₂₅ ro 1,47 ::> +' Q <II _a. x Q ::l '"' ... 200 P = 2,1 bar l:l

....

_ro +' _{Inlet velocity 1 m/s}

,..

<II :::J Q) (f) ::t: _1,32

_:n

175 a.

_'"'

30 40 50 60 70 T T (Oc) sat inlet -Figure 4

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, ageing 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 ••

0,8

t

>. ... • .-1 .-< 0,7 (I) :::J tT ... :::J 0 C .... :::J CD 0,6 a,s Nimonic 75 tube diameter 10/12 mm heated length 4,1 m 1000 Figure 5 40 W/cm , 1500 Mass flux

(kg/m's)-Burn out quality versus mass flux (1976)

2000

An example of this is given by figure 5, in which some results of Boot et al. (1976) are shown.

As a follow up of an exercise of 1973), also the influence of a

the European Two Phase Flow Group (Rome, local heat flux disturbance was studied in Eindhoven (Boot et al., 1977). Some typical results are shown in figure 6. If the heat flux is decreased at some point were burn out is already

present, alternate condensation and superheating induce a propagating perturbation of flow and wall temperatures.

t

'""' _w 450 0 ru ~ ~ ~ m ~ ru Q E ru ~ 400 ~ ~ m 3 ru U .~ ~ C ~ 350 300 50 W/cm' 0,5 0,6 0,7 Figure 6 Inlet quality 0,5 Mass 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 1982. Van der Geld et ale presented a simple method for calculating post-dryout wall temperatures. Temperature values calculated with this model were found to behighlYdependant 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

rearranged in dry-out from measured values of the wall temperature. A computer program was written (R. Clevers, 1984) to perform the computations.

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

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

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

this droplet detection method.

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.

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 drothermocou-plet 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 (1984), Boot, and Van Bommel (1986).

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.

If a droplet hits a surface that has a much higher temperature than the droplet, an insulating vapor film is formed inbetween the surface and the

droplet. This Leidenfrost phenomenon (Leidenfrost.

1756)

or spheroidal

effect (Boutigny,

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 / C1

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 the droplets. In this case the Leidenfrost phenomenon is not important.

Experimental investigations were reported by Ford and Furmidge

(1967),

while

Hoffman

(1975)

reported interesting measurements of a more general nature.

Van der Geld

(1986)

presents new measurements of droplet

cool, curved surfaces (see figure 7) and a numerical

calculation of droplet spreading on flat surfaces. Use

impingement on model for the is made of a collocation method and an algorithme to calculate dynamic contact angles. Although work is still in progress,

towards a prediction method for

superheated, curved surfaces.

Figure 7 (overleaf)

this can be considered as a first step the spreading of droplets on slightly

t =

o

1,34 2,68 4,02 5,36 11,36 51,36 ms demineralised 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)

2THERMO VOID PROBE MEASURING STRATEGY

A measuring device, the "thermo void probe", and a measuring strategy are presented for the measurement of droplet size and droplet velocitie in superheated steam. Experimental results are discussed.

2.1 DETERMINATION OF DROPLET SIZE

A thermocouple, heated by superheated steam, registers a fast temperature fall if a droplet hits the hot junction and evaporates 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,l If a droplet at saturation temperature, Tsat'

evaporates completely on a thermocouple that has temperature T

tc higher than T_sat' a total heat

(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,

8_tx' It is assumed that

- evaporation heat is only extracted from the thermocouple;

- vapor produced 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 R f•

In section 2.1.2.2 the following expression will be derived for local

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

k :

(2.2) 8

Here the material constant

I(

A pC) is obtained by averaging over the

p

corresponding values for chrome1 and a1ume1.

Elimination of

Q

from (2.1) and (2.2) yields

v (2.3.a) r3_h 1

=

C k

e

I

T t , tx (2.3. b) C k

=

3 R~

I

(1T A PCp) / (4H p )

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 h 2 To improve the accuracyof the above

!:..:.e:.

droplet radius estimation, the correction parameters '" '" '" are

'" 00' '" 01' '" 10' introduced in the following way

(2.4)

The parameter <Poo compensates for subcoo1ing of the liquid and superheating

of the vapor produced :

(2.5) <P' 00

where T b is the liquid temperature at the moment the droplet collides with su

the hot junction, and p C

vap p,vap(T - T t) represents the vapor enthalpy

sup sa

yielded by the thermocouple.

This subcooling or superheating enhances the measured, whence '" _"'00 ~ 1.

temperature difference

If heat is also extracted from the surrounding vapor at temperature T_vap'

this can be accounted for by the time averaged value of the parameter

(2.6)

_{<P 0'1}

₌

₁

₊

_{a (T - T ) / (a (T - T}

f1· 1m) )

vap vap film tc tc

Here Tvap denotes the vapor temperature, Tfi1m droplet after spreading on the hot junction,

temperature of the

temperature, and a

vap and a tc corresponding heat transfer coefficients.

Usually the

a

/ a

ratio is much less than 1, and the value of <j) 01 is

vap tc

close to 1. Only if a vapor film occurs between the liquid film and the

thermocouple, (X tc is reduced and <j) 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 showed

that it may be neglected if (T

t c - T sa t) is less than about 20

K.

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

R

tc is about

8

times as large as r.

Section

2.1.4

will deal with thermocouple measurements, while more details

-especially on droplet collision phenomena- are given by Van der Geld

(1986.

see also section

1.3).

The parameter <j) 10 accounts for :

- heat exchange between thermocouple and surrounding vapor;

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

If the droplet temperature is lower than T t' initial heat sa

flows are highest;

- droplet impingement at some distance from the hot junction;

If a droplet impinges at a greater distance from the thermocouple junc-tion, time of evaporation is larger and

a

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 <j) 10 will be presented.

2.1.2 Theoretical analysis of the evaporation process

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' 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 assumptins 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 Newton's cooling of a cylinder; 1D case. 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 section 4. where new thermo void probe

designs are discussed.

At initial moment t

=

0 the cylinder has a uniform temperature T and is

o

placed in a medium with temperature T 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 = (X R / A (Biot number)

(2.7.b) Fo (Fourier number)

in which a denotes the thermal dtffustvity

AI

pC •

p

in time and less is affected by the following two

The ranges of 31 values that are typical for Thermo Void Probe (TVP)

-6 -4

(2.8.a) D £ (26.10 , 5.10 ) m

(2.8.b) a '\,5,35.10-6 m2/s (2.8.c) A '\, 26 W/mK

(2.8.d) a £ (100, 104 ) W/m2K

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 can be neglected in the present case. The following boundary condition is obtained from Fourier's conduction law and Newton's law at the surface:

(2.10) _ d T (R t)

+

~

(Ta - T(R,t) ) = 0

d r ' 1\

From (2.8) Biot numbers are calculated in the range (10

-4

_{10 ), while Fo} -2

approximately equals 8000 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 ) / (T - T ) '\, 1 - Jo(-Rr

12

Bi) exp( -2 Bi Fo)

o a 0

Let T _sur f = T(R,t) and let

B = 2 Bi Fo / t = 2 a a / ( A R) Since J (x)

=

1 - x2

/22

+

x4/(2242 )

o • • •• and since ~

12

Bi <

I.flB:i

< 0,15 , the temperature drop inside the cylinder is small as compared to (T - T _{a sur} f)' The cooling process depends merely on heat transfer between the surrounding medium and the surface of the cylinder. From (2.11) a relaxation time equal to l/B 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.

I

following equations:

_d_ J

d Z 0

2.1.2.2 Spot cooling of a cylinder; rotatoric symmetry. If a spherical

droplet impinges on a thermocouple that has a diameter, D, larger than the

droplet diameter, it will spread out quickly (see section 1.3). Since cooling is mainly uniform and external, and since relevant Biot 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 _o

towards solutions of more general problems, a cooling explosion at time t =

°

and axial location x

=

°

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 local cooling problem has

some bearings to the one discussed in the precious 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 9(r) ~ 0, a local heat transfer

coefficient a(r) can be defined by

(2.12) a(r) = q(r) / 9 (r)

= -

A~

/ 9 (r)

d r

For current Biot number values (less than 0,01), Van der Geld (1986) derives the following estimation for I(R):

(2.13) I(R) ==

~ ~(R)

/ (i)

d l I d

(2.14) d x (R)

«e

dX

(R) and

dt

d l I d (R)

«e

dt

a

(R)

The consistan~y 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) ) a a A R

the following equation is now derived from equations (2.9) and (2.10). It describes the temperature profile along the surface:

d

a

(2.16)

dt

(R) = - B

a

_surf

+

a

dX'

d'

a

(R)

Let 8(x) be Dirac's delta-function and

Q

be the total heat extracted from

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

~ R ~

fdx f

dr 2 r (T(t=O) - T ) /

fdx

(T f(t=O) - T )

-~ 0 0 -~ sur 0

Since Q = 17 P

c"&:

(dx (T f( t=O) - T ), the axial surface temperature

p "- -~ sur 0

profile imposed by an explosive spot cooling at t=O can be written as

(2.17)

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:

2.1.2.3 Uniform cooling; radial temperature drop. Now suppose that heat q is extracted uniformly during the time l' • I f t < l' then:

(2.19) 8 _sur f(x,t) = ₀

J

t dt' q exp(- 4_a Xt' 2

+

- B t) / (1f~ 2 PCp 1'/1f(a

t'»

Using an adapted Laplace transformation the integral in (2.19) was

primitivated to yield:

(2.20)

e

_sur f(x,t) = (q /4pC _p T

/fa-B)

).(exp( {(B?/a»).(-l

+

erf(

1131::+

+

l(x

2

/4a t) ) ) + exp(- I(Bx2/a».(1 + erf( ~ -

l(x

2

/4a t) ) ) )

If x

=

0

(2.21)

If B

=

0, t = l' and x = 0 equation (2.19) yields (2.22) 8 _{s u r K} f(O, 1')

=

q / (1fJ1~ I( 1f ApC _p 1') )

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 (T

-a Tsurf)·

With this prerequisite the weighted thermocouple radius, ~, can be

substituted by R

k• Equation (2.22) with this substitution was already employed in section 2.1.1.

2.1.2.4 Non-uniform, bounded heat flows. Define, for t ~ 1':

(2.23) ~(X,t,T)

=

8(x,t) / 8(0,1')

where 8(x,t) and 8(0, 1') follow from equations (2.20) and (2.21)

respectively. Let an amount of heat, q, be extracted by a heat flow that is uniformly rising in the time interval (0, 1'):

Note that q

=

f

Tdt q(t) and that q(t)

=

(2/ T) ftdt' q / T , i f t

~

T •

. 0 0

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

sink of strength q/T 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) ~"(x,t, T ,2) = (2/ T) ftdt' ~(x,t-t', T)

o

The index 2 will soon become clear.

A more general heat current is defined by

(2.26) q(t)

=

c q t / T2

+

d q / T

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 ~ T:

(2.27) ~"(x, t, T ,c) If t > T then (2.28) ~"(x,t,T,2) with for t > T: (c/2) ~"(x,t, T,2)

+

(1 - c/2) ~(x,t, T) (2/ T)

f

Tdt , ~ (x,t-t', T) o (2.29) ~(X,t,T)

=

(8(x,t) - 8(x,t-T» / 8(O,T)

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

different starting point were combined.

2.1.2.5 Finite droplet size. If a droplet impinges on a surface it will

spread out to form a thin circular liquid film with approximately constant radius R

f (see section 1.3 and its references). Let its centre be at

distance x from the thermocouple welding, and let y measure the axial

o

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 2R_fdY

1(1 -

(Y/R

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 R_f:

(2.31) (2.32)

Of course, i f t exceeds T , then ~ '" (t) = (2/ T )

f

Tdt ' I; I (t ' ) • o

Since I; = e(x,t) / e(O,'1'), the correction parameter 41

10 (see section

2.1.1.2) is given by :

(2.33) '" 3 _ ~'" . ; (t / T )

~10 - ~max· x

where ~'" _max denotes the maximum value of the normalized temperature curve

~" , • (t)

2.1.2.6 Typical cooling curves. The notation for c, d, T and x

o' 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.

Tare therefore less than about

20.

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

-6

2/

-6

2/

may vary from 5,3.10 m s at 380 K up to 6,3.10 m s at 800 K.

The convective heat transfer coefficient, a , depends on thermocouple

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

Nu aD / A Pr pC / A p Re = V D /

v

(Nusselt number) (Prandtl number) (Reynolds number)

More details are given by many authors (see, for

1986). Typical values of a for TVP application

W/m'K,

as already noted in expression (2.8).

example, Van der Geld,

4 range from 100 up to 10

t

8 en ~ I-< 0

...,

u ro 4-C1' 4 c ... U ro QJ I-< { l U1 0 glass

~beL

cellulose acetate

o

2 4 6 Impact energy (10-6 Nm) - - - - I ... After Ford and Furmidge (1967)

data for water on various substances

Figure 8

Spreading factor versus impact energy Influence of static contact angle

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

after collision, R_f, or by the ratio

8

== 2Rf/r. Spreading phenomena are

considered in more detail by Van der Geld (1986). If the impact energy, the

sum of kinetic and surface energies of a droplet, is less than 6.10-6 Nm,

~ varies between 4 and 8 (see figureS, adapted from Ford and Furmidge, 1967).

For impact energies less than 10-9 Nm the spreading is determined by the

static contact angle, W (270 _{for water on nickel) in the following way :}

(2.34) 8 3

== 32 sin'

wi

(1 - COSW)' (2

+

CoSW)

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

-5 -5

is 5.10 m, and a typical value of R

f is therefore 8.10 •

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

of

6

of about 70

%

is still acceptable for droplet size estimation.

b< 0

-0 ...-CD

"

0,2 CD a. 0 _0,4 H U ru H :J +' 0,6 ro H w a. E ru +' u 0,8 ru N °rl "'""

!

~

1,0 0,05 0,10 0,15 0,20 0,25 0,30 Time (5) -Figure 9

Normalized temperature curves; influence of B

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

turned out that if x ~

o ~ , cooling curves with distinct features were

obtained that could be analyzed in a relatively easy way_

... CD ~ a. 0 H U W H ::J +' ro H ru a. E ru +' u ru N orl "'"" ro E

~

H 0 2 0 __ - - - , 0,4 1 ,2 0,05 0,10 0,15 0,20 Figure 10 Xo'" 0 0",0 f B '" 0 0,25 0,30 Time (5) _

The following conclusions concerning cooling curve characteristics were drawn:

- the product B T generally has great influence (see figure 9);

- if the heat flow is not constant, results are notably affected only if

lei>

2/3 (see figure 10);

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

ll) ;

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

Ix I

>

o

0, cooling curves are strongly altered, although the possible influence of c is strongly reduced (see, for example, figure 11).

!-' 0 0-CD ' - 0,2 CD ci a 0,4 H u OJ H :J ...., ro _0,6 H OJ 0.. E OJ ...., u 0,8 OJ N -ri .-i ro E

+

H _1,0 a z _{0,05 0,10} Figure 11 10-4 C = 2/3 B

=

25 Of = 0 C = 2/3 8 = 0 -4 Of = 10 C = 2/3 8 = 0 0=0 C = 0 8 = 0 f 0,15 0,20 0,25 0,30 Time ( 5 ) _

Normalized temperature curves; influence of Xo and D f

2.1.3 Measuring strategy and practical 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 x

o' this slope was quantified by the temperature ratio B1/B2 (see figure 12) and used for estimating x •

o

t

0.4

tx

:...-.-~!---+c 0.3 tX--lIlooi-+IC Figure 12 T Time_ Tavg k

Cooling curve schematics and measuring parameters

For given values of the parameter c and the droplet film width D

f = 2Rf,

equations (2.31) and (2.32) allow for the computation of T and B1/B2 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 values constant. The results for c : 0 and

D

not only t but also Bl/B2 is measured from a cooling curve.

x

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.

1.85 1.8 1.75 t.75

t

1,65 N 1.6 CD "- 1.55 CD _'.5 1.45 I.' 1.35 1.3 1.25 1.2 1 1~ 1.1 1,O!)

[]ill

0,=0 .04 .08 .12 .16 .20 .24 .28 .32 .36.4 ,44 .48 ,52 .56 .6 Figure 13 w = x / f u t 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, in figure 14 the following notations are used:

a thermal diffusivity

(m

2

/s)

B 2a Cl (1

+

I ( d / 2) )

I

d k A

d_k = 2R_k• Effective diameter of the rod that

repre-sents a thermocouple (m)

d <p diameter of thermocouple wire (m)

G

d parameter that accounts for temperature dependance

of (H p)

G

parameter that accounts for temperature dependance

m

of (A p G )

P

G

_k parameter that accounts for temperature dependance

of (R k) I(d

k/2) approximately equals 1

1 radius of thermocouple welding

r

=

rth,I ' the first estimate of droplet radius (m)

r w = r th, 2 = r

I

<p w (m)

T measured mean temperature level (oG)

avg_k

T saturation temperature (oG)

sat

v mean vapor velocity as estimated from mesured

vapor

superficial velocities

(m/s)

vd droplet velocity, either measured directly (section 2.2)

x

B

w

or estimated from v _vapor , r and correlations

(m/s)

=

xo , the location of evaporation as measured from the thermocouple welding (m)

= _Df

I

rw

=

_2Rf

I

rw (-)

=xl14at _x

Figure 14 (overleaf)

Flow chart for the calculation of droplet size

In calculation block 1, a value for r = rth,I is determined from equation

(2.3) with the aid of the parameter G

k that depends on Gd, Gm and Gk• In calculation block 2, <PIo is iteratively determined by minimizing

BEGIN CALCULATION BLOCK 2 v vapor

...

B

...

tx

..

B,/B2 W tx INPUT a

...

CALIBRATION CURVE

f3

MEASURED OR 1-_ _ _ _ O...:f _ _ _ 1--. ESTIMATED r jlPw Gk dk CALIBRATION CURVE Gm

...

CALIBRATION Tav9k CURVE Gd CALIBRATION Tsat CURVE

where each individual quadratic factor was given a lower bound to minimize these three factors together.

The three input parameters B₁/B₂, B2/B3 and Bl/B3 allow for the computation

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

B was set equal to 4a (l / ( A d

k). Since the droplet film diameter, D

f, 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

a

was kept constant

during each iteration cycle of block 2. The value of

a

can be adjusted once

(see figure 14).

The ~arying of ~ by

±

70

%

resulted in a spread of

±

10

%

in the calculated values of q,10.

Only if distinct, 'sharp' thermocouple readings are being analyzed, i.e. only i f x ::;; .; 4a T or w ::;; 0,62, and i f the drople is found to evaporate on the thermocouple welding, i.e. if x

<

0,95 1 , the effective diameter, d

k, is adapted to some value larger than dq, and close to 2· l, 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 d_k is equal to dq, at every

location. The manufacturing of these cylindrical thermocouples is possible (see chapter 4).

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, overleaf).

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 4). 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 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)

THERMAL VOID PROBE

thermocouples in a ceramic tube

mounting unit

unit mounted in a test tube

A TVP unit encompasses two thermocouples. Each electronically conditioned in the way depicted in figure

thermocouple is

16. The optional

thermocouple

(T/e

₂

)

is at constant temperature, and is used to increase the

sensitivity of the device and hence the measurements accuracy if the TVP is operated at high system pressures.

Tic

Opli0f131

fllli

Figure 16

Local puwer suppl y

I

---··I~---Isothermal Dlane

f:;oi,jl.lon I);)rricr

Swit.ch

-board

Thermo void probe electric conditioner

The conditioner has an automatic ice point compensator.

A TVP unit can be mounted on locations where 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-diameter set up

Droplets were produced by adapted capillary tubes, and superheated steam heated up a chromel-alumel thermocouple made of bars with a diameter of 10 mm. The local system pressure was about 1 bar.

just before they were released from a capillary tube.

Droplet sizes and velocities were measured with high speed

cinematography (frame speed 750 pictures per second), and weight of droplets was measured with the aid of a precision balance after gathering a number of droplets on ice that prevented evaporation after the hitting of the thermocouple bar.

Type II: Real TVP-size set up

Droplet detection thermocouples of a thermo void probe (see section 2.1.4.1) were catching droplets produced by liquid atomizers in superheated steam. System pressures ranged up to 10 bar.

Droplet sizes and velocities were measured with high speed

cinematography 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 .. ErrQr estimates. Relative reading errors for input parameters amounted to 3 % for S(B. / B.) / (B. / B.) 1 J 1 J 4 % _{for S(}

_e

_{tx) /}

_e

_tx 1 % for Set ) / t _{x x} 1 % for S(T avgk) / Tavgk

Typical relative errors for calculated parameters amounted to

7 % for S( a.) / a.

2

%

for droplet size determination by measuring weight loss 9 % for S( <P_{10 ) /} <P 10

10

%

for S( <p ) / <p

w w

2.1.4.4 Large diameter TC verification measurements. A typical temperature

history measured with the 10 rom diameter thermocouple bar (type I

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

Type I measurements allowed for repetition of droplet impingement and keeping constant of droplet size, and hence for

t

u o 20 Figure 17 60 100 .140 Time (5)

----0.-Specimen of temperature history; 10 mm thermocouple

reduction of the error in the mean value of the calculated droplet radius. 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

r _welg . ht = 1.46 ± 0,02 mm

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

~2~.~1~.~4~.~5~~T~V~P __ ~v~e~r~i~f~i~c~a~t~i~0~n~~m~e~a~s~u~r~e~m~e~n~t~s~. 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 of 0,04 0,07 mm and two

thermocouples diameter.

t·

120 u 0 OJ '-' ::J ....,

'"

'-' OJ 0. 117 E OJ !--114 0,5 Figure 18 I

I

I

,

i

I

I I

il

\

II

11 II

,( \ I'.

II

! I ! r \

I

I 1 :

I

! !

I

j , • : • J I ' \ I) : r i r ,

iIi

j

I

I (

!/

\

I

I

~

I

L

iI

I

v

II 11 I; .J

i

i

i \ , I ,; 'I

Ii

il

,

~

!

I

\

1 ,5 2 Time (5) ..

Specimen of temperature history; 0,1 mm thermocouple

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

drop

e

tx varied

3 K and 9 K for

radius of 0,04 ± 0,003 mm was calculated. The temperature

between 20 K and 71 K for the smaller couple, and between

the larger couple. 20 to 40

%

of the total evaporation took place at the

smaller thermocouple.

leading

This experiment was repeated without the smallest, I 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 unsuccesful.

2.1.4.6 Void fraction estimation. Figure 18 clearly shows how the vapor void

fraction, € , can be estimated from

(2.36) 1 - €

=

(l/A

t ).

(N(T)/T).(1/vd).4 n

r~Vg

1

3

with At = effective scattering cross sectional area (nd~

14);

N(T)

1 T

= number of hits per second, which can be determined directly from a thermocouple reading (see figure 18;

vd

=

mean droplet velocity;

r _avg

=

mean droplet radius.

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. Another way to measure droplet velocities 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 Measurement difficulties at annular-mist flow. If 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 phase of a thermocouple. High speed cinematographic films 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

The smallest, leading 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

50 100 Time (5) .. Figure 19

Specimen of temperature history; plateau reheating (arrows)

The largest thermocouple wire radius prefarably should be about eight times as large as a typical droplet radius; often several test runs are needed with differently sized thermocouple wires to establish a typical droplet radius and hence the appropriate wire diameter range;

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

2.2 ON DROPLET VELOCITY MEASUREMENTS

Figure 20 clearly shows that under certain test conditions droplet

velocities can be measured with a thermo void probe by a time-of-flight method. The experimental set up corresponding to figure 20 was described in section 2.1.4.2. 122

t

910.1 mm °c 116 a 4 ₈ Time{ms} -Figure 20

Droplet velocity measurement with time-of-flight method

No velocity measurements were possible if the thin thermocouple had about the same size as a droplet.

For comparison, it was attempted to measure droplet velocities with a different technique as well. A 35 mW He-Ne backscatter laser velocitometer was applied. The type of analyzing equipment used demanded a regular, continuous stream of nearly equally sized droplets at about the same

velocity, that unfortunately could not be produced by the liquid atomizers

used. Since no other, more appropriate electronic equipment could be

Boonekamp, 1984; Van Bommel, 1986).

Droplets smaller than a thermocouple were found to hit the couple and then either to bounce away or to stick to it (see figure 21). Although it

demonstrated the operating limits for velocity measurements, the phenomenon

of catching of droplets is the working principle of droplet detection thermocouples of course (see section 2.1).

Van der Geld (1986) discusses a simple model to account for the friction

losses a droplet experiences if it slides over a very thin thermocouple. Resulting velocity losses of a droplet with a radius of 0,02 mm and undisturbed velocity of 2 m/s were calculated in the range 14-52 cm/s.

Figure 21 (overleaf)

t o 0,59 1 ,18 1,77 2,36 ms

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

3.1 MODELING ASSUMPTIONS AND CORRELATIONS

The calculation model aims at calculating post-dryout heat transfer and flow development in vertical evaporator tubes, and at calculation of droplet size and droplet velocity at point of dry-out by minimizing differences between measured and calculated wall temperatures.

The following assumptions are made

- stationary flow and heat transfer;

rotatoric symmetry around the tube axis;

- liquid and vapor mixtures are flowing vertically upward;

- mist flow with highly dispersed droplets occurs directly downstream of the point of dry-out;

- fluid is at saturation temperature;

- calculations at every axial location can be performedon droplets distributed uniformly over a cross-section, and with a mean droplet radius;

- radiative heat transfer between wall and vapor can be neglected;

- convective heat transfer between wall and vapor is adequately described by the following correlation (Moose and Ganic, 1982):

(3.1) = (A ID) 0 023 ReO,8 Pr 1/3 ( I )0,14

ex wv v ' v v \.l v \.l v , w

Rev G xa D l\.lv ; Red = vd(vv - vl)/\.lv Pry

=

_{\.lv Cp/ Av'}

D represents the inner diameter of the tube, and d the droplet diameter; - direct contact heat transfer between tube wall and droplets is adequately described by (Filonenko, Petukhov Popov; see Webb, 1971):

(3.2a) qwd 0,106 G H (1 - x )(v _{a v}

Iv

_l )( Pl/p ) _v f exp(l

+

(3.2b) f

=

(1,58 In Re - 3,28)-2

v

G denotes the total mass flux

(kg/m

2

s)

- (Tw/T

sat)2 )

and T the wall temperature. The

w

exponential factor in equation (3.2a) is the evaporation efficiency;

by the following correlation (Deruaz and Petitpain, 1976): (3.3a) (3.3b) (3.3c) q

=

cr

(T 4 - T 4 t) £ £ 1/( £ 1 + £ (1 - £ 1) r w sa w w £1

=

_{1 - exp(-2,22 (D/d) (1 - (1 \ (v/vl )(} P / P 1)(1

+

- x )/x)- ) ) a a £

=

0,66 w

- heat transfer between droplets and vapor is adequately described by the following correlation:

(3.4) q d = (A /d) 2(1

+

0,276 RedO,S Pr 1/3).(T - T t)

v v v v sa

- the friction coefficient for droplets can be obtained from (White, 1974):

(3.5) Cd = 24/Re + 6/(1 +

/R;)

+ 0,4 (Re < 2.105)

- vapor velocity increases only gradually in axial direction,

whence droplet accelerations relative to the vapor phase need not be considered:

dv is positive and follows from

(3.7) (3/4) Cd (p v/ p 1) dv' /d

+

9,81 (p / PI - 1) =

°

- first order Eulerian integration is permitted

(3.8a) (3.8b) (3.8c) dx x (z

+

dz)

=

x (z)

+

dz • .-a_d (z) a a dd z d(z

+

dz)

=

d(z)

+

dZ'--d (z) dz Z

vI

=

dt

3.2 MODELING EQUATIONS

The temperature viscosity close

conductivity, A

dependencies of vapor mass density,

P ,

vapor dynamic

v

to the wall, 11 ,vapor heat capacity, C , vapor heat

vw pv

v and surface tension,

a

,have to be accounted for. The

essentially one-dimensional flow problem had therefore to be solved

numerically. Equations that were needed for the gradients in axial direction are derived in this section.

Almost by definition of the actual steam quality, x

a' the following

expression for the void fraction, , holds:

(3.9)

_e:

₌

_{G xa / P v v v}

while the slip factor is determined by

(3.10) v / vI

=

(x / (I-x))

I-v a a

e:

This equation together with equations

(3.5)

through to

(3.7)

determine the

velocities Vv and vI'

The equilibrium steam quality, x , is the quality

e

all vapor superheating energy would have been used

I t is clear that dx

(3.12) ~

=

4qt / (D G H)

that would be present if evaporating droplets:

with the total heat flux given by qt.

=

qwv+qwd+qr. Differentiation of

equation (3.11) yields:

(3.13)

~ddTz

=

(~ddX

_z - (H + C (T _{pv v -} T _{sat dz}

~))

from which Tv can be determined if

~:a

is known.

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

(3.14a) n (1 - £ ) / ( nd'/6) v

Differentiation of (3.14b) yields:

In first approximatin only gradual changes of In(n vI PI) are considered, and the gradient of x is approximated by

a

dx

(3.15b) '(i'? = - (1 _ x)

1.

dd

a d dz

In second approximation, equation (3.15a) is calculated, although the second term on the RHS of (3.15a) in all cases prooved to be very small as compared to the first term.

In the same way the gradient of the droplet diameter is treated. Difference

between in- and out flux of iquid in some part of the tube are due to

evaporation:

In first approximation follows dd

(3.17) dz = _{4qt d / (3H D G (1 - xa ) )}

From equations (3.1) through to (3.17) the boundary condition T (z

=

0) =

v

T t' given values of D, T t' _{sa sa}

x(z = 0) and d _o = d(z

=

downstream of the point bf temperatures are known.

Figure 22

qt' G and dz, and from start values of Xo =

0), the vapor temperatures and heat fluxes dry-out can now be calculated if the wall

(overleaf)

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

flow chart fOf the calculation 01 lhe drQPfel dIameter at dry-aul flom measurements with thermocouples moonled exec-rnally at Ihe wall

INPUT: PRESSURE

CAlCUtA liON OF A REGRESSIVE CURVr rOR THE WALL TEMPERATURE

Z: AXIAL COORDINATE FROM 'DRVOUr

Z 0

MASS fLUX DENSITY WALL HEAT FLUX DENSITY MEASURED WALL TEMPERATURES STARTVALUE MASS OUALITY DRYOUr, XO STARTVALUE DROPLET DIAMETER DRYOU1, DO

PROPERTIES OF THE VAPOR PHASE AS A FUNCTION OF PRESSURE AND VAPOR TEMPERATURE VELOCITIES OF VAPOR AND L10UID PHASE

HEAT TRANSPORT BY DIRECT CONTACT, WALL-DROPLETS HEAT TRANSPORT BV RADIATION, WALL-DROPLETS HEAT TRANSFER COEFFICIENT BY CONVECTION, WALL-VAPOR

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

OUTPUT ACTUAL MASS OUAl/TV EOUILIBRIUM MASS OUALITY VOID FRACTION DROPLETS DIAMETER NUMBER OF DROPLETS VAPOR VELOCITY

DROPH rs vn OellY

VAPO'~ 1 f. Mf>[HA f UHf HEAT TRAN!if£fl FLUx DfNSI1I{:S - CONVECTION WALL·VAPOR

CONVECTION VAPOR-DROPLE TS DIRECT CONTACT WAll-DROPLETS