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. Document status and date: Published: 01/01/1986 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected]providing details and we will investigate your claim. Download date: 12. Apr. 2020
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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.
Document status and date:Published: 01/01/1986
Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publication
General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverne
Take down policyIf you believe that this document breaches copyright please contact us at:[email protected] details and we will investigate your claim.
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
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
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)
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
- 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).
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