Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 10.6.2 - Measurement of Evaporation Coefficient of Water Using Nonlinear Sound Wave
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 Material Information
Title: 10.6.2 - Measurement of Evaporation Coefficient of Water Using Nonlinear Sound Wave Multiphase Flows with Heat and Mass Transfer
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Nakamura, S.
Yano, T.
Watanabe, M.
Fujikawa, S.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: evaporation coefficient
sound wave
sound resonance
 Notes
Abstract: We propose a new method for measurement of the evaporation coefficient using nonlinear sound resonance experiment based on a theory of molecular gas dynamics. The evaporation coefficient is determined from the amplitude of the second harmonics generated by the nonlinear effect of resonant sound wave in a cylindrical space bounded by a sound source and a gas-liquid interface. We demonstrate the applicability of the method by showing the results of test experiments executed under several initial temperature conditions. We find that the amplitude of the second harmonics at sound resonance decreased with the decrease in the initial temperature.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Bibliographic ID: UF00102023
Volume ID: VID00266
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: 1062-Nakamura-ICMF2010.pdf

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Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


Measurement of Evaporation Coefficient of Water Using Nonlinear Sound Wave


S. Nakamura*, T. Yanot, M. Watanabe* and S. Fujikawa*


Division of Mechanical and Space Engineering, Hokkaido University, Sapporo, 060-8628, Japan

Department of Mechanical Engineering, Osaka University, Suita, 565-0871, Japan

nshigeto@mech-me.eng.hokudai.ac.jp

Keywords: evaporation coefficient, sound wave, sound resonance





Abstract

We propose a new method for measurement of the evaporation coefficient using nonlinear sound resonance experiment based on a
theory of molecular gas dynamics. The evaporation coefficient is determined from the amplitude of the second harmonics
generated by the nonlinear effect of resonant sound wave in a cylindrical space bounded by a sound source and a gas-liquid
interface. We demonstrate the applicability of the method by showing the results of test experiments executed under several initial
temperature conditions. We find that the amplitude of the second harmonics at sound resonance decreased with the decrease in the
initial temperature.


Introduction


can be shown as


When a space is filled with a sample liquid and its saturated
vapor, the evaporation or condensation takes place at a
vapor-liquid interface depending on conditions of vapor
temperature, vapor pressure and liquid temperature. Such
phase change phenomena, we may see in our daily life, are
applied in engineering fields such as heat exchanger.
However, we cannot predict the mass, momentum and
energy transport across the interface even when the vapor
pressure, vapor temperature and liquid temperature are
known. This is because the boundary condition at the vapor
liquid interface contains two unknown parameters; the
evaporation coefficient and the condensation coefficient.
Since evaporation and condensation are induced by
nonequilibrium behaviors of molecules, the fluid dynamics
scheme based on the assumption of local equilibrium is not
applicable to the phase change problem. For this problem,
molecular gas dynamics is effective. By solving the
Boltzmann equation, which is the governing equation of
molecular gas dynamics, the velocity distribution function
can be known; thus one can obtain vapor conditions, such as
the vapor pressure, the vapor density. However, the kinetic
boundary condition at the vapor-liquid interface (Ishiyama,
Yano & Fujikawa 2005) is required in order to solve the
Boltzmann equation, and this condition contains the
evaporation and condensation coefficients. These
coefficients cannot be derived from the scheme of molecular
gas dynamics. They should be evaluated in a molecular
dynamics simulation or measured in an appropriate
experiment.
The kinetic boundary condition at the vapor-liquid interface


f"out cp*+(1 *)a f* (i vj)>0


where fout is the velocity distribution function for
outgoing molecules, ae is the evaporation coefficient, ac is
the condensation coefficient, p* is the saturated vapor
density for liquid temperature T,, o is defined by the
velocity distribution function for incident molecules and
obtained by solving the Boltzmann equation, f* is the
Maxwell distribution, ( is the velocity of molecules and vi
is the moving speed of the interface in the direction normal
to the interface. The evaporation and condensation
coefficients are defined by molecular mass fluxes at the
interface as follows.


Jevap
Jout


Jcnds
ac Jco
Jcoll


where Jevap is the evaporation mass flux, which depends
only on the temperature of liquid film and independent of
the vapor conditions, Joi is the incident mass flux, Jcnds is
the condensing mass flux, Jref is the reflecting mass flux,
Jout is the outgoing mass flux, which is the sum of Jevap
and Jref, and the asterisk indicates the value for the
equilibrium state at the temperature in consideration.
Clearly, Jut depends only on the temperature of liquid
film. From the definitions of evaporation coefficient and
condensation coefficient, they can vary from 0 to 1, and
they become equivalent in the equilibrium state for a
specified temperature.





Paper No


The aim of this study is to measure the evaporation
coefficient for water using a new nonlinear sound resonance
experiment.

Measurement Method

The measurement of the evaporation and condensation
coefficients has been attempted by many researchers and
scientists (Marek & Straub 2001). The difficulties of
measurements have brought about a large dispersion of
experimental data. Recently, a reliable experimental method
using shock tube has been established by Kobayashi et al.
(Kobayashi, Watanabe, Yamano, Yano & Fujikawa 2008) for
the measurement of condensation coefficient with the use of
data of evaporation coefficient evaluated by molecular
dynamics simulations. On the other hand, an appropriate
measurement method for the evaporation coefficient has not
been established. In this research, we propose a new
measurement method using nonlinear sound wave in a space
between the sample liquid film and a sound source filled
with the saturated vapor of sample liquid. The point of our
method is that the sound induces a sufficiently small shift
from an initial vapor-liquid equilibrium and hence the
difference of the two coefficients may be negligibly small.
That is, all we have to do is to measure only one unknown
parameter, the evaporation coefficient.
As shown in Fig. 1, the Mll. c of sound source is set
parallel to the vapor-liquid interface. Then, the sound source
starts oscillating harmonically. A plane sound wave
propagates through the vapor space and it reflects at the
interface. After a few cycles, a plane standing sound wave is
formed between the sound source and liquid ,!I l!.c When
the distance between the sound source and liquid film is
equal to an integral multiple of half wavelength of the sound,
the amplitude of standing sound wave is amplified by the
sound resonance and several harmonic components are
excited. In this method, by utilizing the second harmonic
component, we can improve the measurement accuracy,
being free from the effect of electromagnetic noise. At the
liquid i!l I!.l very weak evaporation and condensation take
place cyclically by pressure variation of sound resonance
and a certain amount of sound wave incident on the
interface is absorbed into the liquid. We measure the
pressure amplitude at the sound resonance for several
temperature conditions and determine the evaporation
coefficient with the help of theoretical analysis using
molecular gas dynamics.

Experimental Procedure

Figure 1 shows the schematic of experimental setup. It
consists of the sound source, a sound receiver and a liquid
holder. This experimental apparatus is set up inside a
vacuum vessel, and this vessel is placed inside a
thermostatic chamber. By evacuating the noncondensable
gas from the vacuum vessel, we can conduct measurements
near ideal state where the space filled only with the sample
liquid and its saturated vapor. Also by using the thermostatic
chamber, we can control the temperature of experimental


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

system. The receiver is placed inside the liquid holder and
the liquid film is formed on the ll. c of receiver. The
sound source is mounted on a microstage and a rack and
pinion stage. The distance between the sound source and
liquid film i ll!. is controlled by this system to attain a
resonant condition. The microstage is connected to a
stepping motor and this enables us to control the distance
between two Mt!.ic.. electrically. The sound source
installed in this system is Langevin type transducer (Fuji
Ceramics Co., FBL28502HA). PVDF film (thickness: 0.25
mm, diameter: 54 mm) bonded to an acrylic plate
(thickness: 10 mm, diameter: 54 mm) is used as the
receiver.


Figure 1: Schematic diagram of experimental setup.

The experimental procedure is as follows. Firstly, the
receiver is put inside of the liquid holder and the sound
source is fixed to the microstage. We set the scale of
microstage as 1 mm and set the distance where obtained
maximum output voltage by the rack and pinion stage. At
this time, the degree of liquid holder is additionally adjusted
to attain parallelism of i !. c. of sound source and liquid
film. Secondly, pure water is poured on the Mllh.~ of
receiver to form liquid water film. The thickness of liquid
film is thin compared with the wavelength of sound wave
under water. This thickness is measured as the difference of
scale reading of rack and pinion stage before and after
liquid film forming using the effect of sound resonance. In
this paper, we show some results for the film of about 6 mm
thick. In the case of driving frequency 28 kHz and liquid
temperature 305 K, the wavelength of sound wave under
water is about 53 mm. After forming of water liquid film,
we shut the lid of vacuum vessel which contains the
experimental apparatus and start the evacuation of
noncondensable gas from inside of vacuum vessel using
rotary pump. The evacuation is continued until the internal
pressure of vacuum vessel reaches about 2.4 kPa. The
estimation of gas leakage is performed by monitoring the
temperature of liquid film and internal pressure of vacuum






Paper No


vessel. In this paper, the leak level is defined as the
difference between the measured internal pressure and the
saturated vapor pressure obtained from Antoine equation at
the temperature of liquid film. Figure 2 shows variations of
internal pressure P, saturated vapor pressure of water
obtained from liquid film temperature Pw, the leak level
P-Pw, liquid film temperature Tw and vapor temperature
between the sound source and liquid film Tv from the time
when the evacuation is terminated.


0 20 40 60 80 100
Time elapsing from evacuation end (min.)


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

measurements shown in the next section, we ignore the
effect of leak level and we starts measurements 90 minutes
after the evacuation termination, when an almost steady
state is attained in the vessel.
The distance between the sound source and the liquid film is
changed from a half wavelength to one wavelength by every
0.05 or 0.1 mm and we measure the output signals of
receiver for every distance. This control of distance is
realized by the stepping motor. After the measurement, we
retrieve the fundamental frequency (driving frequency)
component f, second harmonics component 2f and third
harmonics component 3f from measured output signals
through frequency analysis. By conducting the
measurements including two resonant conditions and
evaluation of measurement distance between two amplitude
peak points, we can confirm that the increase of output
amplitude at these two points is the effect of sound
resonance. We convert the output voltage amplitude to
pressure amplitude based on the result of calibration test
which is conducted under atmospheric, solid wall and room
temperature conditions. We use Kistler pressure sensor
(Type 701A) for this calibration.


250
120


Figure 2: Variations of internal pressure and temperatures
of vacuum vessel.

In Fig. 2, the abscissa is the time elapsing from evacuation
termination, the left ordinate corresponds to pressure P, Pw,
P-P, and the right ordinate corresponds to temperature T,,
Tv. From Fig. 2, both pressures and temperatures increase
with time. By the evacuation of gases in the vacuum vessel,
the liquid ,il l!. c attains a boiling state and the temperature
of liquid film decreases by the latent heat. Hence the
temperature of inside the vacuum vessel is different by
position. After the evacuation termination, the temperature
of liquid film is increased gradually as shown in Fig. 2
because of heat supply from surrounding area into liquid
film. Then the pressure inside the vacuum vessel also
increases. Although the pressure and temperature vary with
time, the measured pressure P agrees well with the
estimated saturated pressure Pw. Furthermore the difference
between temperature Tw and vapor temperature Tv is
negligible for more than 100 minutes. From these results,
the experimental system attains an equilibrium state
immediately after the end of evacuation and after that this
equilibrium state is maintained. Meanwhile, the leak level
increases with time elapsing. The rate of leak is about 2 %
as against saturated vapor pressure at 90 minutes. The effect
of noncondensable gas on the evaporation and condensation
is studied by Maerefat et al. iM..!.lcc.i, Fujikawa &
Akamatsu 1990). They mentioned that the rate of leak level
should be at most 0.1 % to eliminate the effect of
noncondesable gas. Therefore, the present leak level may be
insufficient and some influences on measurements are
concerned. However, we set performance of test
measurements as the aim of this time. Thus for this test


40


30


20


io"


-- o0
0 5 10 15 20
Input voltage (Vrms)


2

1.5
o
a,
21
1..
&


5 10 15
Input voltage (Vrms)


Figure 3: Results of calibration test under the conditions
of atmospheric, solid wall and room temperature: (a) the
receiver, (b) Kistler pressure sensor.


S Tw(K)
A Tv(K)
P(kPa)
Pw(kPa)
x P-Pw(kPa)
"_. _- L__"2T'.T2_ -- 't I .. ...... .


1






Paper No


Figure 3 shows the results of calibration test. The distance
between the sound source and the liquid film is a half
wavelength. Input voltage changes from 1 to 20 Vrms.
Driving frequency f is 28.2 kHz. The abscissa is input
voltage, the left ordinate is amplitude of f component and
the right ordinate is 2f and 3f components, respectively.
Figure 3(a) shows receiver output and Fig. 3(b) shows
output of Kistler sensor.
Finally the evaporation coefficient is determined by the
pressure amplitude of 2f at the resonant condition for half
wavelength with theoretical analysis using molecular gas
dynamics. This theoretical analysis is conducted on the
same system as experiment. The relation between the
evaporation coefficient and the pressure amplitude for
methanol obtained from the linear analysis performed by
Inaba et al. (Inaba, Fujikawa & Yano 2008) is shown in Fig.


0c


0 0.5 1
Evaporation coefficient ae


Figure 4: Relation between the evaporation coefficient and
pressure amplitude for methanol (linear analysis). The Po is
initial saturated vapor pressure, the To is initial vapor
temperature and the Ma is Mach number.

The abscissa is the evaporation coefficient and the ordinate
is nondimensional pressure amplitude at the sound source.
From this figure, the pressure amplitude of sound wave near
a resonant point decreases and becomes almost equivalent to
the amplitude of sound source oscillation when the
evaporation coefficient becomes unity. By using the
assumption of plane sound wave, the theoretical analysis
contains the nonlinear effect induced by the sound
resonance can be performed accurately. We are going to
finally use the theoretical result containing nonlinear effect
for determination of the evaporation coefficient of water.

Results of Test Experiment

In this section, we show the several results of test
experiment for several conditions of initial temperature. The
rate of leak level is about 2 % and the sample solution is
water. We start the measurements from equilibrium state.
Ishiyama et al. performed molecular dynamics simulation to
determine the evaporation coefficient of water (Ishiyama,
Yano & Fujikawa 2004). According to their study, the
evaporation coefficient of water is 0.99 for 300 K and it


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

decrease with the increase in the liquid temperature. In the
measurement method using nonlinear sound wave, the
amplitude of standing wave will decrease with the decrease
in the liquid temperature, because the saturated vapor
pressure (initial pressure) depends only on the liquid
temperature. Therefore, more accurate measurement is
required for lower temperature conditions. In this test
experiment, we set the liquid temperature from 293 to 308
K and conduct the evaluation of receiver every 5 K. Other
experimental condition is as follows. The driving frequency
is 28.3 kHz, the input voltage is 20 Vrms and the thickness
of initial liquid film is about 6 mm.



25 2f=56.6 kHz
0.8

20 -
I- 0.6

l ^l l l 0.4 ll
0 101-3 469





0 1 2 3 4 5 6 7 8 9
Distance from starting point (mm)

Figure 5: Output amplitude of receiver for initial liquid
temperature 303 K, initial pressure 4.45 kPa.

Figure 5 shows a typical result under the condition of 303 K
(initial liquid temperature) and 4.45 kPa (initial pressure).
The saturated vapor pressure of water calculated from liquid
temperature is 4.37 kPa and the rate of leak level is about
2 % on the measured pressure. The abscissa is the distance
from starting point, the left ordinate is voltage amplitude of
f component and the right ordinate is voltage amplitude of
2f component. We can see the two peaks for 2f shown as a
red colored line near 0.6 and 8.2 mm. These peaks are due
to the sound resonance because the distance between these
peaks 7.6 mm is equal to the half wavelength calculated
using experimental condition. Meanwhile, for the f
component shown as a black line, the amplitude curve is
almost flat including the resonant conditions. This is the
effect of electromagnetic noises generated from the sound
source. As the result, the driving frequency component
obtained from the receiver is always dominated by the
noises. For the second harmonics component, the effect of
electromagnetic noises is small compared with those for f
component. Then we can obtain the output caused by the
pressure variation of sound wave. This is one reason why
we utilize the second harmonics in this measurement
method.
The relation between the initial temperature of liquid film
and voltage amplitude of second harmonics at the first peak






7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


is shown in Fig. 6. The abscissa is liquid temperature and
the ordinate is voltage amplitude of second harmonics
component. Each red circle corresponds to each measured
data. All data obtained from the almost same experimental
conditions except the initial temperature and initial pressure.
From Fig. 6, we can confirm reproducibility of
measurement. If we set the same initial temperature, then we
can basically obtain the same output voltage amplitude. The
voltage amplitude of second harmonics is converted into
pressure amplitude using the result of calibration test. In the
case of initial temperature 288 K, pressure amplitude is
about 9 Pa, about 15 Pa for 293K, about 45 Pa for 298 K
and about 60 Pa for 303K.
Typical results under the condition of initial temperature 288
K, 293 K and 298 K are shown in Fig. 7(a), 7(b) and 7(c),
respectively. In the case of 288 K, about 20 % of the
amplitude of second harmonics at sound resonance is
composed of electromagnetic noises. Thus some effects are
expected for 288 K.


0. 0.8

S0.6

S0.4


0O














275 280 285 290 295 300 305 3
Initialtemperature of liquid film (K)


Figure 6: Relation between the initial temperature and first
peak amplitude of 2f.


S(a) -- f=28.3 kHz
-25--- 2f=56.6 kHz
25 0.8

S20 -
0.6
15
-0.4
$lo

5 0.2



0 1 2 3 4 5 6 7 8 9
Distance from starting point (mm)


(b) 28.3 kHz
25 2f-56.6 kHz
2

> 20

15
-
~~ l
|l0-




0 1 2 3 4 5 6 7 8
Distance from starting point (mm)

30
(c) f-28.3 kHz
25 2f-56.6 kHz


>20

15


10-


0 1 2 3 4 5 6 7
Distance from starting point (mm)


8 9


1


0.8


0.6 ^


0.4


0.2


0












0.4
1







0.8


0.6




0


10 Figure 7: Output amplitude of receiver for (a) initial liquid
temperature 289 K, initial pressure 1.91 kPa, (b) initial
liquid temperature 294 K, initial pressure 2.55 kPa, (c)
initial liquid temperature 299 K, initial pressure 3.41 kPa.

Conclusions

To measure the evaporation coefficient contained in the
kinetic boundary condition of molecular gas dynamics, we
have proposed the new measurement method. This is the
integration of nonlinear sound resonance experiment and
Theoretical analysis using molecular gas dynamics. To
Demonstrate the applicability of this method, we have
conducted the test experiment under the condition of initial
P liquid film thickness 6 mm. The voltage amplitude of
second harmonics 2f has been measured using the receiver
Placed under the liquid water for each distance. As a result,
the increase of amplitude of 2f by nonlinearity of sound
resonance has been confirmed when the distance between
the sound source and the liquid film is equal to a half
wavelength and one wavelength. By utilization of second
harmonics component, it has been clarified that we can
avoid the influence of electromagnetic noises. Also,
variation of amplitude of 2f at the sound resonance for


Paper No






Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

several initial temperature conditions has been examined. As
a result, the decrease of amplitude of 2f by the decrease of
initial liquid temperature has been confirmed at the sound
resonance. We are going to improve the leak level of
vacuum vessel and conduct the experiment for the
determination of evaporation coefficient of water.

Acknowledgements

This work is supported in part by the Japan Society for the
Promotion of Science, Grant-in-Aid for Scientific Research
(A) 21246031. The authors would like to express their
deepest gratitude to this grant.

References

Ishiyama T., Yano T. and Fujikawa S., Kinetic boundary
condition at a vapor-liquid ii!!.il:c Physical Review
Letters, Vol.95, No.8, Art.No.084504, 2005

Marek R. and Straub J., Analysis of the evaporation
coefficient and the condensation coefficient of water,
International Journal of Heat and Mass Transfer, Vol.44,
pp.39-53, 2001

Kobayashi K., Watanabe S., Yamano D., Yano T. and
Fujikawa S., Condensation coefficient of water in a weak
condensation state, Fluid Dynamics Research, Vol.40,
No.7-8, pp.585-596, 2008

Maerefat M., Fujikawa S. and Akamatsu T.,
Non-equilibrium condensation of a vapor-gas mixture on a
shock-tube endwall behind a reflected shock wave, Fluid
Dynamics Research, Vol.6, pp.25-42, 1990

Inaba M., Fujikawa S. and Yano T., Molecular gas dynamics
on condensation and evaporation of water induced by sound
waves, Rarefied Gas Dynamics, AIP Conference
Proceedings, Vol.1084, pp.671-676, 2008

Ishiyama T., Yano T. and Fujikawa S., Molecular dynamics
study of kinetic boundary condition at an interface between
polyatomic vapor and its condensed phase, Physics of Fluids,
Vol.16, No.12, pp.4713-4726, 2004




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