Temperature effect on phase transition radiation of water

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Temperature effect on phase-transition
radiation of water
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M. Q. Brewster1
Fellow ASME
Department of Mechanical
Science and Engineering,
University of Illinois at Urbana-Champaign,
Urbana, IL 61801
e-mail: brewster@illinois.edu
K.-T. Wang
Urbana, IL 61801
W.-H. Wu
Urbana, IL 61801
M. G. Khan
Urbana, IL 61801
Temperature Effect
on Phase-Transition
Radiation of Water
Infrared radiation associated with vapor-liquid phase transition of water is investigated
using a suspension of cloud droplets and mid-infrared (IR) (3–5 lm) radiation absorption
measurements. Recent measurements and Monte Carlo (MC) modeling performed at
60
C and 1 atm resulted in an interfacial radiative phase-transition probability of
5 Â 10À8
and a corresponding surface absorption efficiency of 3–4%, depending on
wavelength. In this paper, the measurements and modeling have been extended to 75
C
in order to examine the effect of temperature on water’s liquid-vapor phase-change radi-
ation. It was found that the temperature dependence of the previously proposed phase-
change absorption theoretical framework by itself was insufficient to account for
observed changes in radiation absorption without a change in cloud droplet number den-
sity. Therefore, the results suggest a strong temperature dependence of cloud condensa-
tion nuclei (CCN) concentration, i.e., CCN increasing approximately a factor of two from
60
C to 75
C at near saturation conditions. The new radiative phase-transition proba-
bility is decreased slightly to 3 Â 10À8
. Theoretical results were also calculated at 50
C
in an effort to understand behavior at conditions closer to atmospheric. The results sug-
gest that accounting for multiple interface interactions within a single droplet at wave-
lengths in atmospheric windows (where anomalous IR radiation is often reported) will be
important. Modeling also suggests that phase-change radiation will be most important at
wavelengths of low volumetric absorption, i.e., atmospheric windows such as 3–5 lm and
8–10 lm, and for water droplets smaller than stable cloud droplet sizes (20 lm diame-
ter), where surface effects become relatively more important. This could include unacti-
vated, hygroscopic aerosol particles (not CCN) that have absorbed water and are
undergoing dynamic evaporation and condensation. This mechanism may be partly re-
sponsible for water vapor’s IR continuum absorption in these atmospheric windows.
[DOI: 10.1115/1.4026556]
Keywords: phase-transition radiation, extinction, transmission, scattering, evaporation,
condensation, water vapor continuum
1 Introduction
Satellite and ground observations of shortwave (SW) solar and
longwave (LW) infrared radiation have made clear the importance
of clouds and water vapor in Earth’s global energy balance and
their tremendous variability [1]. It has been realized that uncer-
tainties associated with different cloud models can change predic-
tions of global warming in global climate modeling significantly.
For example, global average surface temperature could increase
anywhere from 2
C to 5
C, depending on cloud parameters and
behavior assumptions [2]. It has long been known that water
vapor, the most important greenhouse gas in the atmosphere, has
unexplained, anomalously high continuum absorption and emis-
sion of IR radiation in the atmospheric window of 8–12 lm [3].
Hence, it has been realized that a more accurate understanding of
cloud and water-vapor radiation is needed.
The need to upgrade cloud radiation modeling from 1D to 3D
has recently been emphasized and efforts have been made in this
regard [4]. Nevertheless, even 3D radiative transfer modeling
appears to be insufficient to describe observed cloud radiative
transfer—which includes anomalous absorption and emission—and
do so with the accuracy required for meaningful global climate
modeling. It has therefore been suggested that there may be pieces
of basic cloud radiation physics still missing [4]. In an effort to
improve understanding of basic cloud radiation physics, we have
begun and are continuing a study of water’s phase-change radia-
tion [5]. This approach has been taken: (a) because traditional the-
ories are insufficient to explain observed abnormal absorption by
traditional volumetric absorption, (b) because so much water in
the atmosphere is in or near a thermodynamic state of saturation,
constantly undergoing instantaneous, dynamic evaporation and
condensation, both as visible clouds and as invisible aerosol, (c)
because in the latter case of invisible aerosol, droplet sizes are so
small that any surface radiation effect (such as phase-change radi-
ation) is likely to become relatively more important, and (d)
because even for cloud droplets, it is spectral regions of low volu-
metric absorption by water, atmospheric windows, where interfa-
cial radiation is likely to become relatively more important and it
is within these spectral regions that anomalous radiation is often
reported.
Phase-change radiation, while still not a generally accepted or
recognized concept, is not a new idea; various investigators have
reported on it for several decades. While the exact origin of the
idea seems uncertain, it is claimed by Xie et al., [6] that
Perel’man, who later published theories on it [7], suggested this
kind of unusual enhanced absorption as early as the 1960 s; he
proposed an electronic-state transition theory in 1971 [7]. Later
the phrase “phase transition radiation” was introduced in the liter-
ature. Starting in the late 1960 s researchers began reporting evi-
dence of phase transition radiation for vapor-liquid, liquid-solid,
and vapor-solid transitions. In 1968, Potter and Hoffman [8]
1
Corresponding author.
Contributed by the Heat Transfer Division of ASME for publication in the
JOURNAL OF HEAT TRANSFER. Manuscript received March 21, 2013; final manuscript
received January 18, 2014; published online March 10, 2014. Assoc. Editor:
Zhuomin Zhang.
Journal of Heat Transfer JUNE 2014, Vol. 136 / 062704-1Copyright VC 2014 by ASME
Downloaded From: http://heattransfer.asmedigitalcollection.asme.org/ on 04/05/2014 Terms of Use: http://asme.org/terms

reported an abnormal increase of infrared radiation from boiling
water at 1.54 lm and 2.10 lm. The authors referred to this phe-
nomenon as phase transition luminescence and asserted that water
clusters were responsible for radiating latent heat during the boil-
ing process. Carlon, in several papers in the 1970 s, attributed
anomalous radiation emitted by water during vaporization to
water-vapor clusters; these papers are summarized in Ref. [9]. He
also suggested that water-cluster phase-transition radiation might
be responsible for water vapor’s anomalously high atmospheric
continuum. In 1977, Mestvirishvili et al., [10] reported phase tran-
sition radiation for water undergoing liquid-solid and vapor-liquid
transitions at 28–40 lm and 4–8 lm, respectively. Perel’man and
Tatartchenko [11] reported water phase-transition characteristic
wavelengths for solid-vapor, liquid-vapor, and solid-liquid at
2.57 lm, 2.96 lm, and 20 lm, respectively. They also suggested
that the radiation emission previously reported by Mestvirishvili
et al. was due to multi-photon transitions. Recently, Tatartchenko
[12–14] suggested that phase-transition radiation exists for spe-
cific infrared radiation bands of water by analyzing satellite and
radiometer data.
In recent years several researchers have also been trying to
demonstrate uses and applications of phase transition radiation.
By analyzing the relaxation time and the radiation power, Sall’
and Smirnov [15] proposed that phase-transition radiation could
happen in bulk material containing 105
particles that is controlla-
ble and leads to desired properties such as grain size distribution.
Ambrok et al., [16] proposed the possibility of growth-kinetics
modification of particle ensembles in the presence of external,
selective radiative heating. Tatartchenko also proposed several
applications of phase-transition radiation in various articles,
including hailstorm detection [12], triggering of fog formation
and crystallization [17], and infrared lasers [18].
Theoretical analysis of cloud radiation, particularly single parti-
cle interaction, has been thought to be fairly well established. Liq-
uid cloud droplets can be treated optically as homogenous spheres
with known refractive indices. Mie scattering theory applies and
gives droplet absorption and scattering efficiencies as well as scat-
tering phase function. However, due to the mathematical com-
plexity of Mie equations, appropriate approximations are often
adopted, such as the anomalous diffraction approximation (ADA)
by van de Hulst [19]. ADA, which applies for complex refractive
index magnitude |n| $ 1 and droplet size parameter x ¼ pd/k ) 1,
is often used for cloud droplets even for terrestrial infrared radia-
tion. Since water’s visible and infrared refractive index magnitude
deviates by up to 30% from one, a semi-empirical modified ADA
(MADA) has been developed [20], which has a wider range of
applicability for optical constants (1.01 n 2.00 and 0 k 10)
and is valid for all particle size parameters, x.
In this paper absorptive phase-transition radiation was studied
using mid-IR transmission in a unique cloud chamber. Both ex-
perimental measurements and MC simulation (by adopting the
MADA approximation of Mie theory) were used to investigate
phase-transition radiation effects on absorption by water clouds.
In particular the change in behavior with temperature from 60
C
to 75
C was investigated.
2 Experimental Approach
Water clouds were generated inside a constant-pressure (1 atm)
cloud chamber system through which mid-IR transmission/
absorption measurements were made. Thermodynamic conditions
were controlled through electrical heating, convective heating, IR
heat loss through silicon windows, and a moveable piston. The
heat transfer conditions were such that condensation occurred as
droplets in the bulk vapor phase only (not on chamber surfaces)
and when it occurred, near saturation (equilibrium) was main-
tained, i.e., very small super-saturation. The evaporation occurring
during measurements that was thought to contribute to phase-
transition radiation absorption was not net evaporation; rather it
was the instantaneous evaporation that was in dynamic equilib-
rium with instantaneous condensation. That is, quasi-steady, near-
saturated equilibrium conditions existed.
2.1 Experimental Apparatus. A schematic for the cloud
chamber and measurement system is shown in Fig. 1. A liquid-
nitrogen cooled, InSb IR array camera was used with a focal plane
array (FPA) of 320 by 256 pixels. The array could cover the spec-
tral range from 1 lm to 5.2 lm. A bandpass filter of 3.1–4.95 lm
was used. The used portion of the array was set to 240 by 304 pix-
els. Transient time scales were maintained during the experiments
to be of the order of minutes to maintain near equilibrium. A
framing rate of 63 Hz and integration time of 0.20 ms were used
for the experiments.
The cloud chamber is a T-shaped cylindrical chamber made of
aluminum with an outer diameter of 10 cm, thickness of 1.6 mm,
and length of 40.6 cm. The chamber being made mostly of alumi-
num, it was possible to maintain nearly uniform temperature in
the chamber with judicious electrical heating. Two IR windows
made of silicon wafer were used to provide for transmission of IR
radiation from the source to the detector camera. The Si windows
had one side polished, facing toward the exterior, and one unpol-
ished, facing toward the interior. A TeflonVR
-sealed aluminum pis-
ton was placed in the T-branch of the cloud chamber to maintain
constant pressure during the experiments. Conductive heating was
applied at the exterior surface of the cloud chamber by wrapping
nichrome ribbon in such a way as to maintain a nearly uniform
temperature over the cloud chamber. A DC power supply was
used to control the heating rate of the nichrome ribbon wire by
regulating the electric current output. Fiberglass insulation was
used on the exterior of the cloud chamber. Convective heating
with hot air directed at the Si windows was used to keep the Si
windows barely above the dew point. In this way heat could be
lost by IR radiation through the windows to maintain cloud satura-
tion conditions while maintaining the windows free from surface
condensation.
Six thermocouples (K-type) were used to measure temperature
and recorded by computer. Five of them were attached along the
horizontal inner surface of the chamber and one thermocouple
was suspended inside the cloud chamber to indicate gas tempera-
ture and the radial temperature variation. The position of the sus-
pended thermocouple was maintained to avoid surface contact
and to minimize transmitted IR blockage by placing it slightly
above the radiation path. Cloud chamber temperature distribution
was kept nearly uniform by maintaining the maximum tempera-
ture difference less than 1.5
C in most experiments. However, in
a few cases the maximum temperature difference exceeded that
limit but was still less than 1.8
C. Maximum temperature differ-
ence refers to the average of the five thermocouples located at the
chamber wall and the minimum temperature, which was the sus-
pended thermocouple temperature. Another temperature uniform-
ity test was performed at the chamber wall for azimuthal
temperature distribution and it was found that it was even more
uniform than in the longitudinal axial direction. A DC power sup-
ply with was used to heat the nichrome ribbon and to balance the
heat loss from the chamber wall and windows to the surroundings
Fig. 1 Schematic of experimental system for measurement of
water cloud transmissivity
062704-2 / Vol. 136, JUNE 2014 Transactions of the ASME

by conduction, convection and radiation. Nominal voltage and
current during steady-state were 24 V and 1.38 A, respectively.
A blackbody-like TeflonVR
-coated aluminum cavity was used as
an IR radiation source. Blackbody behavior of the IR source was
tested by comparing with the Planck function over a temperature
range from 50
C to 95
C [5]. An optical shutter made from black
anodized aluminum was placed between the cloud chamber and
the IR source and used to switch on and off the IR radiation. Two
calcium fluoride plano-convex lenses (25.4 mm diameter, 100 mm
effective focal length) were used to focus the IR radiation to FPA
of the IR camera.
2.2 Experimental Procedure. Creation of water clouds
inside the cloud chamber system required special care to maintain
a steady uniform temperature at 75
C. A given amount of dis-
tilled water ma (used to determine the cloud droplet water volume
fraction, fv) was dispensed inside the cloud chamber at the begin-
ning of the experiment. The cloud droplet volume fraction could
be determined from
fv ¼
mi þ ma À mv
qlV
(1a)
where the three mass terms on the right-hand side are the initial
water vapor mass in the air in the chamber mi, the water mass
added as liquid ma, and the saturated vapor water mass at the mea-
surement condition mv. After adding water the chamber was
closed and heated to over 100
C to convert the liquid water into
vapor. The chamber was then cooled to 75
C at a few degree per
minute to maintain equilibrium and to avoid surface condensation.
The air temperature in the chamber was maintained at the dew
point temperature and a cloud was formed at saturation condi-
tions. Sufficiently slow cooling rate was used to avoid significant
supersaturation (i.e., well under a fraction of a percent). Unfiltered
lab air was present in the chamber with heterogeneous nucleation
on clouds droplets occurring on natural aerosol in the air. Conden-
sation on the Si windows could easily be detected both by IR
images and unusual photon counts at the array detector. At normal
operation with no condensation on the IR windows the photon
count standard deviation was low as 10–13, which was around
0.4% of total photon counts. Other possible locations of unwanted
condensation such as the manometer probe and the piston inner
surface were verified visually at the end of the measurement,
before cool-down, to assure there was no visible evidence of sur-
face condensation.
Transmission experiments were conducted by measuring the ra-
tio of IR source intensity with the presence and absence of clouds
at the operating temperature (75
C). First, the combined effects
of the IR radiation from optical components and other thermal sig-
nals associated with the chamber and surroundings were measured
as photon counts with the shutter closed without any water cloud
present; this signal is denoted as R0,1. The IR source radiation was
added to this configuration by opening the shutter; this signal is
denoted as R0,2. The same procedure, repeated with the water
cloud present, resulted in signals denoted as R1 and R2. The differ-
ence between the first pair of readings gives a signal proportional
to the radiation from the IR source incident on the water cloud.
The difference between the second pair of readings gives a signal
proportional to the radiation transmitted through the water cloud.
The ratio of these two differences gives the transmissivity of the
water cloud to the IR source radiation, which includes absorption
by the droplets, multiple scattering by the droplets, and reflection
and a small amount of absorption by the chamber walls.
sjk¼3:1À4:95l¼
R2 À R1
R0;2 À R0;1
(1b)
Efforts were made to minimize background noise by recording
photon counts only at the cold spot region of the window. The
cold spot was a reflection of the liquid nitrogen dewar on the win-
dow near the camera. This region was easily recognized in the
camera image.
3 Theoretical Methods and Monte Carlo Simulation
3.1 Droplet Single Scattering and Absorption Properties.
Theoretical thermal radiative heat transfer considerations perti-
nent to water clouds are discussed next to explain the choice of
assumptions for the Monte Carlo radiative transfer simulation.
Inside the cloud chamber the extinction (absorption and scatter-
ing) of externally incident IR radiation was assumed to be affected
by only the water cloud; molecular gases such as water vapor,
nitrogen, and oxygen and unactivated (interstitial) aerosol were
assumed to have negligible effect on absorption or scattering of
radiation from the IR source. This assumption is based on well
established research [21,22]. It is accepted that liquid water’s
mass absorption coefficient is at least three orders of magnitude
larger than that of water vapor and aerosol in the spectral region
of interest.
As noted above the semi-empirical MADA [20] was adopted
for computing single water droplet volumetric absorption and
scattering efficiencies, assuming spherical droplets.
Qe;m ¼ Qef (2a)
Qe ¼ Re 2 þ
4eÀx
x
þ
4 eÀx
À 1ð Þ
x2
!
(2b)
f ¼ 2 À exp ÀxÀ2
3

(2c)
where x ¼ 2kx þ iq and q ¼ 2x n À 1ð Þ. Eliminating complex
notation [23] gives
Qe ¼ 2 þ 4u2
cos 2bð Þ À 4eÀqtan bð Þ
usin q À bð Þ þ u2
cos q À 2bð Þ
Â Ã
(3)
with tan bð Þ ¼ ðk=n À 1Þ and u ¼ ðcos bð Þ=qÞ. The volumetric
absorption efficiency is
Qa;v ¼ 1 þ
2
1
eÀ1
À
2
12
1 À eÀ1
ð Þ (4)
where 1 ¼ 4xk ¼ 2q tan bð Þ is optical depth along droplet diame-
ter. The term “volumetric” is used with the absorption efficiency
above to distinguish it from “surface” absorption, which will be
included below. The scattering efficiency (which is always only a
surface phenomenon) is
Qs ¼ Qe À Qa;v: (5)
The spherical water droplets in the cloud chamber were assumed
to be monodisperse in size throughout the chamber for any given
experimental condition. This assumption was checked previously
by considering poly-dispersity effects [5] and found to be reason-
ably valid for the conditions of this experiment. Hence, volumet-
ric extinction, absorption, and scattering coefficients can be
written as
Ke;a;s ¼ pr2
Qe;a;sN0 ¼
3Qe;a;sfv
4r
(6)
where r is the radius of water droplet, fv is the volume fraction of
water droplets
fv ¼
4
3
pr3
N0 (7)
and N0 is the number density of water droplets, which is the same
as the number density of activated nucleation sites or CCN. At a
Journal of Heat Transfer JUNE 2014, Vol. 136 / 062704-3

given temperature condition, but for different water loadings (dif-
ferent fv) the value of No was assumed to be constant such that the
droplet size from run to run was assumed to vary as fv
1/3
. The refer-
ence value of droplet size was determined by using the best fit of
the simulation results. The reference droplet diameter is referred to
as d* corresponding to fv ¼ 1.6 Â 10À4
. This assumption (of con-
stant CCN at a given temperature) is based on the experiments hav-
ing been done in the same controlled lab environment, assuming
consistency of the aerosol in that environment.
Absorption of radiation at the surface of water droplets was
also incorporated along with volumetric radiation absorption in
order to account for phase-transition absorption. Surface absorp-
tion can occur at multiple surfaces if the volumetric absorption
effect is not too optically thick; however, single surface absorp-
tion is considered first before treating multiple surface contribu-
tions. Phase-transition radiation is radiation emitted or absorbed
by an atom or molecule during a radiative transition that is associ-
ated with phase-change of the atom or molecule. While sepa-
rately, both radiation and phase-change are understood to be
common and pervasive phenomena, their simultaneous occurrence
in the same atom or molecule is not generally thought to be a
common occurrence. Phase-change energy is normally thought of
as appearing in the form of phonons rather than photons, and as
being conducted to or away from the phase-change interface after
or before any (volumetric, not interfacial) thermal radiation
occurs. However, interfacial phase-change radiation, as noted
above, has become increasingly recognized and a more detailed
discussion of its theoretical basis has been made elsewhere [24].
For present purposes, it may be noted that in local thermodynamic
equilibrium the fraction of incident monochromatic radiation of
frequency v absorbed at an interface due to evaporation [24] may
be represented as
fs ¼
a21
4p2
H
Bv
ng
ffiffiffiffiffiffiffiffiffiffiffiffiffi
8pkBT
mg
s
(8)
where H and ng are the population distribution function and parti-
cle number density of the upper (gaseous) state, Bv is the Planck
function, and mg is the molecular mass of water. The coefficient
a21 is the radiative transition probability associated with a liquid
water molecule at the surface evaporating into the gas phase and
is related to the spontaneous Einstein coefficient by
A21 ¼ a21N0r2
ffiffiffiffiffiffiffiffiffiffiffiffiffi
8pkBT
mg
s
(9)
Further details for the surface absorption theoretical framework
can be found in Ref. [5].
The discussion above pertains to the probability of absorption
at an evaporating surface regardless of geometry of the surface,
that is, without regard for the spherical shape of a cloud droplet.
However, for wavelength and droplet size combinations for which
droplet volumetric absorption is not optically thick, surface
absorption may happen at interfaces other than the first one
encountered by the propagating incident radiation. The following
description presumes a geometric optics regime, in which the con-
cept of rays of radiation has meaning and wave effects such as in-
terference do not need to be considered. Cloud droplets of
approximately 20 lm diameter at mid-IR wavelengths (3–5 lm)
would be in this category (x ) 1). Figure 2 shows a schematic of
how multiple encounters with different parts of the evaporating
surface of a droplet gives rise to enhanced opportunity for surface
absorption. As radiant energy encounters the first interface, the
incident energy may be reflected, absorbed or transmitted. This
balance is represented by
rs þ ss þ fs ¼ 1 (10)
where rs is the surface reflection, ss is the surface transmission,
and fs is the surface absorption. Radiation transmitted through the
interface propagates into the droplet where it may be absorbed in-
depth, volumetrically. Inside the water droplet the internal trans-
mittance is designated as sv. Radiation not absorbed inside the
droplet will encounter the droplet surface at another location
where again it may be reflected (internally), absorbed, or transmit-
ted out of the droplet. The contribution to overall droplet absorp-
tion efficiency from the surface absorption effect is the
summation of all the possible surface absorptions
Qa;s ¼ fs þ sssvfs þ sssv rssvð Þfs þ sssv rssvð Þ2
fs þ Á Á Á
¼ fs 1 þ
sssv
1 À rssv
!
(11)
The internal transmittance can be approximated as one minus the
internal volumetric absorption efficiency
sv ¼ 1 À Qa;v (12)
The surface reflectivity can be estimated using the Fresnel inter-
face relations integrated over the curved surface of the droplet.
The integrated reflectivity for a spherical surface subjected to col-
limated irradiation is equivalent to the hemispherical reflectivity
of a flat surface subjected to diffuse irradiation [23]. Thus, the sur-
face reflectivity for a smooth spherical surface with collimated
irradiation is
Fig. 2 Ray tracing (geometric optics) model of multiple chan-
ces for surface absorption
Fig. 3 Transmissivity versus liquid water volume fraction at
75
C, experimental results and MC simulations for fixed proba-
bility constant (a21) and varying d* (lm)

rs ¼
1
p
ð
2p
q0
k cos hdX ¼ 2
ðp=2
0
q0
k cos h sin hdh (13)
where q0
k is the Fresnel directional spectral reflectivity, which can
be calculated from the Fresnel relations [23] assuming unpolar-
ized radiation
R? ¼
n1 cos h1 À uð Þ2
þv2
n1 cos h1 þ uð Þ2
þv2
(14)
Rjj ¼
n2
2 À k2
2
À Á
cos h1 À n1u
Â Ã2
þ 2n2k2 cos h1 À n1v½ Š2
n2
2 À k2
2
À Á
cos h1 þ n1u
Â Ã2
þ 2n2k2 cos h1 þ n1v½ Š2
(15)
q0
k ¼
1
2
R? þ Rjj
À Á
(16)
where R? and Rjj are the perpendicular and parallel Fresnel inter-
face reflectivity, n1 is the refractive index of air, n2 is the refrac-
tive index of liquid water, k2 is the absorption index of liquid
water, and h is incident angle from air to water liquid. Intermedi-
ate parameters u and v are given by
2u2
¼ n2
2 À k2
2 À n2
1 sin2
h1
À Á
þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n2
2 À k2
2 À n2
1 sin2
h1
À Á2
þ4n2
2k2
2
q
(17)
2v2
¼À n2
2 Àk2
2 Àn2
1 sin2
h1
À Á
þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n2
2 Àk2
2 Àn2
1 sin2
h1
À Á2
þ 4n2
2k2
2
q
(18)
Finally, the droplet overall surface absorption efficiency can be
added to the internal volumetric absorption efficiency to give the
total (surface and internal-volumetric) droplet absorption
efficiency.
Qa ¼ Qa;v þ Qa;s ¼ Qa;v þ fs 1 þ
1 À rs À fsð Þ 1 À Qa;v
À Á
1 À 1 À Qa;v
À Á
rs
#
(19)
The single-scattering albedo is then
x0 ¼
Qs
Qe
¼
Qs
Qs þ Qa
¼
Qs
Qs þ Qa;v þ fs 1 þ
1 À rs À fsð Þ 1 À Qa;v
À Á
1 À 1 À Qa;v
À Á
rs
#
(20)
The scattering phase function was calculated using Rayleigh-
Gans scattering theory [19], instead of anomalous diffraction
theory. This allowed anisotropic scattering to be described reason-
ably accurately without trying to resolve the strong forward scat-
tering that corresponds to diffraction, which is not included in the
scattering efficiency anyway. This approach thus treats diffracted
radiation in the simulation as if it were not scattered at all. The
phase function for Rayleigh-Gans scattering is
P hð Þ ¼ C
3
u3
sin u À u cos uð Þ
!2
1 þ cos2
h
À Á
(21)
where u ¼ 2x sin h=2ð Þ and C is a normalization constant.
3.2 Monte Carlo Radiative Transfer Simulation. MC simu-
lation was used for computational analysis of IR radiative transfer
in the cloud chamber because of its versatility and accuracy. The
details of the procedure used were discussed previously [5]. A
cylindrical system was used for the geometry of the cloud cham-
ber. A spectral range of k ¼ 3.1–4.95 lm with a spectral resolution
of 0.01 lm was used. As in the previous study [5] the minimum
number of emitted energy bundles was chosen above 4000 at each
wavelength. The spectral transmissivity is denoted by
sMC;k ¼
Ss;2 À Ss;1
À Á
k
Ss;2 À Ss;1
À Á
k;0
(22)
where Ss refers to the number of photon bundles gathered by IR cam-
era, subscripts 1 and 2 represent the shutter closed and open, respec-
tively, and the subscript 0 indicates the absence of the water cloud.
The spectral transmissivity is calculated at each wavelength and inte-
grated over the working wavelength range for the total transmissivity.
4 Results and Discussion
4.1 Experimental Results for 75
C. In Fig. 3 experimen-
tally measured transmissivity, s, is plotted as a function of liquid
water volume fraction, fv, ranging from 0 to 2 Â 10À4
for
75 6 1.5
C. At small fv, (which also means small d; recall that d
is assumed to vary as fv
1/3
for constant No), the transmissivity is
high and decreases as the liquid water droplet loading increases.
This is because of the way optical depth varies with water loading,
that is, primarily through the effects of d and fv. The volumetric
extinction coefficient (and thus optical depth along the path length
through the chamber) varies (for approximately constant extinc-
tion efficiency and path length) as fv/d, d2
, or fv
2/3
for constant No.
This is the primary dependence exhibited in the data of Fig. 3. In
addition to that primary trend there are secondary variations of
extinction efficiency, albedo, and phase function with droplet size.
Measurements at the smallest values of fv (smallest d) are the
most difficult for several reasons. First, the relative uncertainty in
the measured fv value is the greatest (due to the uncertainty in
measuring a small amount of water to add to the chamber). Sec-
ond, the system seems to be near a stability boundary and main-
taining equilibrium is difficult. Either the droplets themselves are
intrinsically unstable, or the delicate balance of maintaining satu-
rated vapor-liquid mixture conditions in the gas suspension while
keeping the chamber surfaces, including windows, slightly above
the dew point temperature becomes more difficult. The smallest fv
value at which measurements could be made with a cloud in the
chamber was 0.1 Â 10À4
. Below this value the cloud droplets
would rapidly evaporate and the measured transmissivity would
increase from about 0.65 to 1 precipitously, meaning over a nar-
row range of fv that was so small as to be immeasurable.
4.2 Comparison of MC Simulation With Experimental
Results for 75
C. Figure 3 also shows the comparison of MC
simulation with experimental results for the 75
C case. The prob-
ability constant (a21) is 3 Â 10À8
. This value is slightly lower than
that used previously for the 60
C case (5 Â 10À8
) because previ-
ously surface absorption at multiple interfaces was not included
whereas here it is. Droplet diameter was determined as before by
looking for a best fit between the simulation results and the meas-
ured data for transmissivity. The best match was found for
d* ¼ 17 lm. This value is smaller than the value of 20 lm
obtained at 60
C but is consistent with a higher CCN number
density and is still typical of cloud droplet sizes. Table 1 lists vari-
ous reference droplet diameters that were tried. Reference size is
indicated by an asterisk (d*) and means the droplet size at the con-
dition fv
*
¼ 1.6 Â 10À4
. The N0 values found in this study as listed
in Table 1 are of the same order of magnitude (104
cmÀ3
) or
maybe slightly larger than typical values reported for urban areas
with industry [25]. Urbana, IL is a small urban area without much
industry so the indicated N0 in Table 1 are actually a little higher
than typical reported values. However, reported values are for
temperatures near 20
C, whereas our experiments were at 75
C.
The temperature dependence of Kohler theory (the theory of acti-
vation of CCN from aerosol) shows that higher temperatures
allow activation of larger inorganic salt aerosol particles for the

same supersaturation, which means more CCN and higher N0 with
higher temperature for the same supersaturation. We did not mea-
sure supersaturation in our cloud chamber, which would be very
difficult, but we believe that it was both very small and relatively
constant due to the slow cooling rate and the consistency of cool-
ing conditions between experiments. The fact that the results in
Fig. 3 for multiple experiments correlate so well with an assumed
constant N0 indicates that the supersaturation was relatively con-
stant between different experiments.
Initial attempts at simulation were made using the same CCN
number density as the previous 60
C case, 3.8 Â 104
cmÀ3
, with
temperature dependence included only through the terms noted
above in Eq. (8). This approach, however, was not able to
adequately match the experimental data. In order to do so the
CCN number density was increased by approximately a factor of
two to 6.22 Â 104
cmÀ3
. Temperature-dependence of CCN con-
centration has not been discussed as much in the literature as
supersaturation dependence, notwithstanding both explicit and
implicit temperature dependencies in Kohler theory. Recently
temperature dependence of CCN parameters was reported [26];
however, the temperature dependence was also a transformation
of the supersaturation dependence, a variation of which also
existed in the experiment along with a temperature variation. To
be able to theoretically account for the change in CCN with tem-
perature would require knowledge of the aerosol composition and
size distribution, which was not available. In the absence of better
information about CCN number density as a function of tempera-
ture, an increase of CCN with temperature was assumed here that
resulted in a reasonable match with measured data, as shown in
Fig. 3. As noted above, an increase in CCN with temperature,
albeit difficult to quantify without aerosol information, is consist-
ent with Kohler theory for activation of common hygroscopic aer-
osol such as ammonium sulfate.
The MC simulation was used to explore the relative contribu-
tions to IR extinction from scattering and absorption by setting
one or the other of Qa or Qs to zero in Qe ¼ Qa þ Qs for the case
of no surface absorption. The results are shown in Fig. 4 with
three cases: pure scattering, non-absorbing, Qe ¼ Qs; pure absorb-
ing, nonscattering, Qe ¼ Qa; and mixed scattering-absorbing
(Qe ¼ Qa þ Qs). For the pure scattering case the transmissivity is
essentially unchanged from the mixed case for fv 0.15 Â 10À4
,
indicating that for small droplets (d 7 lm) scattering is the dom-
inant extinction mechanism over absorption; the droplets are opti-
cally thin in absorption. For larger fv values absorption begins to
contribute to extinction, even becoming stronger at
fv ¼ 1.5 Â 10À4
. The effect of surface absorption Qa,s can also be
seen because it was neglected even in the mixed scattering-
absorption simulation. For values of fv 0.4 Â 10À4
the transmis-
sivity difference between experimental and MC simulation results
is noticeable and is due to the absence of surface absorption in the
simulation. Further discussion of the how the trade-off between
scattering and absorption determines the shape of the transmissiv-
ity curve has been given previously [5].
Figures 5 and 6 further show the effect of surface absorption.
Figure 5 shows the effect of varying surface absorption probability
constant (a21) for a fixed droplet diameter (d* ¼ 17 lm). The
best value and the one adopted here was a21 ¼ 3 Â 10À8
. This
corresponds to a surface radiative transition probability of one out
of 3.33 Â 107
collisions between water vapor molecules and drop-
lets. In Fig. 6, surface absorption was excluded from the simula-
tions, showing that no value of d* would give as good a match as
Table 1 CCN number density (No) for different d* values at
fv 5 1.6 3 1024
fv d* (lm) CCN (cmÀ3
)
1.6 Â 10À4
15 9.05 Â 104
16 7.46 Â 104
17 6.22 Â 104
18 5.24 Â 104
19 4.46 Â 104
20 3.82 Â 104
21 3.30 Â 104
Fig. 4 MC simulation and experiment results comparison at
75
C in the absence of surface absorption for pure absorbing
(nonscattering), pure scattering (nonabsorbing), and mixed
absorbing and scattering droplets
Fig. 5 MC simulation results for different surface absorption
coefficients (i.e., varying, a21) at 75
C
Fig. 6 MC simulation results for different d* with no surface
absorption contribution (a21 5 0) at 75
C

when surface absorption was included, confirming the importance
of phase transition radiation in the saturated conditions of the
cloud chamber.
Figures 7 and 8 show the theoretical surface and volume
absorption efficiencies over the 3–5 lm spectral range for the lim-
iting volume fractions (droplet diameters), fv ¼ 0.1 Â 10À4
(d ¼ 6.75 lm) and fv ¼ 2 Â 10À4
(d ¼ 18.3 lm). At k ¼ 4 lm
water’s absorption index k is relatively small, which leads to neg-
ligible volumetric absorption; however, the surface absorption
contribution is significant at this wavelength. This shows the im-
portance of including surface absorption in atmospheric spectral
windows. On the other hand near k ¼ 3 lm, surface absorption is
relatively weak compared with volumetric absorption, due to the
strongly absorbing fundamental intramolecular O-H stretching
bands, symmetric (v1) and asymmetric (v3). As wavelength
approaches k ¼ 5 lm, the strongly absorbing fundamental intra-
molecular bending band (6 lm) increases the importance of volu-
metric absorption over surface absorption. This principle can be
extended to the important atmospheric window at 8–12 lm. Inter-
estingly this window is a region where anomalously high contin-
uum absorption by “water vapor” is known to be important but is
still not well understood. It seems at least worth exploring the idea
that phase-change radiation involving aerosol or haze (much
smaller than cloud droplets) could be a contributing mechanism,
along with the current popular dimer and far-wing absorption
theories.
4.3 Comparison With Previous Results at 60
C. The new
transmissivity measurements at 75
C are compared with previous
results at 60
C [5] in Fig. 9. Increasing temperature caused trans-
missivity to drop at all volume fractions above 0.15 Â 10À4
. The
primary reason is thought to be an increase in CCN number den-
sity with temperature, as noted above. An intrinsic increase in sur-
face absorption with temperature also plays a role but apparently
a secondary one compared to the CCN effect. Another change in
transmissivity versus volume fraction (droplet diameter) with tem-
perature is in the slope variations of the curve. At 75
C the curve
is smoother; the slope variations are essentially monotonic. At
60
C the slope of the data has nonmonotonic character with no-
ticeable slope changes such that there is a relatively flat portion at
transmissivity of 0.4 for volume fractions between 0.3 Â 10 À 4
and 0.5 Â 10À4
. These slope changes may seem subtle in looking
at the discrete experimental data points but become more noticea-
ble when looking at continuous theoretical curves.
Figure 10 shows a comparison between the previous 60
C
measurements and a new MC simulation based on the new param-
eters developed here from the 75
C data comparison. The simula-
tion done previously for 60
C was done without including the
possibility of multiple surface absorption. Therefore this effect
was included in the present simulations. A slight change in the
probability constant (a21), from 5 Â 10À8
to 3 Â 10À8
, was made
to compensate for the inclusion of multiple interface absorption,
while other parameters were kept the same as the original simula-
tion. Interestingly, Fig. 10 shows that for 60
C the multisurface
absorption properties led to a good a match with the experimental
results, notwithstanding the more complicated curve shape for the
60
C data.
Fig. 7 Comparison of Qa,v and Qa,s for d 5 6.75 lm
(fv 5 0.1 3 1024
) versus wavelength
Fig. 8 Comparison of Qa,v and Qa,s for d 5 18.3 lm
(fv 5 2 3 1024
) versus wavelength
Fig. 9 Comparison of experimental transmissivity measure-
ment for 60
C and 75
C
Fig. 10 Best simulation fit for 60
C experiment with the pres-
ence of surface absorption

4.4 Transmissivity Prediction at 50
C. In order to gain
some understanding of how the present findings might extrapolate
to atmospheric temperatures and to test the simulation further,
transmissivity at 50
C was simulated by extrapolating the CCN
values used for 60
C (3.82 Â 104
cmÀ3
) and 75
C
(6.22 Â 104
cmÀ3
) to 2.22 Â 104
at 50
C. With lower CCN num-
ber density and smaller optical depth expected at lower tempera-
ture, the simulated transmissivity increases noticeably, as shown
in Fig. 11. Perhaps more interesting is the increase in slope varia-
tions predicted at 50
C. The underlying reason can be seen in Fig.
4 in the trade-off between absorption and scattering effects. At
smaller optical depth conditions the different slope variations seen
in the pure scattering and pure absorbing curves of Fig. 4 become
more noticeable in the composite (scattering plus absorption) result.
This calculation is based on an extrapolation that may not be valid
and these results should not be taken as a firm prediction. Experi-
ments are needed at 50
C to see if this extrapolation has any merit.
5 Summary
Experimental measurements and theoretical MC simulations for
mid-IR transmissivity of saturated water clouds were conducted to
investigate phase-transition radiation absorption by water and to
explore the temperature dependence of this interfacial absorption.
New measurements at 75
C and 1 atm were made and compared
with previous results at 60
C. The MC simulations were used to
quantify the contribution of phase-change radiation absorption.
The MC simulations showed that classical volumetric absorp-
tion and scattering theory by cloud droplets could not account for
measured transmissivities. Additional absorption was needed and
this was provided theoretically by a surface absorption contribu-
tion, attributed to phase-change radiation absorption at the water
droplet-air interface. The results suggested a contribution of 3–4%
absorption per surface in the 3–5 lm range, in agreement with pre-
vious findings. Near 4 lm, where liquid water’s volumetric
absorption is the weakest, even this small amount of surface
absorption is relatively significant. The same would likely hold
true in the 8–12 lm window near 10 lm.
The effect of increasing temperature from 60
C to 75
C was
for transmissivity to decrease at a given water loading (or effec-
tive droplet diameter), i.e., optical depth increased. This change is
primarily attributed to an increase in the number density of drop-
lets associated with an increase in number density of activated
aerosol or CCN with temperature. Theoretical assessment of this
observation with temperature-dependent Kohler theory would be
desirable if aerosol information could be obtained. Based on the
results at 60
C and 75
C a calculation was made at 50
C that
showed even more structure in the transmissvitiy curve versus
droplet volume fraction (diameter) than the 60
C case did. Exper-
imental testing of the 50
C condition is needed.
These findings may be important with respect to explaining
water’s infrared absorption continuum in the 3–5 and 8–10 lm
atmospheric windows. Atmospheres that appear to be cloud-free
or fog-free often still contain aerosol and enough water vapor to
cause deliquescence of a fraction of the aerosol particles, the
smaller and more hygroscopic ones. Thus dynamic evaporation
and condensation of water can occur in atmospheres even without
visible clouds or fog and do so on tiny particles with tremendous
specific surface area. Further theoretical and experimental investi-
gation of phase-transition radiation and water “vapor” continuum
absorption is needed.
Acknowledgment
Support for this work was provided by the National Science
Foundation under Grant Number 1062361 and the Hermia G. Soo
Professorship.
Nomenclature
a21 ¼ probability of radiative relaxation
A21 ¼ Einstein coefficient for spontaneous emission
Bv ¼ blackbody function
C ¼ normalization constant in phase function
d ¼ droplet diameter
d* ¼ reference droplet diameter at fv ¼ 1.6 Â 10À4
(lm)
f ¼ correction factor in MADA theory or interface
absorption
fv ¼ water droplet volume fraction
h ¼ Planck constant
H ¼ population distribution function at gaseous-state
k ¼ imaginary refractive index
kB ¼ Boltzmann constant
Ka ¼ absorption coefficient
Ke ¼ extinction coefficient
Ks ¼ scattering coefficient
M ¼ number of energy bundles
mg ¼ mass of a water molecule
n ¼ real refractive index
N0 ¼ number density of water droplets
ng ¼ number density of water-vapor molecules
P ¼ phase function of scattered energy
Qa ¼ absorption efficiency
Qe ¼ extinction efficiency
Qe,m ¼ corrected extinction efficiency in MADA
Qs ¼ scattering efficiency
r ¼ droplet radius or interface reflectivity
R ¼ polarized Fresnel reflectance
s ¼ path length
T ¼ temperature
u,v ¼ intermediate calculation quantities
x ¼ droplet size parameter, 2pr/k
Greek Symbols
k ¼ wavelength
q ¼ reflectivity
s ¼ transmissivity
Subscripts
a ¼ absorption
e ¼ extinction
g ¼ gaseous state
s ¼ scattering
v ¼ volumetric
Abbreviations
ADA ¼ anomalous diffraction approximation
CCN ¼ cloud condensation nuclei
Fig. 11 MC estimation of transmissivity at 50
C, based on
extrapolation of simulations at 60
C and 75
C

FPA ¼ focal plane array
IR ¼ infrared
LTE ¼ local thermal equilibrium
LW ¼ longwave (IR) radiation
MADA ¼ modified anomalous diffraction approximation
MC ¼ Monte Carlo analysis
NIR ¼ near infrared radiation
SW ¼ shortwave (VIS and NIR) radiation
VIS ¼ visible radiation
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