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Temperatureeffectonphase-transitionradiationofwater

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KibriaKhanRoman

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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. WangDepartment of Mechanical

Science and Engineering,

University of Illinois at Urbana-Champaign,

Urbana, IL 61801

W.-H. WuDepartment of Mechanical

Science and Engineering,

University of Illinois at Urbana-Champaign,

Urbana, IL 61801

M. G. KhanDepartment of Mechanical

Science and Engineering,

University of Illinois at Urbana-Champaign,

Urbana, IL 61801

Temperature Effecton Phase-TransitionRadiation of WaterInfrared radiation associated with vapor-liquid phase transition of water is investigatedusing a suspension of cloud droplets and mid-infrared (IR) (3–5 lm) radiation absorptionmeasurements. Recent measurements and Monte Carlo (MC) modeling performed at60 �C and 1 atm resulted in an interfacial radiative phase-transition probability of5� 10�8 and a corresponding surface absorption efficiency of 3–4%, depending onwavelength. In this paper, the measurements and modeling have been extended to 75 �Cin 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 forobserved 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 from60 �C to 75 �C at near saturation conditions. The new radiative phase-transition proba-bility is decreased slightly to 3� 10�8. Theoretical results were also calculated at 50 �Cin 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 beimportant. Modeling also suggests that phase-change radiation will be most important atwavelengths of low volumetric absorption, i.e., atmospheric windows such as 3–5 lm and8–10 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 areundergoing 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 andlongwave (LW) infrared radiation have made clear the importanceof clouds and water vapor in Earth’s global energy balance andtheir 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 increaseanywhere from 2 �C to 5 �C, depending on cloud parameters andbehavior assumptions [2]. It has long been known that watervapor, the most important greenhouse gas in the atmosphere, hasunexplained, 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 ofcloud and water-vapor radiation is needed.

The need to upgrade cloud radiation modeling from 1D to 3Dhas recently been emphasized and efforts have been made in thisregard [4]. Nevertheless, even 3D radiative transfer modelingappears to be insufficient to describe observed cloud radiativetransfer—which includes anomalous absorption and emission—anddo so with the accuracy required for meaningful global climatemodeling. It has therefore been suggested that there may be pieces

of basic cloud radiation physics still missing [4]. In an effort toimprove understanding of basic cloud radiation physics, we havebegun 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 bytraditional volumetric absorption, (b) because so much water inthe atmosphere is in or near a thermodynamic state of saturation,constantly undergoing instantaneous, dynamic evaporation andcondensation, both as visible clouds and as invisible aerosol, (c)because in the latter case of invisible aerosol, droplet sizes are sosmall 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 itis within these spectral regions that anomalous radiation is oftenreported.

Phase-change radiation, while still not a generally accepted orrecognized concept, is not a new idea; various investigators havereported on it for several decades. While the exact origin of theidea seems uncertain, it is claimed by Xie et al., [6] thatPerel’man, who later published theories on it [7], suggested thiskind of unusual enhanced absorption as early as the 1960 s; heproposed an electronic-state transition theory in 1971 [7]. Laterthe 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]

1Corresponding author.Contributed by the Heat Transfer Division of ASME for publication in the

JOURNAL OF HEAT TRANSFER. Manuscript received March 21, 2013; final manuscriptreceived January 18, 2014; published online March 10, 2014. Assoc. Editor:Zhuomin Zhang.

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reported an abnormal increase of infrared radiation from boilingwater at 1.54 lm and 2.10 lm. The authors referred to this phe-nomenon as phase transition luminescence and asserted that waterclusters were responsible for radiating latent heat during the boil-ing process. Carlon, in several papers in the 1970 s, attributedanomalous radiation emitted by water during vaporization towater-vapor clusters; these papers are summarized in Ref. [9]. Healso suggested that water-cluster phase-transition radiation mightbe responsible for water vapor’s anomalously high atmosphericcontinuum. In 1977, Mestvirishvili et al., [10] reported phase tran-sition radiation for water undergoing liquid-solid and vapor-liquidtransitions at 28–40 lm and 4–8 lm, respectively. Perel’man andTatartchenko [11] reported water phase-transition characteristicwavelengths for solid-vapor, liquid-vapor, and solid-liquid at2.57 lm, 2.96 lm, and 20 lm, respectively. They also suggestedthat the radiation emission previously reported by Mestvirishviliet 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 andradiometer data.

In recent years several researchers have also been trying todemonstrate 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 couldhappen 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-kineticsmodification of particle ensembles in the presence of external,selective radiative heating. Tatartchenko also proposed severalapplications of phase-transition radiation in various articles,including hailstorm detection [12], triggering of fog formationand 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 sphereswith known refractive indices. Mie scattering theory applies andgives 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 oftenadopted, such as the anomalous diffraction approximation (ADA)by van de Hulst [19]. ADA, which applies for complex refractiveindex magnitude |n|� 1 and droplet size parameter x¼ pd/k� 1,is often used for cloud droplets even for terrestrial infrared radia-tion. Since water’s visible and infrared refractive index magnitudedeviates by up to 30% from one, a semi-empirical modified ADA(MADA) has been developed [20], which has a wider range ofapplicability 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 studiedusing mid-IR transmission in a unique cloud chamber. Both ex-perimental measurements and MC simulation (by adopting theMADA approximation of Mie theory) were used to investigatephase-transition radiation effects on absorption by water clouds.In particular the change in behavior with temperature from 60 �Cto 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 conditionswere controlled through electrical heating, convective heating, IRheat loss through silicon windows, and a moveable piston. Theheat transfer conditions were such that condensation occurred asdroplets 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 occurringduring measurements that was thought to contribute to phase-transition radiation absorption was not net evaporation; rather itwas 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 cloudchamber and measurement system is shown in Fig. 1. A liquid-nitrogen cooled, InSb IR array camera was used with a focal planearray (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 lmwas used. The used portion of the array was set to 240 by 304 pix-els. Transient time scales were maintained during the experimentsto be of the order of minutes to maintain near equilibrium. Aframing rate of 63 Hz and integration time of 0.20 ms were usedfor the experiments.

The cloud chamber is a T-shaped cylindrical chamber made ofaluminum 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 inthe chamber with judicious electrical heating. Two IR windowsmade of silicon wafer were used to provide for transmission of IRradiation from the source to the detector camera. The Si windowshad one side polished, facing toward the exterior, and one unpol-ished, facing toward the interior. A Teflon

VR

-sealed aluminum pis-ton was placed in the T-branch of the cloud chamber to maintainconstant pressure during the experiments. Conductive heating wasapplied at the exterior surface of the cloud chamber by wrappingnichrome ribbon in such a way as to maintain a nearly uniformtemperature over the cloud chamber. A DC power supply wasused to control the heating rate of the nichrome ribbon wire byregulating the electric current output. Fiberglass insulation wasused on the exterior of the cloud chamber. Convective heatingwith hot air directed at the Si windows was used to keep the Siwindows barely above the dew point. In this way heat could belost by IR radiation through the windows to maintain cloud satura-tion conditions while maintaining the windows free from surfacecondensation.

Six thermocouples (K-type) were used to measure temperatureand recorded by computer. Five of them were attached along thehorizontal inner surface of the chamber and one thermocouplewas 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 contactand to minimize transmitted IR blockage by placing it slightlyabove the radiation path. Cloud chamber temperature distributionwas kept nearly uniform by maintaining the maximum tempera-ture difference less than 1.5 �C in most experiments. However, ina few cases the maximum temperature difference exceeded thatlimit but was still less than 1.8 �C. Maximum temperature differ-ence refers to the average of the five thermocouples located at thechamber wall and the minimum temperature, which was the sus-pended thermocouple temperature. Another temperature uniform-ity test was performed at the chamber wall for azimuthaltemperature distribution and it was found that it was even moreuniform than in the longitudinal axial direction. A DC power sup-ply with was used to heat the nichrome ribbon and to balance theheat loss from the chamber wall and windows to the surroundings

Fig. 1 Schematic of experimental system for measurement ofwater cloud transmissivity

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by conduction, convection and radiation. Nominal voltage andcurrent during steady-state were 24 V and 1.38 A, respectively.

A blackbody-like TeflonVR

-coated aluminum cavity was used asan IR radiation source. Blackbody behavior of the IR source wastested by comparing with the Planck function over a temperaturerange from 50 �C to 95 �C [5]. An optical shutter made from blackanodized aluminum was placed between the cloud chamber andthe IR source and used to switch on and off the IR radiation. Twocalcium fluoride plano-convex lenses (25.4 mm diameter, 100 mmeffective focal length) were used to focus the IR radiation to FPAof the IR camera.

2.2 Experimental Procedure. Creation of water cloudsinside the cloud chamber system required special care to maintaina steady uniform temperature at 75 �C. A given amount of dis-tilled water ma (used to determine the cloud droplet water volumefraction, fv) was dispensed inside the cloud chamber at the begin-ning of the experiment. The cloud droplet volume fraction couldbe determined from

fv ¼mi þ ma � mv

qlV(1a)

where the three mass terms on the right-hand side are the initialwater vapor mass in the air in the chamber mi, the water massadded as liquid ma, and the saturated vapor water mass at the mea-surement condition mv. After adding water the chamber wasclosed and heated to over 100 �C to convert the liquid water intovapor. The chamber was then cooled to 75 �C at a few degree perminute to maintain equilibrium and to avoid surface condensation.The air temperature in the chamber was maintained at the dewpoint temperature and a cloud was formed at saturation condi-tions. Sufficiently slow cooling rate was used to avoid significantsupersaturation (i.e., well under a fraction of a percent). Unfilteredlab air was present in the chamber with heterogeneous nucleationon clouds droplets occurring on natural aerosol in the air. Conden-sation on the Si windows could easily be detected both by IRimages and unusual photon counts at the array detector. At normaloperation with no condensation on the IR windows the photoncount standard deviation was low as 10–13, which was around0.4% of total photon counts. Other possible locations of unwantedcondensation such as the manometer probe and the piston innersurface 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 cloudsat the operating temperature (75 �C). First, the combined effectsof the IR radiation from optical components and other thermal sig-nals associated with the chamber and surroundings were measuredas photon counts with the shutter closed without any water cloudpresent; this signal is denoted as R0,1. The IR source radiation wasadded to this configuration by opening the shutter; this signal isdenoted as R0,2. The same procedure, repeated with the watercloud present, resulted in signals denoted as R1 and R2. The differ-ence between the first pair of readings gives a signal proportionalto the radiation from the IR source incident on the water cloud.The difference between the second pair of readings gives a signalproportional to the radiation transmitted through the water cloud.The ratio of these two differences gives the transmissivity of thewater cloud to the IR source radiation, which includes absorptionby the droplets, multiple scattering by the droplets, and reflectionand a small amount of absorption by the chamber walls.

sjk¼3:1�4:95l¼R2 � R1

R0;2 � R0;1(1b)

Efforts were made to minimize background noise by recordingphoton 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 thecamera 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 ofassumptions 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 affectedby only the water cloud; molecular gases such as water vapor,nitrogen, and oxygen and unactivated (interstitial) aerosol wereassumed to have negligible effect on absorption or scattering ofradiation from the IR source. This assumption is based on wellestablished research [21,22]. It is accepted that liquid water’smass absorption coefficient is at least three orders of magnitudelarger than that of water vapor and aerosol in the spectral regionof interest.

As noted above the semi-empirical MADA [20] was adoptedfor computing single water droplet volumetric absorption andscattering efficiencies, assuming spherical droplets.

Qe;m ¼ Qef (2a)

Qe ¼ Re 2þ 4e�x

xþ 4 e�x � 1ð Þ

x2

� �(2b)

f ¼ 2� exp �x�23

� �(2c)

where x ¼ 2kxþ iq and q ¼ 2x n� 1ð Þ. Eliminating complexnotation [23] gives

Qe ¼ 2þ 4u2 cos 2bð Þ � 4e�q tan bð Þ u sin q� bð Þ þ u2 cos q� 2bð Þ� �

(3)

with tan bð Þ ¼ ðk=n� 1Þ and u ¼ ðcos bð Þ=qÞ. The volumetricabsorption efficiency is

Qa;v ¼ 1þ 2

1e�1 � 2

121� e�1ð 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 efficiencyabove to distinguish it from “surface” absorption, which will beincluded below. The scattering efficiency (which is always only asurface phenomenon) is

Qs ¼ Qe � Qa;v: (5)

The spherical water droplets in the cloud chamber were assumedto be monodisperse in size throughout the chamber for any givenexperimental condition. This assumption was checked previouslyby 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 bewritten as

Ke;a;s ¼ pr2Qe;a;sN0 ¼3Qe;a;sfv

4r(6)

where r is the radius of water droplet, fv is the volume fraction ofwater droplets

fv ¼4

3pr3N0 (7)

and N0 is the number density of water droplets, which is the sameas the number density of activated nucleation sites or CCN. At a

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given temperature condition, but for different water loadings (dif-ferent fv) the value of No was assumed to be constant such that thedroplet 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 ofthe simulation results. The reference droplet diameter is referred toas d* corresponding to fv¼ 1.6� 10�4. 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, assumingconsistency of the aerosol in that environment.

Absorption of radiation at the surface of water droplets wasalso incorporated along with volumetric radiation absorption inorder to account for phase-transition absorption. Surface absorp-tion can occur at multiple surfaces if the volumetric absorptioneffect 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 absorbedby 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 becommon and pervasive phenomena, their simultaneous occurrencein the same atom or molecule is not generally thought to be acommon occurrence. Phase-change energy is normally thought ofas appearing in the form of phonons rather than photons, and asbeing conducted to or away from the phase-change interface afteror before any (volumetric, not interfacial) thermal radiationoccurs. However, interfacial phase-change radiation, as notedabove, has become increasingly recognized and a more detaileddiscussion of its theoretical basis has been made elsewhere [24].For present purposes, it may be noted that in local thermodynamicequilibrium the fraction of incident monochromatic radiation offrequency v absorbed at an interface due to evaporation [24] maybe represented as

fs ¼a21

4p2

H

Bvng

ffiffiffiffiffiffiffiffiffiffiffiffiffi8pkBT

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 Planckfunction, and mg is the molecular mass of water. The coefficienta21 is the radiative transition probability associated with a liquidwater molecule at the surface evaporating into the gas phase andis related to the spontaneous Einstein coefficient by

A21 ¼ a21N0r2

ffiffiffiffiffiffiffiffiffiffiffiffiffi8pkBT

mg

s(9)

Further details for the surface absorption theoretical frameworkcan be found in Ref. [5].

The discussion above pertains to the probability of absorptionat 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 whichdroplet volumetric absorption is not optically thick, surfaceabsorption may happen at interfaces other than the first oneencountered by the propagating incident radiation. The followingdescription 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 ofapproximately 20 lm diameter at mid-IR wavelengths (3–5 lm)would be in this category (x� 1). Figure 2 shows a schematic ofhow multiple encounters with different parts of the evaporatingsurface of a droplet gives rise to enhanced opportunity for surfaceabsorption. As radiant energy encounters the first interface, theincident energy may be reflected, absorbed or transmitted. Thisbalance 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 theinterface 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 thedroplet will encounter the droplet surface at another locationwhere 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 thesummation of all the possible surface absorptions

Qa;s ¼ fs þ sssvfs þ sssv rssvð Þfs þ sssv rssvð Þ2fs þ � � �

¼ fs 1þ sssv

1� rssv

� �(11)

The internal transmittance can be approximated as one minus theinternal volumetric absorption efficiency

sv ¼ 1� 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 reflectivityof a flat surface subjected to diffuse irradiation [23]. Thus, the sur-face reflectivity for a smooth spherical surface with collimatedirradiation is

Fig. 2 Ray tracing (geometric optics) model of multiple chan-ces for surface absorption

Fig. 3 Transmissivity versus liquid water volume fraction at75 �C, experimental results and MC simulations for fixed proba-bility constant (a21) and varying d* (lm)

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rs ¼1

p

ð2p

q0k cos hdX ¼ 2

ðp=2

0

q0k cos h sin hdh (13)

where q0k is the Fresnel directional spectral reflectivity, which canbe calculated from the Fresnel relations [23] assuming unpolar-ized radiation

R? ¼n1 cos h1 � uð Þ2þv2

n1 cos h1 þ uð Þ2þv2(14)

Rjj ¼n2

2 � k22

cos h1 � n1u

� �2þ 2n2k2 cos h1 � n1v½ 2

n22 � k2

2

cos h1 þ n1u

� �2þ 2n2k2 cos h1 þ n1v½ 2(15)

q0k ¼1

2R? þ Rjj

(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 liquidwater, and h is incident angle from air to water liquid. Intermedi-ate parameters u and v are given by

2u2 ¼ n22 � k2

2 � n21 sin2 h1

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2

2 � k22 � n2

1 sin2 h1

2þ4n22k2

2

q(17)

2v2¼� n22�k2

2�n21 sin2 h1

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2

2�k22�n2

1 sin2 h1

2þ 4n22k2

2

q(18)

Finally, the droplet overall surface absorption efficiency can beadded to the internal volumetric absorption efficiency to give thetotal (surface and internal-volumetric) droplet absorptionefficiency.

Qa ¼ Qa;v þ Qa;s ¼ Qa;v þ fs 1þ1� rs � fsð Þ 1� Qa;v

1� 1� Qa;v

rs

" #(19)

The single-scattering albedo is then

x0 ¼Qs

Qe¼ Qs

Qs þQa¼ Qs

QsþQa;v þ fs 1þ1� rs � fsð Þ 1�Qa;v

1� 1�Qa;v

rs

" #

(20)

The scattering phase function was calculated using Rayleigh-Gans scattering theory [19], instead of anomalous diffractiontheory. 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 thescattering efficiency anyway. This approach thus treats diffractedradiation in the simulation as if it were not scattered at all. Thephase function for Rayleigh-Gans scattering is

P hð Þ ¼ C3

u3sin u� u cos uð Þ

� �2

1þ cos2 h

(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 transferin the cloud chamber because of its versatility and accuracy. Thedetails of the procedure used were discussed previously [5]. Acylindrical system was used for the geometry of the cloud cham-ber. A spectral range of k¼ 3.1–4.95 lm with a spectral resolutionof 0.01 lm was used. As in the previous study [5] the minimum

number of emitted energy bundles was chosen above 4000 at eachwavelength. The spectral transmissivity is denoted by

sMC;k ¼Ss;2 � Ss;1

k

Ss;2 � Ss;1

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 liquidwater volume fraction, fv, ranging from 0 to 2� 10�4 for75 6 1.5 �C. At small fv, (which also means small d; recall that dis assumed to vary as fv

1/3 for constant No), the transmissivity ishigh 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 volumetricextinction coefficient (and thus optical depth along the path lengththrough 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. Inaddition to that primary trend there are secondary variations ofextinction efficiency, albedo, and phase function with droplet size.

Measurements at the smallest values of fv (smallest d) are themost difficult for several reasons. First, the relative uncertainty inthe measured fv value is the greatest (due to the uncertainty inmeasuring 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 areintrinsically unstable, or the delicate balance of maintaining satu-rated vapor-liquid mixture conditions in the gas suspension whilekeeping the chamber surfaces, including windows, slightly abovethe dew point temperature becomes more difficult. The smallest fvvalue at which measurements could be made with a cloud in thechamber was 0.1� 10�4. Below this value the cloud dropletswould rapidly evaporate and the measured transmissivity wouldincrease 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 ExperimentalResults for 75 �C. Figure 3 also shows the comparison of MCsimulation with experimental results for the 75 �C case. The prob-ability constant (a21) is 3� 10�8. This value is slightly lower thanthat used previously for the 60 �C case (5� 10�8) because previ-ously surface absorption at multiple interfaces was not includedwhereas here it is. Droplet diameter was determined as before bylooking for a best fit between the simulation results and the meas-ured data for transmissivity. The best match was found ford*¼ 17 lm. This value is smaller than the value of 20 lmobtained at 60 �C but is consistent with a higher CCN numberdensity and is still typical of cloud droplet sizes. Table 1 lists vari-ous reference droplet diameters that were tried. Reference size isindicated by an asterisk (d*) and means the droplet size at the con-dition fv

*¼ 1.6� 10�4. The N0 values found in this study as listedin Table 1 are of the same order of magnitude (104 cm�3) ormaybe slightly larger than typical values reported for urban areaswith industry [25]. Urbana, IL is a small urban area without muchindustry so the indicated N0 in Table 1 are actually a little higherthan typical reported values. However, reported values are fortemperatures 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 temperaturesallow activation of larger inorganic salt aerosol particles for the

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same supersaturation, which means more CCN and higher N0 withhigher temperature for the same supersaturation. We did not mea-sure supersaturation in our cloud chamber, which would be verydifficult, but we believe that it was both very small and relativelyconstant due to the slow cooling rate and the consistency of cool-ing conditions between experiments. The fact that the results inFig. 3 for multiple experiments correlate so well with an assumedconstant N0 indicates that the supersaturation was relatively con-stant between different experiments.

Initial attempts at simulation were made using the same CCNnumber density as the previous 60 �C case, 3.8� 104 cm�3, withtemperature dependence included only through the terms notedabove in Eq. (8). This approach, however, was not able toadequately match the experimental data. In order to do so theCCN number density was increased by approximately a factor oftwo to 6.22� 104 cm�3. Temperature-dependence of CCN con-centration has not been discussed as much in the literature assupersaturation dependence, notwithstanding both explicit andimplicit temperature dependencies in Kohler theory. Recentlytemperature dependence of CCN parameters was reported [26];however, the temperature dependence was also a transformationof the supersaturation dependence, a variation of which alsoexisted in the experiment along with a temperature variation. Tobe able to theoretically account for the change in CCN with tem-perature would require knowledge of the aerosol composition andsize distribution, which was not available. In the absence of betterinformation about CCN number density as a function of tempera-ture, an increase of CCN with temperature was assumed here thatresulted in a reasonable match with measured data, as shown inFig. 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 settingone or the other of Qa or Qs to zero in Qe¼QaþQs for the caseof no surface absorption. The results are shown in Fig. 4 withthree cases: pure scattering, non-absorbing, Qe¼Qs; pure absorb-ing, nonscattering, Qe¼Qa; and mixed scattering-absorbing(Qe¼QaþQs). For the pure scattering case the transmissivity isessentially unchanged from the mixed case for fv< 0.15� 10�4,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 tocontribute to extinction, even becoming stronger atfv¼ 1.5� 10�4. The effect of surface absorption Qa,s can also beseen because it was neglected even in the mixed scattering-absorption simulation. For values of fv> 0.4� 10�4 the transmis-sivity difference between experimental and MC simulation resultsis noticeable and is due to the absence of surface absorption in thesimulation. Further discussion of the how the trade-off betweenscattering 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 probabilityconstant (a21) for a fixed droplet diameter (d*¼ 17 lm). Thebest value and the one adopted here was a21¼ 3� 10�8. This

corresponds to a surface radiative transition probability of one outof 3.33� 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 atfv 5 1.6 3 1024

fv d* (lm) CCN (cm�3)

1.6� 10�4 15 9.05� 104

16 7.46� 104

17 6.22� 104

18 5.24� 104

19 4.46� 104

20 3.82� 104

21 3.30� 104

Fig. 4 MC simulation and experiment results comparison at75 �C in the absence of surface absorption for pure absorbing(nonscattering), pure scattering (nonabsorbing), and mixedabsorbing and scattering droplets

Fig. 5 MC simulation results for different surface absorptioncoefficients (i.e., varying, a21) at 75 �C

Fig. 6 MC simulation results for different d* with no surfaceabsorption contribution (a21 5 0) at 75 �C

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when surface absorption was included, confirming the importanceof phase transition radiation in the saturated conditions of thecloud chamber.

Figures 7 and 8 show the theoretical surface and volumeabsorption efficiencies over the 3–5 lm spectral range for the lim-iting volume fractions (droplet diameters), fv¼ 0.1� 10�4

(d¼ 6.75 lm) and fv¼ 2� 10�4 (d¼ 18.3 lm). At k¼ 4 lmwater’s absorption index k is relatively small, which leads to neg-ligible volumetric absorption; however, the surface absorptioncontribution is significant at this wavelength. This shows the im-portance of including surface absorption in atmospheric spectralwindows. On the other hand near k¼ 3 lm, surface absorption isrelatively weak compared with volumetric absorption, due to thestrongly absorbing fundamental intramolecular O-H stretchingbands, symmetric (v1) and asymmetric (v3). As wavelengthapproaches k¼ 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 beextended 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 isstill not well understood. It seems at least worth exploring the ideathat phase-change radiation involving aerosol or haze (muchsmaller than cloud droplets) could be a contributing mechanism,along with the current popular dimer and far-wing absorptiontheories.

4.3 Comparison With Previous Results at 60 �C. The newtransmissivity 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� 10�4. Theprimary 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 apparentlya secondary one compared to the CCN effect. Another change intransmissivity versus volume fraction (droplet diameter) with tem-perature is in the slope variations of the curve. At 75 �C the curveis smoother; the slope variations are essentially monotonic. At60 �C the slope of the data has nonmonotonic character with no-ticeable slope changes such that there is a relatively flat portion attransmissivity of 0.4 for volume fractions between 0.3� 10� 4

and 0.5� 10�4. These slope changes may seem subtle in lookingat 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 �Cmeasurements 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 thepossibility of multiple surface absorption. Therefore this effectwas included in the present simulations. A slight change in theprobability constant (a21), from 5� 10�8 to 3� 10�8, was madeto 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 multisurfaceabsorption properties led to a good a match with the experimentalresults, notwithstanding the more complicated curve shape for the60 �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

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4.4 Transmissivity Prediction at 50 �C. In order to gainsome understanding of how the present findings might extrapolateto atmospheric temperatures and to test the simulation further,transmissivity at 50 �C was simulated by extrapolating the CCNvalues used for 60 �C (3.82� 104 cm�3) and 75 �C(6.22� 104 cm�3) to 2.22� 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 shownin 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. Atsmaller optical depth conditions the different slope variations seenin the pure scattering and pure absorbing curves of Fig. 4 becomemore noticeable in the composite (scattering plus absorption) result.This calculation is based on an extrapolation that may not be validand 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 formid-IR transmissivity of saturated water clouds were conducted toinvestigate phase-transition radiation absorption by water and toexplore the temperature dependence of this interfacial absorption.New measurements at 75 �C and 1 atm were made and comparedwith previous results at 60 �C. The MC simulations were used toquantify 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 formeasured transmissivities. Additional absorption was needed andthis was provided theoretically by a surface absorption contribu-tion, attributed to phase-change radiation absorption at the waterdroplet-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 volumetricabsorption is the weakest, even this small amount of surfaceabsorption is relatively significant. The same would likely holdtrue in the 8–12 lm window near 10 lm.

The effect of increasing temperature from 60 �C to 75 �C wasfor transmissivity to decrease at a given water loading (or effec-tive droplet diameter), i.e., optical depth increased. This change isprimarily attributed to an increase in the number density of drop-lets associated with an increase in number density of activatedaerosol or CCN with temperature. Theoretical assessment of thisobservation with temperature-dependent Kohler theory would bedesirable if aerosol information could be obtained. Based on theresults at 60 �C and 75 �C a calculation was made at 50 �C thatshowed 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 explainingwater’s infrared absorption continuum in the 3–5 and 8–10 lmatmospheric windows. Atmospheres that appear to be cloud-freeor fog-free often still contain aerosol and enough water vapor tocause deliquescence of a fraction of the aerosol particles, thesmaller and more hygroscopic ones. Thus dynamic evaporationand condensation of water can occur in atmospheres even withoutvisible clouds or fog and do so on tiny particles with tremendousspecific surface area. Further theoretical and experimental investi-gation of phase-transition radiation and water “vapor” continuumabsorption is needed.

Acknowledgment

Support for this work was provided by the National ScienceFoundation under Grant Number 1062361 and the Hermia G. SooProfessorship.

Nomenclature

a21 ¼ probability of radiative relaxationA21 ¼ Einstein coefficient for spontaneous emissionBv ¼ blackbody functionC ¼ normalization constant in phase functiond ¼ droplet diameter

d* ¼ reference droplet diameter at fv¼ 1.6� 10�4 (lm)f ¼ correction factor in MADA theory or interface

absorptionfv ¼ water droplet volume fractionh ¼ Planck constantH ¼ population distribution function at gaseous-statek ¼ imaginary refractive index

kB ¼ Boltzmann constantKa ¼ absorption coefficientKe ¼ extinction coefficientKs ¼ scattering coefficientM ¼ number of energy bundles

mg ¼ mass of a water moleculen ¼ real refractive index

N0 ¼ number density of water dropletsng ¼ number density of water-vapor moleculesP ¼ phase function of scattered energy

Qa ¼ absorption efficiencyQe ¼ extinction efficiency

Qe,m ¼ corrected extinction efficiency in MADAQs ¼ scattering efficiency

r ¼ droplet radius or interface reflectivityR ¼ polarized Fresnel reflectances ¼ path lengthT ¼ temperature

u,v ¼ intermediate calculation quantitiesx ¼ droplet size parameter, 2pr/k

Greek Symbols

k ¼ wavelengthq ¼ reflectivitys ¼ transmissivity

Subscripts

a ¼ absorptione ¼ extinctiong ¼ gaseous states ¼ scatteringv ¼ volumetric

Abbreviations

ADA ¼ anomalous diffraction approximationCCN ¼ cloud condensation nuclei

Fig. 11 MC estimation of transmissivity at 50 �C, based onextrapolation of simulations at 60 �C and 75 �C

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FPA ¼ focal plane arrayIR ¼ infrared

LTE ¼ local thermal equilibriumLW ¼ longwave (IR) radiation

MADA ¼ modified anomalous diffraction approximationMC ¼ Monte Carlo analysisNIR ¼ near infrared radiationSW ¼ shortwave (VIS and NIR) radiationVIS ¼ visible radiation

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