A new theoretical method for calculating temperature and water vapor saturation ratio in an expansion cloud chamber

Authors

  • Nobuhiro Moteki,

    Corresponding author
    1. Department of Earth and Planetary Science, Graduate School of Science, University of Tokyo, Tokyo, Japan
    • Corresponding author: N. Moteki, Department of Earth and Planetary Science, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan. (moteki@eps.s.u-tokyo.ac.jp)

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  • Yutaka Kondo

    1. Department of Earth and Planetary Science, Graduate School of Science, University of Tokyo, Tokyo, Japan
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Abstract

[1] The expansion cloud chamber is a widely used apparatus for investigating the dynamics of condensational growth of aerosols and clouds. Theoretical calculations of temperature T and water vapor saturation ratio S are necessary for quantitative interpretations of experimental data obtained from the expansion cloud chamber. In this paper, we revisit the thermodynamics associated with the underlying assumptions for calculating the time-dependent temperature T(t) and saturation ratio S(t) in an expansion chamber as a function of experimentally observable parameters. We introduce an intuitive and robust method, the virtual path (VP) method, by which changes in the thermodynamic state of a moist air parcel containing cloud droplets are schematically represented on a thermodynamic diagram. The validity of the VP method is confirmed by comparisons with the differential equation (DE) method, which is a numerical simulation of real physical processes according to the time evolution equations involving T and S. In contrast to the conventional DE method, the governing equations of the VP method do not involve time t, an irrelevant parameter in the framework of classical thermodynamics. The VP method is advantageous compared to the DE method because the former is applicable to the raw experimental data acquired with a finite time resolution, allowing a robust calculation of the T and S values and the errors that are only caused by the measurement errors of the input data.

1 Introduction

[2] Predicting the rate of condensation (evaporation) of water vapor onto (from) cloud droplets under a given value of saturation ratio S is an important subject in the physics and chemistry of aerosols and clouds, but the quantitative aspect remains controversial [Mozurkewich, 1986; Kolb et al., 2010]. The expansion chamber is one of the experimental techniques for measuring the rate of droplet growth under a given saturation ratio [e.g., Wagner, 1975; Winkler et al., 2006]. In previous studies using the expansion chamber, the typical time of the expansion process was set to be only several milliseconds so that the expansion completed before the formation of cloud droplets [Wagner, 1975; Wagner and Strey, 1981; Fladerer and Strey, 2003; Winkler et al., 2006]. It has been believed that the temperature after expansion can be predicted by the Poisson equation without consideration of any latent heat released by expansion that occurs so quickly. However, according to classical thermodynamics, the Poisson equation is mathematically deduced by supposing that infinitesimal work dW done by the air parcel continues to be equal to pdV during the expansion, where p and V are pressure and volume, respectively. This expression of dW as a function of macroscopic parameters is correct only if the internal pressure continues to be homogeneous and balances the external pressure during the expansion process. To satisfy this physical condition, the speed of expansion has to be much slower than the velocity of sound, which is the typical speed of instantaneous relaxation of spatially inhomogeneous pressure. As the expansion speed becomes faster than the sound velocity, the work done by the air parcel becomes smaller because the air parcel no longer exerts an outward force per unit area on the moving boundary equivalent to the internal equilibrium pressure. During this expansion process, the dW decreases to appreciably less than pdV, where the p is macroscopic internal pressure defined at any position far inward from the moving boundary. If the limit of the expansion speed is much faster than the sound velocity, the outward force at the moving boundary decreases to zero, and the total work W done by the air parcel during the expansion will vanish. This extreme is physically equivalent to the free expansion of a gas in a vacuum, under which conditions the temperature of the gas does not change [Reif, 1965]. In previous studies, the Poisson equation was repeatedly used for calculating the temperature change associated with expansion without careful consideration of the underlying assumption of the Poisson equation and the limitation on the expansion speed [Wagner, 1975; Wagner and Strey, 1981; Fladerer and Strey, 2003; Winkler et al., 2006]. As a recent example, Winkler et al. [2006] expanded the gas within their chamber with a volume V ~ 35 L (they referred to Rudolf et al. [2001] for the volume of their chamber) within Δt ~ 5 ms, corresponding to the typical expansion speed V1/3t ~ 7 × 101 m s−1. This is not negligibly small compared to the sound velocity in air ~3 × 102 m s−1. Although they used the Poisson equation to predict temperature changes associated with expansion, the magnitude of changes would be overestimated because of a possible breakdown of the quasi-static approximation, which is an important assumption in the Poisson equation.

[3] In this paper, we carefully deduce the time evolution equation for the temperature of an expanding moist air parcel containing cloud droplets without any mathematical approximations, along with clarifications of the physical assumptions including the quasi-static approximation. To satisfy the requirement of slow expansion speed, we construct theoretical formulations so that condensation or evaporation of water vapor can occur during the expansion process. This is one of the major differences from the previous expansion chamber studies; to our knowledge, all previous expansion chamber studies assume completion of the expansion before the onset of the condensation of water vapor [Wagner, 1975; Wagner and Strey, 1981; Fladerer and Strey, 2003; Winkler et al., 2006]. We introduce a new intuitive and robust method, the virtual path (VP) method, for calculating the thermodynamic state of a moist air parcel in an expansion cloud chamber. The VP method is more concise compared to the conventional method using the time evolution equations adopted in the existing numerical models of cloud dynamics, because the governing equations of the former do not involve time t, an irrelevant parameter under the framework of classical thermodynamics. In section 2, we derive fundamental differential equations for predicting the time evolution of the temperature and the saturation ratio of a moist air parcel as a function of observable parameters in an expansion cloud chamber. In section 3, we introduce physical concepts and mathematical formulas of the VP method. In section 4, we confirm the validity and advantages of the VP method through comparisons with the differential equation (DE) method—the numerical simulation of real physical processes—by solving the time evolution equations derived in section 2. The symbols and subscripts of physical quantities used in the following sections are summarized in Table 1.

Table 1. List of Symbols and Subscripts
SymbolsDescriptionsUnits
clSpecific heat capacity of liquid waterJ K−1 kg−1
cpSpecific heat capacity at constant pressureJ K−1 kg−1
DpDiameter of cloud dropletsm
eWater vapor pressurePa
lvSpecific enthalpy of vaporization of waterJ kg–1
mMasskg
ndNumber concentration of cloud droplets(kg dry air)−1
pPressurePa
qMass mixing ratiokg (kg dry air)−1
rRadial distance from the originm
RIdeal gas constantJ K−1 kg−1
SSaturation ratio-
tTimes
TTemperatureK
φSpecific entropyJ K−1 kg−1
ρwDensity of liquid waterkg m−3
κThermal diffusivity of airm2 s−1
εRatio of molecular weights of water vapor and dry air (=0.622)-
ΨEntropy of air parcelJ K−1
 
SubscriptsMeaningsApplied Variables
dDry airm, p, φ, R, cp, q
vWater vaporm, p, φ, R, cp, q
lLiquid waterm, φ, q
wTotal waterq
sSaturation conditione, qv
0Initial conditionp, T
obsObserved valuep, q

2 Time Evolution of Temperature and Saturation Ratio

[4] In this section, we derive several differential equations for predicting the time evolution of the temperature and the saturation ratio of a moist air parcel, as a function of observable parameters in expansion cloud chambers. The observable parameters are assumed to be the droplet number concentration per unit mass of dry air nd, the temperature and dew point of the moist air before expansion T0 and Td, respectively, the time-dependent pressure p(t), and the droplet diameter Dp(t) during expansion. The constant-angle Mie scattering technique is an established experimental method used to observe nd and Dp(t) in expansion chambers [Wagner, 1985; Szymanski and Wagner, 1990].

2.1 Time Evolution of Temperature

[5] We determine the differential equation necessary to predict the time evolution of temperature from the principle of entropy conservation under quasi-static and adiabatic conditions. Adiabatic conditions for an expanding air parcel are satisfied within a limited time window where the effects of thermal conduction from the surroundings are negligible. The duration of this time window can be estimated by numerical simulations of the thermal conduction under the given boundary conditions. Without vapor condensation or evaporation, the quasi-static condition is satisfied while the equality dW = pdV is true during the expansion process. If vapor condensation or evaporation occurs during expansion, then the thermodynamic description of a moist air parcel by single macroscopic parameters (e.g., pressure, temperature, and water vapor mixing ratio) requires that the volume of the nonequilibrium region near cloud droplets is negligibly small compared to the whole volume of the moist air parcel. In this paper, we implicitly assume this requirement as in the existing models of cloud physics [Straka, 2009; Lamb and Verlinde, 2011]. Because of this, the accuracy of the quasi-static approximation, or equivalently the invariance of total entropy under the condition dW = pdV, is likely better for smaller |S − 1| values in the presence of cloud droplets. Quantitative discussion on the limit of the applicability of a quasi-static approximation requires solving the Boltzmann transport equations [Vincenti and Kruger, 1965] to evaluate the local entropy as a function of time and space inside the moist air parcel, including cloud droplets, which is beyond the scopes of this paper and the current theoretical framework of cloud physics. In this paper, we assume that the total entropy of a moist air parcel is invariant during the expansion even if condensation or evaporation occurs, as in conventional models of cloud physics [Straka, 2009; Lamb and Verlinde, 2011].

[6] Total entropy Ψ of the moist air parcel can be expressed as

display math(1)

where m and φ are mass and specific entropy and the subscripts v, d, and l indicate water vapor, dry air, and liquid water, respectively. If we differentiate equation ((1)) with time t, while considering the mass conservation of dry air (dmd/dt = 0) and total water (dmv/dt = −dml/dt), we have

display math(2)

[7] If we divide this equation by md, using the definition of enthalpy of vaporization lv

display math(3)

and the definitions of mass mixing ratios of water vapor qv and liquid water ql,

display math(4)

we have

display math(5)

[8] If we use the thermodynamic formulations of infinitesimal heat dQ and work dW for a quasi-static process (i.e., dQ = Tdφ and dW = pdV), and we apply the equation of state of ideal gas for dry air and water vapor, the time derivative of the entropy of each component can be written as

display math(6)
display math(7)
display math(8)

where cpd and cpv are the specific heats at constant pressure, Rd and Rv are the ideal gas constants, pd and e are the partial pressures of dry air and water vapor, respectively, and cl is the specific heat of liquid water. We neglect to include the compressibility of liquid water in the derivation of equation ((8)), which is valid for the pressure of atmospheric interest. If we substitute equations ((6))–((8)) into equation ((5)), introducing symbol cp for the specific heat of a moist air parcel

display math(9)

and use the following equality for the time derivative of vapor pressure e:

display math(10)

we have

display math(11)

where ε in equation ((10)) is the molecular weight ratio of water to dry air (=Rd/Rv). Assuming quasi-static processes, the total entropy of the moist air parcel does not change (dΨ/dt = 0), and the left-hand side of equation ((11)) vanishes. By rearranging the resulting equation, we have the temperature differential equation:

display math(12)

[9] The time evolution of the temperature of a moist air parcel can be calculated by numerical integration of equation ((12)) under the given initial conditions T = T0 at t = 0, provided that all the experimental parameters on the right-hand side (RHS) are known as a function of time. The water vapor mixing ratio qv, dry air pressure pd, and their time derivatives on the RHS of equation ((12)) are immediately calculated from the observable parameters, as shown in sections A1 and A2.

2.2 Time Evolution of the Saturation Ratio

[10] The saturation ratio S is defined as the ratio of actual vapor pressure to equilibrium vapor pressure at a given temperature. If we have temperature T and water vapor mixing ratio qv, the equation for the saturation ratio is immediately defined from the relation

display math(13)

where we have used equation ((A5)) in the second equality. A trivial way to calculate temperature is by taking a numerical integration of its time evolution equation ((12)) from t = 0 to the time of interest.

[11] Instead of equation ((13)), we can also calculate the saturation ratio based on the time evolution equation. By differentiating the defining equation of saturation ratio with time, we have

display math(14)

where the time derivative of equilibrium vapor pressure des/dt can be expressed as

display math(15)

because to an excellent approximation, es is a function of temperature only [Bohren and Albrecht, 1998]. The des/dT term in equation ((15)) can be expressed as the Clausius-Clapeyron equation:

display math(16)

[12] By substituting equation ((15)) with ((16)) into ((14)), we have the time evolution equation of the saturation ratio

display math(17)

where the water vapor mixing ratio qv, dry air pressure pd, and their time derivatives on the RHS are calculated immediately from observable parameters as shown in Appendix A. The saturation ratio is calculated by numerically integrating the differential equation ((17)) to the time of interest, under the initial condition S = es(Td)/es(T0) at t = 0.

[13] At this stage, we have two methods for calculating the time-dependent saturation ratio. The DE method involves calculating both temperature and saturation ratio by means of numerical integrations of the time evolution equations (i.e., equations ((12)) and ((17))). The second method requires calculating the temperature by numerically integrating the time evolution equation and calculating the saturation ratio directly from equation ((13)); we call this second method the differential equation only for temperature (DEoT) method. In this paper, we will use the DE and DEoT methods later to validate and demonstrate the advantages of the VP method, which will be developed in the next section.

3 Virtual Path Method

[14] In the DE and DEoT methods explained in section 2, we simulate real changes of temperature with time by numerically integrating the time evolution equation of temperature ((12)). Under the assumption of a quasi-static adiabatic process, the temperature is a thermodynamic state variable uniquely determined by the initial and final states regardless of any intermediate paths. Therefore, it is physically permissible to track any other artificial intermediate paths, the virtual paths that can be more advantageous for physical interpretations and numerical accuracy compared to the real path tracked by numerical integration of the time evolution equations. Among infinitely many possible virtual paths, we choose a natural process consisting of segments of dry adiabatic and moist adiabatic paths as explained below along with illustrations on skew T-log p diagrams (Figure 1). We consider the quasi-static adiabatic expansion of a moist air parcel with an initial state that has the following parameters p = p0, T = T0, and qv = qw, where the qw is the total water mixing ratio.

Figure 1.

Virtual thermodynamic paths used for calculating the temperature T for (a) supersaturated and (b) subsaturated parcels of moist air containing condensed water schematically illustrated on a skew T-log p diagrams.

[15] If liquid water does not exist in the final state, temperature T is immediately estimated by tracking the dry adiabatic path from the initial pressure p0 to the final pressure pobs.

[16] If liquid water does exist in the final state with water vapor mixing ratio qvobs (<qw), water vapor can be supersaturated (S > 1) or subsaturated (S < 1) with respect to liquid water (equilibrium condition S = 1 can be regarded as a limit of |S − 1| → 0 for super or subsaturation). From the initial pressure at p0, we track the dry adiabatic path down to the pressure pc where saturation is reached so that qvs(T,p) = qw. This point is physically equivalent to the lifting condensation level defined in meteorology. From pc, we track the moist adiabatic path down to pressure p0′ where the saturation water vapor mixing ratio equals the observed water vapor mixing ratio (qvs(T,p) = qvobs). Along the moist adiabatic path, we condense water vapor so that the value of qv decreases from the initial value qw to the final value qvobs. From this point, we track the dry adiabatic path again from p0′ to pobs. The temperature of the final state T is determined at the end of the second dry adiabatic path where p reaches pobs. If p0′ at the starting point of the second dry adiabatic path is greater than pobs, water vapor is supersaturated (Figure 1a); the opposite is the case when water vapor is subsaturated (Figure 1b). This completes the qualitative explanation of the VP method. In the following subsections, we derive mathematical expressions for the three segments of the VP method: (1) the first dry adiabatic path, (2) the moist adiabatic path, and (3) the second dry adiabatic path.

3.1 The First Dry Adiabatic Path

[17] The first dry adiabatic path satisfies the conditions ql = 0, qv = qw, and dqv/dt = 0. The time evolution equation for temperature, equation ((12)), under these constraints is

display math(18)

where cp is defined here as

display math(19)

[18] According to equations ((A6)) and ((A9)), equation ((18)) can be rewritten as a function of observable parameters p and dp/dt instead of pd and dpd/dt:

display math(20)

[19] If we define a constant (invariant during the first dry adiabatic path)

display math(21)

and separate the variables in equation ((20)), we have

display math(22)

[20] Integrating equation ((22)) from the initial state to the final state, we have

display math(23)

where the final pressure p must be greater than the pc so that

display math(24)

where the saturation mixing ratio qvs(T,p) is defined as

display math(25)

[21] The inequality ((24)) must be satisfied on the first dry adiabatic path.

3.2 Moist Adiabatic Path

[22] Once the water vapor reaches saturation at the end of the first dry adiabatic path, we begin to track the moist adiabatic path from this point. The water vapor mixing ratio qv on the moist adiabatic path continues to be the saturation water vapor mixing ratio qvs(T,p). By imposing the saturation condition qv = qvs on equation ((12)), we have

display math(26)

where the constant cp is defined as

display math(27)

[23] The time derivative of qvs as it appears on the RHS of equation ((26)) can be written as

display math(28)

[24] If we use equations ((15)) with ((16)) as the des/dt on the RHS, we have

display math(29)

[25] If we substitute equation ((29)) into the dqvs/dt on the RHS of equation ((26)) and rearrange the resulting equation, we have

display math(30)

where the parameter β is dependent on temperature and pressure and is defined as

display math(31)

[26] By differentiating the equality pd = p − es with respect to time and using equation ((15)) with ((16)) for the des/dt, we have

display math(32)

where the parameter γ is dependent on temperature and is defined as

display math(33)

[27] If we substitute equation ((32)) into ((30)) and rearrange the terms of the resulting equation, we have

display math(34)

and in time-independent form, the equation becomes

display math(35)

[28] The moist adiabatic path is calculated by numerical integration of equation ((35)). The following inequality describing the water vapor mixing ratio continues to be satisfied on the moist adiabatic path

display math(36)

[29] The first and second equalities are true at the start and end points of the moist adiabatic path, respectively. The observed water vapor mixing ratio qvobs is calculated from equation ((A3)), from the total water mixing ratio qw, and from the observed liquid water mixing ratio qlobs calculated from equation ((A2)).

[30] Note that the moist adiabatic path defined here is slightly different from the wet adiabatic lines drawn on a skew T-log p diagram conventionally used for meteorological analysis. The wet adiabatic lines represent the pseudoadiabatic process in which it is assumed that the condensed water is immediately removed by precipitation and does not contribute to the specific heat of the moist air parcel [Bohren and Albrecht, 1998].

3.3 The Second Dry Adiabatic Path

[31] Now, we define new symbols T0′ and p0′ as the initial temperature and initial total pressure of the second dry adiabatic path. The second dry adiabatic path is characterized by conditions qv = qvobs and dqv/dt = 0. By imposing these constraints on equation ((12)), we have

display math(37)

for which the cp is defined here as

display math(38)

[32] Substituting equations ((A6)) and ((A9)) into the pd and dpd/dt in equation ((37)), we have

display math(39)

[33] If we introduce a constant α′ (invariant during the second dry adiabatic path) defined as

display math(40)

and separate the variables in equation ((39)), we have

display math(41)

[34] Integrating equation ((41)) from (T0′, p0′) to the final state (T, pobs)

display math(42)

where the T is the temperature of the final state. If water vapor is supersaturated, p0′ > pobs and T < T0′ in equation ((42)), then T is less than Tmoist that is expected if supersaturation (S > 1) is relaxed to equilibrium (S = 1) by condensation of excess water vapor at constant pressure pobs (Figure 1a). Similarly, if water vapor is subsaturated, p0′ < pobs and T > T0′ in equation ((42)), then T is greater than the temperature Tmoist that is expected if subsaturation (S < 1) is relaxed to the equilibrium condition (S = 1) by evaporation of excess liquid water at constant pressure pobs (Figure 1b).

[35] In this section, we have considered both supersaturated and subsaturated conditions for completeness, although supersaturated conditions may more frequently occur in expansion cloud chamber applications.

4 Experimental Data

[36] In order to obtain expansion cloud chamber data that can be used as realistic input parameters to test our methods for calculating temperature and saturation ratio, we have designed and conducted a simple expansion cloud chamber experiment. Sample air was taken from the room; this sample contained water vapor and aerosols which act as cloud condensation nuclei. We used a 5 L round-bottomed flask as the chamber, and the chamber pressure was slowly reduced from an initial pressure p0 = 1000 to ~750 hPa about 4 s after opening an exhaust valve connecting the chamber to the vacuum pump (model VP0435A, Nitto-Koki, Japan). An electric pressure transducer (model 276, SETRA systems, USA) was used to measure pressure p(t) in the chamber during expansion. The dew point Td and temperature T0 of the air sample at initial conditions were 19.0°C and 26.2°C, respectively, as measured by a mirror-type electric dew point sensor (model DPH-203, Tokyo Opto-Electronics, Japan). The concentration of total aerosols (smallest particle diameter ~10 nm) in the sample air was measured by a condensation particle counter (CPC model 3022A, TSI, USA) in units of cm−3. After conversion of the unit using the ambient pressure and temperature, the aerosol concentration was determined to be 2.8–2.9 × 109 particles per kilogram of dry air (i.e., (kg dry air)−1). In our experiment, the droplet number concentration nd was not measured directly. We assumed the values of nd so as to be less than the concentration of total aerosols, because the homogeneous nucleation of water vapor which requires S > ~4 [Wagner and Strey, 1981] is very unlikely in our experiment, as to be expected from the typical value of S shown in the next section. A droplet diameter Dp(t) greater than ~8 µm was estimated by analyzing a colored image of the corona produced by the cloud droplets illuminated by a beam of incandescent light based on the Fraunhofer diffraction theory, assuming that blue-, green-, and red-colored corona rings correspond to the wavelengths of 0.45, 0.54, and 0.65 µm, respectively. In this method, the measurement error of Dp was roughly defined as the standard deviation of Dp values determined from the corona rings of all observed combinations of color and order. An application of this method for polystyrene latex spheres of 10 and 25 µm nominal diameters (JSR Corp., Japan) suspended in water resulted in estimated diameters of 9.7 ± 0.1 and 25.2 ± 0.5 µm, respectively, confirming reasonable accuracy of this method. Colored images of the corona every 0.15 s during expansion were acquired by a commercial USB camera connected to a computer during each run. The duration of the time window and the region inside the chamber where the adiabatic condition is satisfied have been estimated by numerical simulations of the thermal conduction under realistic initial and boundary conditions. The full details of the simulation are shown in section A3. Based on the simulation results, we have limited the region for observing corona rings to within a half radius from the center of the flask, so that the possible effects of thermal conduction from the wall are negligible within the duration of an expansion (~4 s). In Figure 2, we show the measured time-dependent droplet diameter Dp(t) and pressure p(t) with their empirical functions in the form of y(t) = a + b tc with the parameters a, b, and c have been determined by a least squares method. In this paper, we skip the technical details of the experiment because rigorous evaluation of accuracy and precision is not of primary importance for the data, which is only used for demonstrating the theoretical calculations. The full details of our experimental system and a calibration procedure for measurements of Dp from colored images of the corona are discussed in more detail in H. Aoki et al. (Corona-imaging colorimetric method for accurate measurement of the size of water droplets in an expansion chamber, submitted to Aerosol Science and Technology, 2013).

Figure 2.

Time course of cloud droplet diameter Dp and pressure p measured in an expansion chamber for a sample of room air and their empirical functions.

5 Results and Discussion

[37] In this section, by comparing the VP method to the DE method, we validate and demonstrate the advantages of the VP method as a way to calculate the time-dependent temperature and saturation ratio. We use the experimental data described in section 4 as the input parameters necessary for these calculations. For the purpose of comparisons between the VP and DE methods, rather than inputting the discrete observed data, it is necessary to input the empirical functions of p(t) and Dp(t) (Figure 2) derived from fitting the observations. The VP and DE methods agreed to within the last four or five significant digits when the temperature and saturation ratios as a function of time were calculated, assuming an nd = 2.5 × 109 (kg dry air)−1 (Figure 3). Additionally, the precise agreement between the VP and DE methods was unchanged when the assumed value of nd varied, even though S(t) markedly changes depending on nd (Figure 4). In the case of nd = 2.55 × 109 (kg dry air)−1, S(t) decreased to be less than 1 for t > ~3 s. The complete agreement with the DE method under both supersaturated and subsaturated conditions demonstrates the validity of the newly introduced VP method. Because S(t) is obviously sensitive to the input value of nd (Figure 4), an accurate measurement of nd is mandatory for quantitative experiments involving the condensation dynamics for which the absolute value of S(t) is critically important.

Figure 3.

Time courses of temperature T and saturation ratio S calculated by the VP, DE, and DEoT methods for the experimental data shown in Figure 2. The DEoT curve was completely overlapped by those of VP and DE in (b).

Figure 4.

Time courses of saturation ratio S calculated with the VP, DE, and DEoT methods under the same conditions as Figure 3 except for the assumed values of (a) nd = 2.45 × 109 (kg dry air)−1 and (b) nd = 2.55 × 109 (kg dry air)−1.

[38] According to thermodynamics theory, assuming a quasi-static adiabatic process, temperature and saturation ratio at the final state depend only on the initial state, irrespective of the thermodynamic path. Therefore, any changes of Dp(t) in the past do not affect the temperature T(t) or saturation ratio S(t) of the present and future. We compared the VP, DE, and DEoT methods for the default Dp(t) and a modified Dp(t) (Figure 5). In the time domain when t ≥ 1.2 s, both T(t) and S(t) calculated by the VP method are independent of the change to Dp(t) when t < 1.2 s, indicating the correctness of the numerical results by the VP method. In contrast, T(t) and S(t) calculated by the DE and DEoT methods were changed markedly at t ≥ 1.2 s, depending on the Dp(t) at t < 1.2 s. The results produced by the DE and DEoT methods for a modified Dp(t) are not physically possible, and computational errors occur due to the discontinuity of the input parameters qv and dqv/dt at t = 1.2 s. Numerical integrations of differential equations ((12)) and ((17)) accurately proceed provided that the input parameters remain continuous with time. The magnitude of the numerical error in S(t) is smaller for the DEoT method than that of the DE method because the DEoT method only differentiates temperature. These comparisons demonstrate an advantage of using the VP method compared to the direct simulations of the DE method: The numerical results of the VP method are correct regardless of any behavior of the Dp(t) in the past even while input parameters are discontinuous with time.

Figure 5.

Time courses of (a) temperature T and (b) saturation ratio S calculated with the VP, DE, and DEoT methods for the default Dp(t) and a modified Dp(t) that is zero when t < 1.2 s and is equal to the default Dp(t) when t ≥ 1.2 s.

[39] The applicability of the VP method to discrete input data allows us to calculate the T and S values directly from the raw data with measurement error (Figures 2 and 6). We can recognize the significance of artificial errors in S(t) caused by using the empirical Dp(t) function instead of the raw Dp(t) data (Figure 6b). The use of the VP method with raw experimental data is advantageous for avoiding systematic errors caused by the use of empirical functions and for calculating the magnitude of the error propagated from the input data.

Figure 6.

Time courses of (a) temperature T and (b) saturation ratio S calculated with the VP method using the raw Dp(t) data with measurement error and the empirical Dp(t) function. The magnitude of the S(t) error is affected solely by the measurement error of Dp(t).

6 Conclusions

[40] In this paper, we have elucidated the underlying assumptions and principal limitations in the theory of expansion cloud chambers, which had not been discussed carefully. After the clarification of the theoretical basis, we developed the intuitive and robust VP method for calculating the time-dependent temperature T and saturation ratio S of a moist air parcel during quasi-static adiabatic expansions. The VP method uses virtual thermodynamic paths consisting of dry adiabatic and moist adiabatic paths so that the physical interpretations are better than those from simulations of real thermodynamic processes by using the time evolution equations (DE method). In contrast to the DE method, the VP method is theoretically concise and practically flexible because the time t, a physically irrelevant parameter, is not involved in the governing equations. The absence of t allows the use of temporary discrete experimental data of droplet diameter Dp and pressure p as inputs, avoiding the possible systematic errors caused by using the empirical functions approximating the data. By means of the VP method, we can calculate the magnitudes of the errors in T and S directly propagated from the measurement errors of Dp and p. Because of these advantages, we recommend the VP method rather than the DE method for use in future expansion chamber experiments.

Appendix A

A1 Water Vapor Mixing Ratio

[41] We present explicit formulations of water vapor mixing ratio qv and its time derivative dqv/dt as a function of observable parameters, the droplet number concentration per unit mass of dry air nd and the droplet diameter Dp. Total water mixing ratio qw is calculated from the measured dew point Td and the pressure before expansion p0 with the following equation:

display math(A1)

where ε is the molecular weight ratio of water to dry air (=0.622) and es(T) is the equilibrium vapor pressure of water as a function of temperature, which can be calculated from the Clausius-Clapeyron equation. We calculate the liquid water mixing ratio ql from Dp and nd using the following equation:

display math(A2)

where the ρl is the density of liquid water. The water vapor mixing ratio qv and its time derivative dqv/dt are calculated by

display math(A3)

and

display math(A4)

respectively. We use equations ((A1))–((A3)) and ((A4)) for calculating the water vapor mixing ratio qv and its time derivative dqv/dt, respectively, from directly observable parameters in an expansion chamber.

A2 Dry Air Pressure

[42] We derive explicit formulations of dry air pressure pd and its time derivative dpd/dt as a function of known parameters. By rearranging the defining equation of water vapor mixing ratio qv = εe/pd = εe/(p − e), we have

display math(A5)

[43] If we substitute this expression into the defining equation of total pressure p = pd + e, we have the explicit expression for dry air pressure as a function of known parameters

display math(A6)

[44] The time derivative of the dry air pressure is

display math(A7)

[45] If we substitute equation ((10)) into the second term on the RHS of equation ((A7)) and rearrange the resulting equation, we have

display math(A8)

or

display math(A9)

once equation ((A6)) is substituted into the pd in equation ((A8)). We use equations ((A6)) and ((A9)) for calculating the dry air pressure and its time derivative from known parameters.

A3 Effects of Thermal Conduction From the Wall

[46] We have estimated the temporal and spatial ranges where the adiabatic assumption is satisfied in our expansion chamber experiments by simulating thermal conduction from the wall during cooling by expansion. Because of the near-spherical shape of our expansion chamber (round-bottomed flask), we assume that the geometry for the numerical simulation is spherically symmetrical. The governing equation is the following thermal conduction equation with a source term

display math(A10)

where r is the radial distance from the center of the flask and κ is the thermal diffusivity of air depending on pressure and temperature [Shoji, 1995]. The source term on the RHS of equation ((A10)) corresponds to the cooling by quasi-static expansion. We imposed the boundary condition T = 300 K at the inner wall of the flask (r = 0.1 m) and the initial condition of T = 300 K everywhere at t = 0. In equation ((A10)), we did not include the heating term due to condensation so that we could estimate the possible upper limit of the effect of thermal conduction from the inner wall on the saturation ratio. The empirical p(t) function shown in Figure 3 was used as input data for this calculation. Figure A1 shows the calculated radial profiles of T(r,t) at t = 1, 2, 3, and 4 s and the corresponding change of the saturation ratio compared to the case neglecting thermal conduction. Within the duration of an expansion experiment t < 4 s, errors of the saturation ratio greater than 0.001 are unlikely inside a spherical volume with r < 0.05 m; therefore, it is safe to confine the region for observing cloud droplets to within a half radius from the center of the flask, provided that the assumption that a saturation ratio error less than 0.001 is unimportant.

Figure A1.

Numerical simulation of the radial profiles of temperature T (dashed lines), and the relative change of saturation ratio S (solid lines) from that neglecting thermal conduction Sad inside the expansion chamber (r < 0.1 m) at four different times t =1, 2, 3, and 4 s.

Acknowledgments

[47] This work was supported by the JSPS KAKENHI grant 23221001, and the Global Environment Research Fund of the Japanese Ministry of the Environment (A-1101).

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