Validation of Activity Coefficient Models Using Resonances in Light

Apr 24, 2008 - theoretically predicted, and the times at which resonances appear are calculated from Mie theory. By minimizing the errors between the ...
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Ind. Eng. Chem. Res. 2008, 47, 3963–3973

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GENERAL RESEARCH Validation of Activity Coefficient Models Using Resonances in Light Scattering from Evaporating Multicomponent Droplets HaoHua Tu and Asit K. Ray* Department of Chemical and Materials Engineering, UniVersity of Kentucky, Lexington, Kentucky 40506-0045

We present a technique for determining parameters of any activity coefficient model from the times at which resonances appear in light scattering spectra from an evaporating multicomponent droplet of known initial composition. Using individual component droplet evaporation data along with assumed values for the parameters of a chosen activity coefficient model, the size and the composition of the droplet as functions of time are theoretically predicted, and the times at which resonances appear are calculated from Mie theory. By minimizing the errors between the calculated and observed resonance appearance times, the optimum parameter values of the activity coefficient model are established. The technique has been applied to examine binary and ternary systems consisting of diethyl phthalate, dipropyl phthalate, and dimethyl phthalate. In addition, the technique has been validated by comparing the prediction from the optimum parameter values with independently determined activity coefficient data for binary mixtures of dioctyl phthalate and dimethyl phthalate. 1. Introduction Vapor–liquid equilibrium (VLE) data are critical in designing many industrial processes and modeling atmospheric phenomena. Over the past two decades, several investigations have focused on the extraction of VLE data from experiments on evaporation and growth of single microdroplets that were suspended in electrodynamic balances. These studies are based either on quasi-steady-state evaporation of constant composition droplets or on unsteady-state evaporation or growth of droplets. Ray et al.1 and Ray and Venkatram2 have employed quasisteady-state droplet evaporation technique to determine activity coefficients of water-glycerol and ethanol-glycerol systems. In these studies, a glycerol droplet evaporated under a quasisteady-state condition while the activity (i.e., partial pressure) of the volatile component (e.g., water or ethanol) in the gas phase remained constant. The droplet maintained a dynamic equilibrium with respect to the volatile constituent in the gas phase and, thus, evaporated as a constant-composition droplet. They measured the droplet size as a function of time and its composition during the evaporation period from the resonances observed in light-scattering spectra. They deduced the activity coefficient of the volatile component from the knowledge of the droplet composition and the gas-phase activity, and they deduced that of the nonvolatile component from the size-versustime data. Even though the size and composition data independently provide activity coefficients of both components, the applicability of the technique is limited to binary systems whose components differ markedly in volatility. In addition, each experiment at a specific gas-phase activity provides activity coefficient data only at one composition. Recently, Tu and Ray3 have extended the concept of dynamic equilibrium to measure the activity coefficient of water in glycerol from the unsteady-state growth of a glycerol droplet in a diffusion cloud chamber. By introducing a constanthumidity salt solution in the chamber, they allowed the partial pressure of water vapor in the gas phase surrounding the droplet * Corresponding author. Tel.: +1-859-257-7990. Fax: +1-859-3231929. E-mail: [email protected].

to develop with time in a well-defined manner, while the droplet maintained a dynamic equilibrium with the water vapor during its growth. Using the resonances observed in the light scattering by the droplet, they determined the droplet size and its composition as functions of time and the activity coefficient of water in glycerol from the knowledge of the droplet composition and the water vapor saturation ratio in the gas phase. Using this technique, the activity coefficient of the volatile component over a wide range of compositions can be determined from a single experiment, but this technique provides only the activity coefficient of the volatile component of a binary system containing volatile and nonvolatile components. Experiments on multicomponent droplets evaporating under unsteady-state, nonequilibrium conditions (e.g., evaporation in vapor-free atmosphere) can also provide activity coefficient data over a wide range of compositions, since the composition of an evaporating droplet changes continuously because of the stripping of the volatile constituent. Allen et al.4 were the first to use nonequilibrium binary droplet evaporation data to determine activity coefficients of several binary organic mixtures. They have used balancing voltage (i.e., relative mass change) data and angular light-scattering data along with resonances observed in scattering intensity-versus-time spectra to infer the size and the composition of an evaporating droplet. Their technique requires time-consuming analysis of an extensive amount of data, in addition to visual matching of experimental spectra with theoretical spectra. Aardahl et al.5 have used simultaneous elastic and Raman scattering data from evaporating microdroplets to examine thermodynamics of binary mixtures of halogenated hydrocarbons but suggested that inelastic scattering data can be used to determine activity coefficients of systems containing more than two components. Light-scattering data can be used for simultaneous determination of the size and the refractive index of a droplet; thus, the composition of a binary droplet can only be deduced from the refractive index. This fact has limited microdroplet-based thermodynamic studies to binary systems, except for a study by Widmann and Davis.6 They avoided direct determination of

10.1021/ie070851g CCC: $40.75  2008 American Chemical Society Published on Web 04/24/2008

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activity coefficients but showed that experimental size-versustime data obtained from evaporating droplets consisting of up to five components agree well with predictions based on the UNIFAC model for activity coefficients. It is impractical to measure VLE data of systems containing many components. In most cases, VLE properties of such a system are predicted from experimental VLE data on binary systems using excess Gibbs free energy models.7 One needs to, however, evaluate unknown parameters and validate the Gibbs free energy model for a multicomponent system using experimental data. Light-scattering techniques based on resonances can be used for that purpose. Resonances (i.e., intensity peaks) appear in a scattering intensity-versus-time spectrum from an evaporating or a growing droplet that is illuminated by a fixed-wavelength laser beam. Since each resonance provides a unique relation between the size and the refractive index of the scattering droplet, these entities can, in principle, be determined as functions of time from the observed resonances. Early studies have determined size and refractive index by visually matching experimental scattering spectra with those calculated from Mie theory. This method provides accurate results when the size or the square of the size varies linearly with time,1,2,8–10 as in the case of evaporation of constant-composition droplets. For nonlinear size variation of a droplet that occurs during unsteadystate evaporation or growth, the visual matching techniques are not reliable, unless supported by secondary data such as relative mass change values from balancing dc voltages.4,11,12 Recently, Tu and Ray13 have developed an automated procedure for analysis of time-dependent resonant spectra from single microdroplets that undergo dynamic size and refractive index changes. The procedure can accurately determine size and refractive values at the resonances when the relations for the variations of size and refractive index as functions of time or the identities of the observed resonances are known. In a later study, they14 have unambiguously validated the technique using resonances from evaporating droplets of an ideal binary solution. They accurately recovered the properties (i.e., refractive index and evaporation rate) of the individual components of the binary system by using the technique. It is the purpose of this study to demonstrate that resonances observed during evaporation of a multicomponent droplet can be used to validate an activity coefficient model or to determine optimum values of parameters of an activity model that best describes the activity coefficients of the components of the droplet. The method relies on the fact that, for a given initial radius and composition of a multicomponent droplet, we can predict its size and composition as functions of time, if the evaporation rates of single component droplets of the constituents, as well as the activity coefficients as functions of composition, are known. In this study, we determine the evaporation rates of the individual components of a multicomponent droplet from the single-component-droplet evaporation data and assume an activity coefficient model and the values of the parameters associated with the model. We predict theoretical resonance appearance times from Mie theory using the droplet size and composition history computed from the droplet evaporation theory. We compare predicted resonance appearance times with the times resonances were observed in the experimental spectra from the droplet and estimate optimum values for the parameters of the activity coefficient model from the minimum in the alignment errors between the predicted and observed resonance appearance times. The reliability of this technique is ensured by the complete agreement between experimental and theoretically calculated spectrum shapes. The

technique has been validated with experimental data from binary droplets of diethyl phthalate (DEP)-dipropylphthalate (DPP), DEP-dimethylphthalate (DMP), DPP-DMP, and DMPdioctylphthalate (DOP) systems. The technique yields highly reproducible activity coefficient relations, and the results from DMP-DOP droplets show excellent agreement with previously reported activity coefficient data obtained using a different technique. The techniques has also been applied to data from ternary droplets of the DMP-DEP-DPP system and used to check whether the parameters of binary activity coefficient models for DMP-DEP, DMP-DPP, and DEP-DPP systems can be extended to the ternary system. 2. Droplet Evaporation Theory There are a number of rigorous models (e.g., Newbold and Amundson15) that accurately describe the evaporation of a multicomponent droplet. These models can be simplified without any loss of precision for various situations. In this section, we briefly review the model used to describe the evaporation of an experimental droplet involved in this study. The experimental droplets consisted of mixtures of relatively nonvolatile components. For such slowly evaporating droplets, the droplet phase can be assumed to be well-mixed, and the evaporation process can be considered isothermal. In addition, quasi-steady-state mass flux relations without the convection term (i.e., dilute solution) can describe species transport from the interface of a droplet to the gas phase. Under these conditions, for evaporation in a vapor-free gas phase, an interfacial mass balance gives the rate of change of species i in a droplet of radius of a, containing S species as γixiP0i (T∞) dni ) -4πaDim dt RT∞

for i ) 1 to S

(1)

where ni is the number of moles of i in the droplet, Dim is the diffusion coefficient of i in the surrounding gas phase, Pi0(T∞) is the vapor pressure of i at the gas-phase temperature T∞, γi is the activity coefficient at the droplet-phase mole fraction xi, and R is the universal gas constant. For a single-component droplet (i.e., ni ) 4πa3/3Vi, xi ) 1, and γi ) 1), the above relation can be integrated to obtain a2 ) a20 -

2DimViP0i (T∞) t RT∞

(2)

where a0 is the initial droplet radius, and Vi is the liquid-phase molar volume of i. The slope of the linear relationship between the square of the droplet size and time, si )

2DimViP0i (T∞) RT∞

(3)

is termed as the evaporation rate of a pure droplet of component i. For a multicomponent droplet, the droplet composition and the radius are related to the number of moles of each species present in the droplet through the following relations, xi )

ni

(4)

S

∑n

k

k)1

and a)

(

S

∑

)

3 nV 4π i)1 i i

1⁄3

(5)

Ind. Eng. Chem. Res., Vol. 47, No. 11, 2008 3965

Figure 1. Experimental system based on an electrodynamic balance inside a diffusion cloud chamber. Table 1. Relevant Properties of the Compounds Studied physical properties

a

experimentally determined values

compound

M.W.

density, F, kg/m3

refractive indexa, m

evaporation rate, si, µm2/s

DMP DEP DPP DOP

194.19 222.24 250.29 390.56

1190 1118 1078 981

1.5101 ((0.0001) 1.4972 ((0.0001) 1.4919 ((0.0001) 1.4815b

0.2180 ((0.0022) 0.0778 ((0.0008) 0.0178 ((0.0002) 5 × 10-4

Parenthetical values indicate standard deviations from measurements of this study. b Estimated value.

j i is the liquid-phase partial molar volume of the ith where V species at the droplet composition xi. Using eqs 3-5, eq 1 can be rewritten as

(

S

∑

)

dni 3 ) -2π xV dt 4π k)1 k k

1⁄3

siγini S

∑n

Vi

for i ) 1 to S

(6)

k

k)1

Now we have S equations in the form of eq 6 for a droplet of S components. Once the droplet initial conditions (i.e., initial size a0 and composition xi0), the evaporation rate of a singlej k, and component droplet of i, si, the partial molar volumes V the activity coefficients, γi, as functions of composition are known, S simultaneous differential equations, arising from 6, can be solved to obtain the number of moles of each component, ni, as a function of time. The size and composition history of the droplet can be obtained from eqs 44 and 55. Equations 4-6 form the theoretical basis of this study. The evaporation rate si, for each component i, can be obtained accurately from an experiment on a pure droplet of component i, using a technique described by Tu and Ray.13 The partial molar j k, as functions of composition can be ascertained volumes, V from experiments where known amounts of components are mixed, and the final volume of the mixture is measured. When j k equals the liquida system behaves like an ideal solution, V phase molar volume, Vk, of pure k. Now, the remaining unknowns, the activity coefficients, γi, as functions of composition, in the system of equations given by eq 6, must be resolved before the droplet size and composition as functions of time can be predicted. Thus, in this study, the resolution of the activity coefficient relations is reduced to the interpretation of lightscattering data that sensitively reflects the size and composition history of the scattering droplet. This will be further discussed in the data reduction section below. 3. Experimental Section A schematic of the experimental system used in this study is shown in Figure 1. Tu16 has provided a complete description of the system, and we present a synopsis of the pertinent features. A four-ring electrodynamic balance (EDB) is used to suspend a charged droplet. The EDB is housed inside a modified thermal diffusion cloud chamber that is maintained at a constant uniform

temperature by circulating constant-temperature fluid through the top and bottom plates. The chamber has a number of ports that are used for droplet introduction and viewing, laser beam entrance and exit, light-scattering measurements, gas circulation, and evacuation. A 20 mW vertically polarized He-Ne laser beam (λ ) 632.8 nm) is used to illuminate a droplet, and two photomultiplier tubes (PMTs) that are placed in the planes parallel (i.e., PMT 1) and perpendicular (i.e., PMT 2) to the plane of polarization are used to detect light scattered by the droplet at scattering angles of about 90°. PMT 1 and PMT 2 independently detect transverse electric (TE) and transverse magnetic (TM) mode resonances,17 respectively. In a typical experiment, charged droplets were generated from a solution of known composition by electrical atomization, and a droplet was isolated and trapped in the EDB by manipulation of the ac and dc potentials. Care was taken to avoid any partial saturation of the gas phase by the droplet constituents. Dry air was allowed to flow past the suspended droplet for a few minutes, and this ensured removal of vapor from the chamber. The droplet was allowed to evaporate in the vapor-free atmosphere, and two PMTs were used to record lightscattering intensities from the droplet as functions of time. In this study, we have examined droplets generated from mixtures of DEP, DPP, and DMP in various combinations, that include single-component, binary-component, and ternarycomponent droplets, in addition to binary droplets of DMP-DOP. All the experiments were conducted at 25.3 °C, and the relevant physical properties of the chemicals used in this study are listed in Table 1. 4. Light Scattering Scattering intensity-versus-time data obtained from an evaporating droplet were analyzed using Lorenz-Mie theory18–22 to obtain the size and refractive index of the droplet. When a homogeneous spherical particle of radius a is illuminated by monochromatic light of wavelength λ, the intensity of light scattered elastically at a point by the particle is a function of the refractive index of the particle m and its size parameter, which is defined as X ) 2πa/λ. The variation of the size parameter causes modulation in the scattering intensity, and a scattering intensity-versus-size parameter spectrum is marked by the presence of peaks that are referred to as morphology-

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appearance times, we aligned each resonance peak observed at time ti,o, with a theoretical resonance peak located at Xi, for an assumed value of m. We regressed Xi versus ti data generated by such alignments of all the observed peaks to eq 7 and computed alignment errors from the relation Npt

S(X0, m, R, β) )

∑ (X

2

i

- X02 + Rti - βti2)2

(8)

i)1

Figure 2. Comparison of observed scattering-intensity spectra from a DPP droplet with spectra calculated for size parameter relation and refractive index value obtained from alignment procedure.

dependent resonances (MDRs). A resonance is labeled by a set of three MDR parameters (n, l, and µ), where n and l denote the mode and order numbers, respectively, and µ indicates polarization and assumes a value of either 1 for TM mode or 2 for TE mode. The value of X, at which a resonance appears, depends solely on m and can be computed precisely from the theory. By identifying the resonances observed in an experimental scattering intensity-versus-size parameter spectrum, and by aligning them to computed resonances, the size and refractive index of the scattering particle can be obtained with high precision.13,17,23 The variation of size during evaporation of a droplet brings about a number of resonances in the scattering intensity-versustime spectrum from the droplet. Figure 2 shows examples of TE and TM mode experimental scattering intensity spectra recorded by PMT 1 and PMT 2 during the evaporation of a pure DPP droplet in dry air. The experimental intensity spectra provide the times {t1,o, t2,o, . . . , tN,o} at which resonances are observed in the spectra, and the problem involves determination of size, a(t), and refractive index, m(t), as functions of time. Tu and Ray13 have described the procedure used to interpret resonances observed in scattering spectra. In the following sections, we briefly describe the procedures used to analyze data from single-component and multicomponent droplets with specific examples. 4.1. Single-Component Droplet Evaporation. During quasisteady-state evaporation of a single-component droplet, the radius of the droplet changes according to eq 2. Because of the presence of minute amounts of nonvolatile impurities, the evaporation rate diminishes as time progresses. We incorporated the effect of this factor through a second-order term used to describe the size parameter as a function of time X2 ) X02 - Rt + βt2

(7)

where X ) 2πa/λ is the size parameter. To determine the initial size parameter X0, the refractive index m of the droplet, and the evaporation rate parameters R and β from the resonance

For the same refractive index, we repeated the procedure by aligning each observed peak to all possible theoretical peaks (i.e., resonances with widths in the range 1 × 10-4 e ∆X1/2 e 2 × 10-2), and then we repeated the entire procedure by incrementing the refractive index value, so as to include all possible values for the refractive index. We selected values of X0, R, β, and m that yielded the global minimum in the alignment errors as the optimum values for the given spectra from a droplet and authenticated the results by visually comparing the experimental spectra with the theoretical spectra computed with the optimum parameter values. For example, for the experimental spectra shown in Figure 2, we chose a set of 24 sharp resonance peaks, labeled 1-24, from the intensity peaks observed in the TE mode spectrum and another set of 25 peaks from the TM mode spectrum. We applied the alignment procedure to these 49 observed peaks and obtained the following optimum values, X0 ) 205.819, R ) 1.7653, β ) 4.58 × 10-6, m ) 1.4919 (9) with a standard relative error in the size parameter of about 7 × 10-4. Figure 2 compares the experimental scattering spectra with the theoretical spectra that were computed using the optimum parameter values in eq 9. The experimental spectra replicate the theoretical spectra even to minute details, thus validating the results of the alignment procedure. In addition to the optimum parameter values, the alignment procedure establishes the identity, that is, the MDR parameters (n, l, and µ), of each of the resonances. In Figure 2, we have indicated the mode and order numbers, n and l, respectively, of some of the theoretical resonance peaks aligned to the experimental peaks. It should be noted that we always obtained negligibly small values for β for all of our data. Although the diffusion theory gives β ) 0, the alignment procedure, however, often fails in the absence of the term involving β in eq 7. This is because each observed resonance specifies size so precisely that a small deficiency in the relation used to align the resonances causes misalignment of peaks. For all the single-component droplets examined, we observed that the values of parameter R and refractive index are highly reproducible for a given component and that β values are negligibly small (i.e., N). Therefore, each experimental resonance observed at time ti,o is then aligned with a calculated resonance whose appearance time, ti,c, is closest to the observed time, that is, ti,c ) tj,c, where j is selected from min |ti,o - tj,c| for j ) 1 to M. After aligning all the observed resonances with a subset of the most probable

theoretical resonances, we calculate the standard error in appearance times from the following relation,

∑ N

E(X0, A, c ) )

1 (t - t )2 N i)1 i,o i,c

(12)

j ) [A1, A2, . . . , An] and jc ) [c1, c2, . . . , where X0 ) 2πa0/λ, A j , and cn]. We repeat the procedure by varying the parameters X0, A jc to obtain the global minimum in alignment errors. The values of j , and jc at the global minimum are chosen as the optimum X0, A values, and the corresponding subset of calculated resonances provides identification of the observed resonances. We validate the entire procedure by matching the shape of the experimental spectra with the theoretical spectra computed using the X(t) and m(t) data j , and jc. obtained for the optimum values of X0, A We illustrate the procedure through a TE mode spectrum shown in Figure 3. The spectrum was obtained from a droplet of the binary DMP-DPP system, initially containing 50 vol % DMP. From the resonances observed in the spectrum, we selected 63 sharp resonances that are indicated by short vertical lines in Figure 3. For this binary system, we have experimentally studied the dependence of the molar volume and the refractive index on the composition by examining solutions generated by mixing the two components at various ratios. We found that the ideal mixing rule (i.e., no volume change due to mixing) was accurate within 0.2%. Similarly, the refractive index measurements show that the following volume-fraction-weighted relation can accurately describe the refractive index of the mixture m ) m1f1 + m2(1 - f1)

(13)

where subscripts 1 and 2 stand for DMP and DPP, respectively, and m1 and f1 are the refractive index and the volume fraction of component 1, respectively. Later, we will present a sensitivity analysis on the volume and refractive index mixing rules. Since we know the initial composition of the droplet, x1,0 ) j 1 ) M1/F1 ) 0.1632 m3/ 0.5871, the partial molar volumes, V 3 j kmol and V2 ) 0.2322 m /kmol, and the evaporation rates of pure DMP and DPP droplets, s1 ) 0.2180 µm2/s and s2 ) 0.0178 µm2/s (see Table 1), respectively, we can solve eqs 4-6, for given initial droplet size a0, and relations for the activity coefficients, γ1 and γ2, in terms of the composition. It should

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better with the observed ones when the nonideality of the binary system is taken into account. Since DMP and DPP are homologous compounds with similar molecular size, shape, and chemical nature, we take slight nonideality of the binary system into account by incorporating the one-suffix Margules model for the activity coefficients: ln γ1 ) Ax22

Figure 4. Results obtained by applying the alignment procedure to resonances observed in the spectrum shown in Figure 3. Each curve represents standard resonance alignment error as a function of the initial radius for a given value of one-suffix Margules parameter A.

be noted that the initial composition is the composition of the parental solution used to generate the droplet. First, we tested an ideal solution behavior (i.e., γi ) 1) for the observed data. For this situation, we only need the initial droplet size a0, to calculate the size, a(t), and the composition, x1(t), history, from eqs 4-6. For an assumed value of a0, we numerically solved the two differential equations arising from eq 6 over the time interval involved in the experimental spectrum shown in Figure 3 and computed refractive index, m(t), as a function of time from the composition x1(t) using eq 13. From the size parameter, X(t), and refractive index, m(t), versus time data thus generated, we calculated all the theoretical resonances and aligned them with the observed resonances using the alignment procedure outlined previously. Then we computed the standard alignment error from eq 12. Since the alignment error, E(X0), for the present situation is a function of the initial size only, we repeated the alignment procedure by incrementing the initial size. Figure 4 shows alignment error, E, versus the initial radius, a0, data obtained by analyzing the 63 observed resonances under the assumption of an ideal solution behavior. The results show that there is a distinct minimum in the alignment error that occurs at an initial radius of a0 ) 22.1652 µm (i.e., the optimum value), with a standard error of about 14.06 s. In Figure 3, we have plotted the theoretical spectrum calculated from the droplet size and the composition versus time data obtained from the solution of eq 6, using the optimum initial size and the ideal solution behavior. The positions of the theoretical resonances are indicated under the theoretical spectrum. Although the theoretical spectrum closely replicate the shape of the experimental spectrum, a closer inspection reveals that the theoretical resonances are not well-aligned with the observed resonances, as indicated by the high alignment error. These misalignments between the theoretical and observed resonances suggest that the ideal solution behavior does not adequately represent the DMP-DPP system, and the theoretical resonances may align

(14)

Now the problem involves optimization of the initial size, a0, and the activity coefficient model parameter A. For a given value of A, we varied the initial size a0, and at each value of a0, we applied the alignment procedure and calculated the alignment error, E(X0, A). We then repeated the procedure by incrementing the value of A. The calculated E versus a0 data for various values of A are shown in Figure 4. The results show that, for each value of A, there is a minimum in the error that occurs at a specific initial size. The minimum value decreases as A increases from the ideal solution behavior (i.e., A ) 0) and then goes through a minimum before increasing. In a three-dimensional plot of E-A-a0, the global minimum occurs at a0 ) 22.2342 µm and A ) 0.0997, with a standard error of about 2.9 s, which is lower than the error for an ideal solution behavior by a factor of about 5. It should be noted that the standard error of 2.9 s, at the optimum parameters, represents an error of