Ind. Eng. Chem. Res. 2003, 42, 4389-4397
4389
Analysis of the Stefan Tube at Supercritical Conditions and Diffusion Coefficient Measurements E. Irmak Ozguler, Sermin G. Sunol, and Aydin K. Sunol* Department of Chemical Engineering, University of South Florida, 4202 East Fowler Avenue, ENB 118, Tampa, Florida 33620
An analysis of the Stefan tube for binary diffusion coefficient measurement of a system comprised of a solid solute and a supercritical fluid has been carried out. The prototype system used to evaluate the approach is naphthalene-supercritical carbon dioxide. An initial theoretical calculation with the available data shows that the diffusion process would be in the unsteadystate region for a practical duration of the experiment. The development of a theoretical model was carried out for systems where the transport of the species is dominated only by diffusion as well as for systems involving convective transport along with diffusion. Tractable analytical solutions are developed for finite control volume, relaxing the semi-unbounded boundary assumption often used. The binary diffusion coefficients were determined at temperatures of 35, 50, and 65 °C for the pressures 80, 100, and 120 atm. The quantification of the convective term showed little effect on the diffusion coefficient for the supercritical naphthalene-carbon dioxide system. The measured diffusion coefficients are in good agreement within each other and with the literature and therefore validate Stefan tube’s applicability for diffusion coefficient measurements at supercritical conditions. 1. Introduction Over the last 2 decades, supercritical processes and supercritical solvents, particularly supercritical carbon dioxide, have gained wider use and more interest.1-12 Therefore, for the design and operation of such processes, the availability of reliable transport properties at the relevant conditions is vital. However, the estimates from available correlations and predictive methods vary greatly and are system dependent, as shown in Table 1, which depicts the variation of diffusion coefficient estimates at the experimental conditions at which this work has been carried out. Therefore, a fundamentally correct description and quantification of physical, thermodynamic, and transport properties of components of interest, particularly in this less understood but promising field, is necessary.13 Funazukuri14 discussed the shortcomings of the diffusion coefficients for various mixtures of carbon dioxide, which obstruct the development of a general predictive model. Matthews and Akgerman15 and Eaton and Akgerman16 developed predictive equations based on the rough-hard-sphere theory to account for an extensive density range of 69.9-500.2 kg/m3 with high accuracy. However, they discuss the need for more experimental data to be able to extend their approach beyond this range for a wider spate of systems. There are a number of experimental techniques developed for the measurement of the diffusion coefficients at supercritical conditions. Liong et al.17 in their review paper discussed the shortcomings and advantages of the existing experimental techniques. Among these, the Taylor-Aris dispersion technique is the most widely employed technique. On the basis of the theory of dispersion of flow in pipes, introduced by Taylor18 and further developed by Aris,19 binary diffusion coefficients * To whom correspondence should be addressed. Tel.: (813) 974-3566. Fax: (813) 974-3651. E-mail:
[email protected].
in the infinite-dilution region can be obtained. Even though chromatographic techniques have proved to be versatile in a number of physicochemical measurements,20,21 the diffusion coefficient measurements with supercritical chromatography have been shown to have some inherent shortcomings. These include the presence of dead volumes, occurrence of secondary flow, excessive pressure drop, and adsorption of the solute onto the capillary walls.22,23 The steady-state technique used by Debenedetti and Reid24 and Tuan et al.25 and the light scattering technique used by Saad and Gulari26 can be used to measure diffusion coefficients at finite concentrations. The steady-state techniques require that steady-state and fully developed velocity profiles be attained before the experiment can proceed. Iomtev and Tsekhanskaya27 and Knaff and Schlunder28 employed diffusion coefficient measurement methods, which were only applied to solid supercritical systems and, therefore, are referred to as the solid dissolution methods. The mathematical description of the systems, for the determination of the diffusion coefficients, were treated as unsteady-state27 and quasisteady-state28 diffusion from a flat plate. These methods of diffusion coefficient experiments, where partial saturation occurs, allow measurements for infinite-dilution and finite concentrations depending on the duration of the experiment. The other technique applied for the diffusion coefficient measurements in solid supercritical systems is the NMR fixed-field-gradient spin-echo Bessel function analysis technique used by Lamb et al.29 The Stefan tube, originally developed from Stefan’s work in 1889,30 was adapted and used for diffusion coefficient measurements by Arnold31 and Slattery and Mhetar32 at near ambient conditions. Slattery and Mhetar developed the concentration profiles for the diffusing solute for a semi-infinite region of the diffusion
10.1021/ie020938w CCC: $25.00 © 2003 American Chemical Society Published on Web 08/19/2003
4390 Ind. Eng. Chem. Res., Vol. 42, No. 19, 2003 Table 1. Diffusion Coefficients for Naphthalene-Supercritical Carbon Dioxide from Different Correlations DAB × 105 (cm2/s) exptl conditions T (°C) P (atm) 35 50 65
80 100 120 80 100 120 80 100 120
Wilke-Chang
modified Wilke-Chang
Hadyuk-Minhas
modified Chapman-Enskog
16.0349 14.4745 13.3130 34.8243 25.2098 20.5161 63.0658 43.7350 28.0218
18.6352 16.8218 15.4719 40.4714 29.2979 23.8431 73.2927 50.8271 32.5658
16.0292 14.9132 14.0592 28.7167 22.8678 19.7766 45.1876 34.9109 25.5079
22.3638 10.7934 9.61783 57.0987 27.3145 15.8027 73.5677 46.3650 29.5253
medium. The significance of the Stefan tube arises from the basics of diffusion in the Stefan tube being closely related to several research areas such as drying of porous materials and considerations on the meso- and macropore level transport, which are very often formulated in terms of fluxes and driving forces for the transfer of the species.33 The system may be considered either steady state or unsteady state depending on the properties of the diffusing species, the medium of diffusion, and the duration of the experiments. The experimental parameter to be measured is concentration or mass changes in order to calculate the diffusion coefficient from the governing relationships of a material balance for a solute for an infinitesimal element within the solvent phase. The resulting equation to be solved is effectively a one-dimensional mass-transfer problem coupled with the boundary conditions. Among others, one important feature of the Stefan tube diffusion approach is that one can operate closer to equilibrium as well as others by adjusting the boundary conditions. This region has been suggested to result, theoretically, in the most accurate measurement of transport properties.13 Alizadeh et al.34 have concluded, after their detailed analysis of the experimental systems, that a small perturbation of the system from equilibrium can be introduced by the pulse which will meet the theoretical constraint that a system has to be as close to equilibrium as possible to make accurate measurements under the conditions. However, the lack of very high-resolution detection equipment, keeping the system at a thermodynamic stable state throughout the diffusion process, and the rigorous description of the phenomena are challenging tasks. Meeting all three of these requirements poses a challenge for all fluids and fluid mixtures. In this paper, we explore the applicability of the Stefan tube in the diffusion coefficient measurements of nonvolatile solids in supercritical fluids. The carbon dioxide-naphthalene system is used as the model system. The effect of possible natural convection on the diffusion process is incorporated in the modeling stage and is further analyzed. 2. Theoretical Background In this problem, solid solute (naphthalene) diffuses from the bottom of the tube into the supercritical fluid (carbon dioxide) that fills the rest of the tube and is swept away by the pure supercritical fluid (carbon dioxide) that is flowing from the top of the tube (Figure 1). The diffusion medium is bounded by the z ) 0 and z ) h planes, in which the z ) 0 plane starts above a thin film region where the supercritical fluid (carbon dioxide) is saturated with the solid solute (naphthalene).
Figure 1. Schematic representation of the Stefan tube for the study of the diffusion of naphthalene in supercritical carbon dioxide.
The entire system is maintained at isothermal and isobaric conditions throughout the experiment. The Stefan tube, with the model system, was analyzed initially with available literature diffusion data35 to determine the mode and duration of operation, i.e., steady or unsteady. The assumptions involved in the development of the system of governing equations with the initial and boundary conditions are as follows: 1. The diffusion coefficient of naphthalene in supercritical carbon dioxide (DAB) is independent of its concentration. Furthermore, the total concentration, c, of the system is assumed to be constant. These assumptions are justified because the naphthalene concentration in the supercritical carbon dioxide is quite low. 2. The solid-supercritical gas boundary is assumed to be stationary (i.e., is not time-dependent). The level change was about 2% for the cases analyzed. Slattery and Mhetar32 also concluded that the moving boundary effect is negligible. 3. The concentration is assumed to be equal to the equilibrium solubility of naphthalene in supercritical carbon dioxide at the solid-supercritical fluid interface. Under the experimental conditions, the mixture of naphthalene-supercritical carbon dioxide is below the mixture upper critical end point that ensures equilibrium between the solid and dense fluid for all pressures.36 4. The solute transported to the tube entrance is completely removed by the carbon dioxide flowing over the tube to ensure that the upper boundary condition is maintained throughout the experiment. This assumption is secured by maintaining high supercritical carbon dioxide flow rates through the top of the tube. 5. In the Stefan tube, the velocity profile is assumed to be flat. In other words, the variation of the velocity in the radial direction is neglected. This is based on the analysis provided by Whitaker.37
Ind. Eng. Chem. Res., Vol. 42, No. 19, 2003 4391
Hence, for the development of the composition profiles, the general equation of continuity for species A in a nonreacting mixture is
∂cR ) -(∇‚NR) ∂t
R ) 1, 2, 3, ..., n
(1)
where NR is the combined molar flux vector of species. NR can be written as NR ) J/R + cRvˆ *, where J/R is the molar flux with respect to the molar average velocity and vˆ * is the molar average velocity. Because the diffusion is only in the z direction, the equation of continuity reduces to
∂yA ∂2yA ∂yA + vˆ /z ) DAB 2 ∂t ∂z ∂z
(2)
diffusion as
NAz0(t) ) yA0[NAz0(t) + NBz0(t)] - cDAB
t)0
yA ) yAi ) 0
/ ) vˆ z0
eq BC1 z ) 0 yA ) yA0 ) yA BC2 z ) h yA ) yAh ) 0
}
for t > 0
In eq 2, the term vˆ /z has to be defined and inserted in the equation before proceeding with the solution. The equation of continuity for a short control volume for compressible flow can be written as
dF dv dA + )0 F v A
(3)
where F is the density (gmol/cm3) of the fluid, v is the velocity (cm/s) of the fluid, and A is the cross-sectional area (cm2) normal to the flow. Because there are no radial area changes, the fractional changes in the density are compensated for by the fractional velocity changes. For the system under study, density changes are small (about 1% maximum) for carbon dioxidenaphthalene system at the experimental conditions. Therefore, the velocity change has been assumed to be small enough for the velocity to be considered constant. Furthermore, the velocity is assumed to be equal to its value at the solid-fluid interface. / vˆ /z ) vˆ z0
(4)
/ vˆ z0
(average molar velocity) can be calculated by The the relation
∫0t vˆ z0/ (t) dt ) ∫0t dt f
/ vˆ z0
f
(5)
/ (t) vˆ z0
is the time-dependent molar velocity and where can be described by
1 / (t) ) [NAz0(t) + NBz0(t)] vˆ z0 c
(6)
Here, NBz0, the flux of carbon dioxide in the stagnant phase, is zero. Therefore, / (t) NAz0(t) ) cvz0
∫0t
(
-
f
|
DAB ∂yA 1 - yA0 ∂z
z)0
)
dt
tf
[
∫0t
f
(
-
|
DAB ∂yA 1 - yA0 ∂z tf
(9)
)
z)0
]
dt
∂yA ∂2yA ) DAB 2 (10) ∂z ∂z
This is the partial differential equation, which describes the transport of naphthalene in supercritical carbon dioxide, in the Stefan diffusion tube. The diffusion medium is the vertical, stagnant supercritical carbon dioxide phase above the solid naphthalene phase. The transported naphthalene is removed from the top of the tube by the supercritical carbon dioxide flowing from the top of the tube horizontally. Because natural convection is negligible, two approaches are suggested to solve eq 2. 1. The term vˆ /z in eq 2 is assumed to be constant and / / . vˆ z ) vˆ z0 2. The convective term is completely neglected and assumed to be zero. 2.1. Solution of Equation 2 for Constant Molar Average Velocity. When the dimensionless variables
θ)
DABt yA z , η) , τ) 2 yA 0 h h
are introduced into eq 2, the following equation is obtained:
∂θ ∂θ ∂2θ +B ) ∂τ ∂η ∂η2
(11)
where B ) vˆ /z h/DAB is the dimensionless convective term. The general form of the solution will be
θ(η,τ) ) θ∞(η) - θt(η,τ)
(12)
where θ∞(η) is the steady-state part of the solution and θt(η,τ) is the transient part of the solution, which fades out as time goes to infinity. The steady-state and transient parts of this general solution are solved separately and then combined for the final solution. At infinite time, the system attains steady state; therefore, the concentration profile, θ∞(η), at τ∞ reduces to a second-order ordinary differential equation with the boundary conditions defined earlier. Solving and applying the boundary conditions, one obtains
(7)
NAz0 can be expressed by the Fick’s first law of binary
(8)
where tf is the total duration of the diffusion experiment. / , if we insert eq 9 into eq 2, we obtain Because vˆ /z ) vˆ z0
∂yA + ∂t
for 0 < z < h
z)0
/ , and then inserting Equating eqs 7 and 8, one gets vz0 into eq 5, one obtains the molar average velocity, which is given by
where the initial and boundary conditions are
IC
|
∂yA ∂z
θ∞ )
exp(Bη) - exp(B) 1 - exp(B)
(13)
4392 Ind. Eng. Chem. Res., Vol. 42, No. 19, 2003
The transient part of the general solution is a secondorder, homogeneous partial differential equation with homogeneous boundary conditions. The solution of this equation was carried out by the separation of variables, which resulted in ∞
θt(η,τ) )
∑ n)1
8nπ
B2 + 4n2π2
[(
exp
)]
B2
exp -n2π2 +
4
[]
τ
B η sin nπη (14) 2
When the transient solution (14) and the steady-state solution (13) are substituted into eq 12, the general solution that includes a significant amount of convection will be obtained:
θ(η,τ) )
exp(Bη) - exp(B) 1 - exp(B)
∞
-
∑ n)1
) ]
θ(η,τ) ) (1 - η) -
2
exp(-n2π2τ) sin nπη ∑ n)1nπ
(16)
Equation 16 was developed independently with the method of Laplace transform for a system with no convection, resulting in the same relation as that in eq 16. The concentration profiles were drawn from eq 16 with the available literature data to decide the duration of the experiments. For this purpose, only the noconvection case concentration profiles were employed through eq 16. No literature data that consider convective forces were found. 2.3. Determination of Diffusion Coefficients for Systems That Include Free Convection. The rate of mass loss of naphthalene from a surface area, S, can be calculated by
∂M ) S(MW)NAz0 ∂t
(17)
where M is the amount of evaporated naphthalene at time t and MW is the molecular weight of naphthalene. The flux of naphthalene at the surface of the solid phase can be written from eq 8, by rearranging as
NAz0 ) -
|
cDAB ∂yA 1 - yA0 ∂z
(18)
z)0
Substituting eq 18 into eq 17, one obtains
|
cDAB ∂yA ∂M ) -S(MW) ∂t 1 - yA0 ∂z
z)0
(19)
(20)
η)0
Integrating eq 20 between τ ) 0 and τ ) τf, where τf ) DABtf/h2 and tf is the duration of the experiments, i.e., 15 h, one obtains the total mass loss, Mf, which is
Mf )
∂θ ∫0M dM ) -Sh(MW)c1 - yA ∫0τ (∂η |η)0) dτ yA0
f
f
(21)
0
Using eq 15, the integral term of eq 21 is determined and the final form of the total mass loss equation is reached.
2
2.2. Solution of Equation 2 for Zero Average Molar Velocity. When the free convection in eq 2 is completely neglected (vˆ /z f 0), B approaches zero. The concentration distribution can be obtained by taking the limit of eq 15. ∞
|
yA0 ∂θ ∂M ) -Sh(MW)c ∂τ 1 - yA0 ∂η
8nπ
B + 4n2π2 B2 B exp -n2π2 + τ + η sin nπη (15) 4 2
[(
In terms of the dimensionless numbers defined earlier, eq 19 describing the rate of mass loss will be
Mf )
( [
Sh(MW)cyA0 DABtf 1 - yA0
2
h
B
+
B
e -1
[(
1 - exp -n2π2 +
∞
∑
2 2
32π n
n)1
) ]
B2 DABtf 4
h2
16n4π4 - B4
])
(22)
Equation 22 includes a convective dimensionless number, B, that should be calculated before determining the diffusion coefficient, DAB. To do so, one starts by / , given by eq 9, into the dimensionless inserting vˆ z0 / term B ) vˆ z h/DAB, resulting in the following equation:
B)
-DAB h DAB tf(1 - yA ) 0
(∫ | ) tf∂yA
0
∂z
z)0
dt
(23)
When the integral term of eq 23 is solved by using the dimensionless numbers and eq 15, the relation for the dimensionless term B is written as
B)
-h2yA0
[
tfBDAB
tfDAB(1 - yA0) h2(1 - eB) ∞
exp -n2π2 +
∑ 32π2n2
n)1
[(
-
) ]
B2 DABtf 4
h2
B4 - 16n4π4
-1
]
(24)
As can be seen from eqs 22 and 24, the terms B and DAB are implicit functions of each other. This causes the complexity for the determination of DAB. An iterative procedure for the determination of DAB was adopted and was carried out using MATLAB. 2.4. Determination of Diffusion Coefficients for Systems That Do Not Include Free Convection. The solution procedure of DAB in this section is similar to that in the previous section. Until eq 21, there is no difference in derivation. Because the convection term is completely neglected for the calculation of DAB, the concentration profile term ∫τ0f(∂θ/∂η)|η)0 dτ in eq 21 is
Ind. Eng. Chem. Res., Vol. 42, No. 19, 2003 4393
Figure 2. Schematic diagram of the system for measurement of diffusion coefficients in the Stefan tube.
evaluated by using eq 16. If convection is completely neglected, eq 21 will become
[
yA0 DABtf Mf ) Sh(MW)c + 1 - y A0 h 2 ∞
2
∑
n)1
[
]
DABtf 1 - exp -n2π2 h2 n2π2
]
(25)
The procedure to compute DAB from eq 25 is simpler than the case involving convection. An iterative procedure is not warranted and is solved using MATLAB. 3. Experimental Section 3.1. Materials. Carbon dioxide was 99.9% pure (Air South Gas Inc., Anna Mariette, GA). Naphthalene (CAS 91-20-3) was certified, crystalline, 99.998% pure and was purchased from Fisher Scientific Co., Pittsburgh, PA. 3.2. Experimental Apparatus. A detailed schematic description of the experimental system is shown in Figure 2. The diffusion tube is a 1/4 in. o.d. and 1/8 in. i.d. stainless steel tube (Swagelok, Solon, OH), which is capped at one end and open to supercritical carbon dioxide that flows as shown in Figure 1. The total length of the diffusion tube was 11.5 cm, and the diffusion medium was 9.6 cm, with the naphthalene loaded. A medium-pressure digital density meter (model DMA60, Anton Paar GmbH, Austria), with a remote measuring cell (model DMA512, Anton Paar GmbH, Austria) for flow-type measurements, is used for the monitoring of the density of the system. The density meter was calibrated with two fluids of known densities for the temperature and pressures for which the diffu-
sion experiments are going to be carried out, based on the procedure outlined by the manufacturer. The diffusion tube and the measuring cell of the density meter are placed in a temperature-controlled environment, which is insulated on all surfaces. The rest of the instrument of the remote measuring cell of DMA512 is placed outside of the temperature-controlled chamber sending the measuring cell signals to ensure constant-amplitude oscillations. The temperature-controlled environment is established and stabilized by circulating the water through the density meter remote cell and the circulation bath prior to the experiment. During this initial period, the temperature readings at several locations are tracked. When the variation between the temperature readings at these locations and the desired experimental temperatures is within (0.02 °C, the system was assumed to be ready for the experiments to commence. The carbon dioxide is delivered to the system with the aid of a syringe pump (model 100DX, ISCO, Inc., Lincoln, NE), which is being cooled by circulating a refrigerant through its jacketed feed section. The carbon dioxide, which is delivered at a constant flow rate, is preheated in an approximately 5-ft-long tubing that passes through the hot water bath reservoir (The Model B81 Haake, Thermo Electron Corp., Waltham, MA) and then the temperature-controlled environment before entering the diffusion tube. This provided sufficient time and distance for carbon dioxide to reach the system temperature. The pressure-monitoring system includes the syringe pump’s proportional-integral-derivative controller at the pump exit, an inline pressure gauge in the temperature-controlled environment that is upstream to the diffusion tube entrance, and the automated backpressure regulator (Thar Technologies, Inc., Pittsburgh, PA) that is at the exit of the system. The temperature measurement and monitoring throughout the system is accomplished by high-precision platinum RTD probes (Weed Instruments, Co., Inc., Round Rock, TX) and a digital thermometer (model
4394 Ind. Eng. Chem. Res., Vol. 42, No. 19, 2003 Table 2. Collected Data and the Calculated DAB Values for Convection and Nonconvection Systems
T (°C)
P (atm)
35
80
35
100
35
120
50
80
50
100
50
120
65
80
65
100
65
120
evaporated naphthaleneb (mg)
measured densityc (g/cm3)
mole fraction of naphthalene solubilityd
molar densitye (gmol/cm3)
2.2 2.1 4.4 4.3 5.1 5.1 0.4 0.4 2.1 2.1 5.3 5.2 0.4 0.4 1.0 1.0 2.4 2.5
0.4648 0.4650 0.7272 0.7268 0.7792 0.7800 0.2269 0.2254 0.4098 0.4097 0.5995 0.5988 0.1850 0.1856 0.2752 0.2752 0.3953 0.3951
0.004 986 0.004 986 0.008 135 0.008 135 0.010 650 0.010 650 0.001 226 0.001 226 0.004 869 0.004 869 0.010 435 0.010 435 0.001 347 0.001 347 0.002 838 0.002 838 0.007 285 0.007 285
0.010 461 0.010 466 0.016 270 0.016 261 0.017 352 0.017 369 0.005 144 0.005 110 0.009 226 0.009 223 0.013 355 0.013 340 0.004 192 0.004 206 0.006 219 0.006 219 0.008 859 0.008 854
convection included (Ba * 0, vz * 0) vz × 107 DAB × 104 (cm/s) (cm2/s) B × 103 14.710 15.409 30.529 31.220 48.242 48.289 2.449 2.436 12.974 12.971 34.323 34.941 2.463 2.469 6.277 6.277 24.424 24.412
3.840 3.662 4.935 4.826 5.364 5.359 1.419 1.429 4.154 4.155 7.242 7.114 1.741 1.735 2.934 2.934 4.944 4.947
2.505 2.281 1.552 1.484 1.067 1.065 5.562 5.631 3.074 3.075 2.025 1.954 6.786 6.747 4.487 4.487 1.943 1.945
no convection (B ) vz ) 0) DAB × 104 (cm2/s) 2.510 2.286 1.554 1.486 1.068 1.066 5.581 5.650 3.079 3.081 2.021 1.950 6.804 6.766 4.505 4.505 1.945 1.947
a Dimensionless convective term (hv /D ), where h is the height of the stagnant gas layer, v is the average molar velocity, and D z AB z AB is the diffusion coefficient. b For each temperature and pressure combination, two experiments were conducted. c The measured density of the binary mixture, Cg. d The solubility mole fraction of naphthalene was calculated using the SRK EOS to fit literature data. e The molar density of the binary mixture was calculated by using the relation C ) Cg/MWs, where MWs is the molecular weight of the solution and is calculated by MWs ) yA0MWN + (1 - yA0)MWCO2, where MWN is the molecular weight of naphthalene, MWCO2, the molecular weight of CO2, and yA0, the equilibrium solubility molar fraction of naphthalene.
2180A, Fluke Co., Everett, WA) equipped with a 10channel selector (model Y2000, Fluke Co., Everett, WA). The supercritical carbon dioxide-naphthalene mixture is swept from the Stefan tube’s top and passes through the density meter, and then the mixture flows to the automated backpressure regulator. The carbon dioxide-naphthalene gas mixture expands past the backpressure regulator, with carbon dioxide bubbling through a collection vessel filled partially with water that collects the solids, naphthalene. An analytical balance (model AND FR-300, Itin Scale Co., Inc., Brooklyn, NY) of precision 10-4 was used in the determination of the solid naphthalene weight loss, which was done by weighing the diffusion tube before and after each experiment. 3.3. Experimental Procedure. The first step was to load the solid naphthalene into the diffusion tube from the end that is capped during the experiments. The procedure was carried out with extreme care by placing a solid rod of outer diameter that is almost as wide as the inner diameter of the tube, i.e., 1/8 in., in the tube to block the other diffusion direction end. Next, a very small spatula was used to transfer powdered naphthalene in the order of 0.01 g at a time to the tube. After each transfer, another rod, of the same size as the one inside the tube, is used to compress the naphthalene. This filling procedure was continued until approximately 1 g of naphthalene was loaded into the Stefan tube. This charging procedure was done only once for the nine data points, which were duplicated. The height of the diffusion space was measured and noted before each experiment, even though it remained essentially constant because of the very small amount evaporated. The weighing of the tube and naphthalene contents was carried out before and after each experiment with an analytical balance. At the startup of an experiment, the temperaturecontrolled environment is brought to a steady-state temperature distribution by setting the circulator temperature. The diffusion tube is weighed and then
connected to the supercritical carbon dioxide line in the temperature-controlled environment. The connection points are checked for any leaks. The backpressure regulator is set to the experiment pressure, which keeps the system pressure at the desired set value. The system is filled with carbon dioxide via the syringe pump, and carbon dioxide is pumped into the system until the desired experimental pressure is attained. This is conducted rather slowly in order not to disturb the already attained temperature equilibrium of the temperature-controlled environment. Then, the pump is refilled and pressurized to the system pressure. The flow rate for 15 h of operation is calculated, and the run is started. The 15 h provided a detectable weight change due to diffused naphthalene, sufficient time to offset inaccuracies due to startup dynamics, and avoided the excessive times to reach steady state. During each experiment, the temperatures at different locations, pressure of the system, and density meter output were observed and recorded. 4. Results and Discussion Table 2 depicts the data collected from the experimental runs at 35, 50, and 65 °C for 80, 100, and 120 atm. The third column shows the solid naphthalene mass loss, evaluated by weighing the tube filled partially with compressed naphthalene, before and after the experiments. The reported mass loss is the average of duplicate experiments. The fourth column is the density of the naphthalene-supercritical carbon dioxide mixture measured by the density meter described in the Experimental Section. The evaluated equilibrium solubility mole fractions of naphthalene in supercritical carbon dioxide are given in the fifth column. The sixth column is the molar density of the naphthalenesupercritical carbon dioxide mixture, converted from the values listed in the fourth column. These are the data required for the evaluation of the binary diffusion coefficients from Stefan tube experiments.
Ind. Eng. Chem. Res., Vol. 42, No. 19, 2003 4395
4.1. Calculation of the Naphthalene Solubility. The solubility information is of critical importance for the steady state and the solid dissolution techniques of diffusion coefficient measurement techniques. Therefore, for the evaluation of eq 22 or eq 25, the equilibrium solubility mole fractions of naphthalene in the supercritical carbon dioxide system under the experimental conditions of, 35, 50, and 65 °C and 80, 100, and 120 atm are needed. To that end, available solubility data are regressed using the Soave-Redlich-Kwong equation of state (SRK EOS) with conventional mixing rules. The binary interaction coefficients kij are further regressed to incorporate temperature dependence. A kij fit and a correlation for the kij temperature dependency were necessary because experimental conditions were different from the experimentally available solubility data. For this purpose, the SRK and Peng-Robinson (PR) EOSs are used with the van der Waals one-fluid mixing and combining rules, and the best fit was chosen based on the correlation coefficient, R2, that includes kij for deviation of aij from the geometric mean rule. The literature solubility data were from the works of several research groups such as Tsekhanskaya et al.,38 McHugh and Paulaitis,36 Knaff and Schlunder,28 and Lamb et al.,29 who worked with different methods and at temperature and pressure ranges of 35-58.5 °C and 60500 atm, respectively. The correlation between fitted and experimental data was high, with a correlation coefficient (R2) ranging from 0.947 to 0.991 for SRK EOS and from 0.939 to 0.989 for PR EOS. Because correlation with SRK EOS results in slightly higher correlation coefficient values at all temperatures, it was chosen for the regression of the saturation solubility mole fractions needed for the calculation of the diffusion coefficients. 4.2. Analysis of the Calculated Diffusion Coefficients. The results of the calculations are listed from the seventh to the tenth columns in Table 2. The seventh column gives the values of the nondimensional constant, B, involving the molar average velocity, defined by the equation B ) vˆ /z h/DAB and calculated by eq 24 with a MATLAB program. The molar average velocities are listed in the eighth column, which are readily calculated from equation B ) vˆ /z h/DAB. The ninth column gives the diffusion coefficients calculated from eq 22 with a MATLAB program for systems with free convection. The last column lists the diffusion coefficients calculated by neglecting the free convection through eq 25 again with a MATLAB program. After the diffusion coefficients and the convective terms were obtained, the concentration profiles were redrawn to examine the validity of the 15-h experiment and to evaluate the merits of convection- and noconvection-based approaches. Generally, the binary diffusion coefficients in supercritical fluids decrease with increasing pressure at constant temperature and increase with increasing temperature at constant pressure. This general trend was observed for the diffusion coefficients determined for this study and will be discussed further in the following section. 5. Error Analysis An error analysis has been carried out considering the impact of the variations of the following variables on the diffusion coefficients: (i) deviations from the set temperatures in the amount of (0.2 °C; (ii) deviations from the set pressures in the amount of (1 atm; (iii)
Figure 3. Diffusion coefficients of naphthalene in supercritical carbon dioxide for different pressures and temperatures.
deviations from the true determination of the weight loss (amount evaporated) in the range of -0.05 to +0.04 mg. Calculations for each data point demonstrated how the diffusion coefficient could vary from the mean value, which is the value calculated by no variations in temperature, pressure, and mass loss. The structure of the error analysis has been designed such that a combination effect of the temperature, pressure, and mass loss deviations has been assessed for different pairs of maximum positive, maximum negative, and no shift conditions from the set values. Following the calculations, a histogram for each data point was drawn showing the distribution of the diffusion coefficient around the mean in terms of its frequency. From these histograms, the value of 2 standard deviations for each data point has been calculated. This value corresponds to the 95% confidence level that constitutes the error bar for each data point. Figure 3 shows the average diffusion coefficients of naphthalene in supercritical carbon dioxide, obtained from two runs, calculated for the deviations, the error for the 95% confidence interval, and the upper and lower bounds for each data point versus pressure for different temperatures. 6. Comparison of Diffusion Coefficients between the Literature and the Data Obtained in This Study Generally, DAB decreases as the pressure increases for constant temperatures, and this conforms with the trend reported in the literature. The diffusion coefficients of naphthalene in supercritical carbon dioxide at 65 °C and the data point at 80 atm for 50 °C were not available in the literature. However, these values compared well with values obtained via a number of available correlations. In particular, the free-volumetype model with the modifications introduced by Matthews and Akgerman39 and Liong et al.40 was found to give the best fit to the diffusion coefficients determined in this work. The greater isothermal pressure sensitivity of the diffusion coefficients at lower pressures, especially at 80 atm, and at higher temperatures, especially at 65 and 50 °C, is due to the sharp density or viscosity changes of carbon dioxide in the aforementioned supercritical range. The differences between the diffusivities are more profound at lower pressures for the same temperatures, where this difference relatively fades away as the pressures are increased. This trend gets more dramatically expressed as the critical pressure is approached
4396 Ind. Eng. Chem. Res., Vol. 42, No. 19, 2003
solute. Because the boundary conditions will be different from those of the work that was carried out here, a generalized solution is required for the solution of the governing equations. 7. Conclusions
Figure 4. Diffusion coefficients of naphthalene as a function of the carbon dioxide density.
because the Fickian diffusivities are known to go to infinity as the experimental conditions are forced to the vicinity of critical pressures at higher temperatures. The effect of isobaric temperature changes is also worth noting in the sense that as the temperatures are increased for the same pressures, the supercritical solvent becomes a poorer solvent, that is, less liquidlike and more gaslike. These gaslike characteristics of the supercritical fluid are due to a decrease in its density, resulting in higher diffusivities. The relationship between the diffusion coefficients and the density of the supercritical solvent is demonstrated in Figure 4, for the data obtained in this work. As expected and mentioned before, there is a large dependence of diffusivity on the density of the solvent. As the density of the solvent increases, the path traveled by the solute molecule within the solution becomes more erratic, reflected as a steep decrease in the diffusion coefficient. In such a dense environment, the relative sizes, and thus the molecular diameters, become more of the limiting factor to the fate of the diffusion process compared to the average intermolecular path to be traveled. This increase in the collisional transfer of momentum and energy and the number of collisions per second is the reason underlying the sharp decrease of the diffusion coefficient with a slight density increase of the supercritical solute-solvent systems.41 There is some evidence of the concentration dependency of the diffusion coefficients in supercritical fluids in the literature. As the solute becomes more concentrated, the collisional transfer of momentum and energy and the number of collisions per second decrease.37 This is due to the closeness of the solute molecules, which creates a shield between the solute and solvent molecules and reduces the effect of attractive forces between them. This phenomenon will be reflected as a reduced diffusion coefficient of the solute at finite concentrations in the supercritical solvent, compared to the magnitude of their diffusion coefficient at infinitely dilute concentrations. The effect of concentration on the diffusion coefficients and the weakness of the Fick representation of it have been known for quite some time;42-44 however, the lack of experimentally determined diffusion coefficients at finite concentrations hinders the development of reliable representations of the diffusion coefficients for wider ranges and supercritical mixtures. Therefore, one may assume the phenomenological applicability of the Stefan tube in measuring the diffusion coefficients and studying their concentration dependence at finite concentrations, with the upper limit being the solubility of the
The analysis of the Stefan tube at supercritical conditions for the diffusion coefficient measurements has been carried out. The binary diffusion coefficients for the system naphthalene-supercritical carbon dioxide were determined at 35, 50, and 65 °C for the pressures, 80, 100, and 120 atm. The models developed both for systems with convection and for systems without convection were employed to quantify and compare the convective effect on the diffusion coefficients. Results showed no significant effect of convection on the diffusion coefficients for the naphthalenesupercritical carbon dioxide system. The concentration profiles drawn with the measured data confirmed the initial determinations of the unsteady state of the diffusion process. The determined diffusion coefficients agree well with the literature and confirm the behavior of a supercritical diffusion coefficient under the influence of temperature and pressure changes, which result in density changes of the supercritical solvent. Acknowledgment This work was, in part, supported by the RAMP project. E.I.O. also expresses her gratitude for the financial support from Philoquest and the University of South Florida. Nomenclature DAB ) diffusion coefficient of naphthalene in supercritical carbon dioxide, cm2/s F ) total density, g/cm3 c ) total concentration, gmol/cm3 t ) time, s N ) combined molar flux vector, gmol/cm2-s J ) molar flux with respect to the molar average velocity, gmol/cm2‚s vˆ * ) molar average velocity, cm/s y ) mole fraction in the supercritical gas phase h ) height of the stagnant supercritical phase above the solid in the Stefan tube, cm θ ) dimensionless concentration η ) dimensionless distance τ ) dimensionless time B ) dimensionless free convection term n ) dummy variable M ) total mass, g S ) total cross-sectional area, cm2 MW ) total molecular weight, g/gmol aij ) energy parameter of the equation of states kij ) binary interaction coefficient used with the equation of states for mixtures Subscripts A ) naphthalene B ) supercritical carbon dioxide R ) component index z ) coordinate axis 0 ) at z ) 0 f ) duration of the diffusion experiments nc ) no convection i ) component index j ) component index
Ind. Eng. Chem. Res., Vol. 42, No. 19, 2003 4397 Sub-subscripts z ) coordinate axis 0 ) at z ) 0 h ) at z ) h Superscript eq ) equilibrium
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Received for review November 20, 2002 Revised manuscript received June 3, 2003 Accepted June 17, 2003 IE020938W
