SUBSCRIPTS m
=
refers to bulk (original) phase
literature Cited
Berg, W., Proc. Roy. SOC.,Ser. A 164, 79 (1938). Humphreys-Owen, S., Proc. Roy. SOC.,Ser. A 197, 218 (1948). Kirwan, D. J., Pigford, R. L., A.I.Ch.E. J . 15, 442 (1969). Mudge, L. K., Heideger, W. J., A.I.Ch.E. J . 16, 602 (1970).
Nishijima, Y . , Oster, G. J., J. PoZym. Sci. 19, 337 (1956). Searle, F. C., Phil. Mag. 37,361 (1946). Secor, R., A.1.Ch.E. J. 11,452 (1965). Traher, A. D., B.S. Ch.E. Thesis, University of Virginia, 1971. Tolansky, S., "Multiple Beam Interferomet.ry," Oxford University Press, London, 1948. Vignes, A., IND. ENG.CHEM.,FUNDAM. 5 , 189 (1966). RECEIVED for review June 27, 1972 ACCEPTEDJanuary 31, 1973
Measurement of Gaseous Diffusion Coefficients Using the Stefan Cell James M. Pommersheim" Bucknell University, Lewisburg, Pa. 17837 Bruce A. Ranck York Research Corp., Stamford, Conn.
A modification of the standard treatment of the quasi-steady-state stagnant film diffusion of a solvent in a finite Stefan cell gives a means to determine diffusivities which i s independent of the diffusional path length correction factor. The best experimental value of the diffusivity pressure product for pentane in nitrogen at 31.4"C i s 0.0864 f 0.001 9 atm cm2/sec.
T h e Stefan or Arnold diffusion cell (Arnold, 1944) and its modifications have been widely used as a simple means of determining diffusivities of evaporation solvents (Altshuller and Cohen, 1960; Larson, 1964; Lee and Wilke, 1954; l f c Kelvey and Hoelscher, 1957). Both metal and glass systems, have been used. I n most investigations the amount of solvent evaporated was determined discontinuously, either by weight or volume, repeating runs at different liquid levels rather than from continuous sightings of liquid interface levels for the same run. Two studies (Pommersheim, 1971; Schwertz and Brow, 1951) obtained liquid level data continuously from single runs. However, in We former study attention was focused on diffusion in a closed system, while in neither study was a n effort made to determine the diffusional path length correction factor caused by mixing at the top of the diffusion tube. The present paper describes a convenient method for treating interface data t o obtain solvent diffusivities from a Stefan cell and s h o w the results of testing this method on a single evaporative system. The method should apply to any system where heat transfer does not limit evaporation rates. Theory
Consider the Stefan diffusion cell shown in Figure 1 coiltaining a liquid solvent -4.Cell temperature and pressure are held as constant as possible. A steady flow of insoluble gas B passing across the top of the cell establishes a stagnant film above the liquid. Evaporated A diffuses upward through this film and mixes with pure B at the cell top. At time t the cell has interface depth z t , measured from the top of cell to 246
Ind. Eng. Chcm. Fundam., Vol. 12, No. 2, 1973
the bottom of the liquid meniscus. For a glass cell, data ( z t , t ) can be continuously collected using a cathetometer. Using a quasi-steady-state analysis the ideal solution governing the interface depth can be shown to be (Beniiett and Myers, 1962; Pommersheim, 1971; Welty, et al., 1969)
This equation was first derived by Schwertz aiid Brow (1951)
as a limiting form of a n unsteady-state development. hlthough it is suitable for estimating diffusivities, it neglects end-length corrections (Lee and Wilke, 1954) which can make the true path length differ from the physically measured one. The major correction occurs because of the mising effect a t the cell top, with only a minor correction to the measured length attributed to meniscus curvature a t the interface (Altshuller and Cohen, 1960). At high inert velocities turbulent eddies a t the cell top would be expected to decrease the diffusion length by partially destroying the stagnant film. However, Altshuller and Coheii (1960), using inert velocities between 100 aiid 2000 rnl,/miii, could find no correlation between inert flow rate and Az, the end leiigtli correction factor. At low velocities mising is poor and a plume of solvent may form. I n addition the assumption of unidirectional molar transport would also become invalid. 130th effects introduce an additional end-length resistance to evaporation, so that a t a suitably low velocit,y A2 (as measured positively down from the cell top) should decrease with further decrease in inert flow, eventually turning negative. With 110 flow aiid
-
ATM.
WATER MANOMETER
CATHETOMETER
WATER
BATH
Figure 1 . Equipment sketch
small tube diameter a theoretical smallest negative value of r d / 8 for Az is predicted (Xiidrew, 1955). Correction for solvent vapor changes a t the cell top can be made by calculating P A d f ) , the time-dependelit' solvent part'ial pressure, based oil the assumptioil that t,he solvent vapor mixes uniformly with the inert gas. In order to determine diffusivities, Lee and Wilke (1954) began with the steady, one-dimensional integrated A-flus equation
A quasi-steady-state analysis can also be used t o determine solvent diffusivities. Equating the liquid evaporation rate of A to t h e vapor flus produced P 1 -dzi
dt
_1 -Da
Az 1 DAB z i
1 +-DAB
(3)
which predicts that a straight line should result when 1/D, (found from the first equality ill eq 2) is plotted against l / z i . Using t,his relation they determined the diffusivities of a number of evaporation solvents. From eq 3 it can be seen t h a t t h e intercept, DAB, must be ohtaiiied by extrapolation. Also a standard linear leastsquares analysis gives disproportionate weight t o those points obtained a t early times where errors are larger. These errors include those associat'ed with the initial unsteady state, with the larger evaporation rat'es, and with t h e finite values of PAD.
Lee aiid Wilke's (1954) t'reatmeiit is valid a t each new steady state which occurs as the liquid descends. However, it does not allow for a continuous change of ~ V and A ~is in error to that extent. For example, if data were collected periodically, the error would be larger a t smaller times where higher evaporation rates attend.
DABP In ( '30 (4) ) RT(zi - Az) P - P'A
The total concentration has been replaced by P / R T , its ideal gas law equivalent. Separating variables and integrating
2 The apparent diffusivity L), corresponds t o t h e physically measured (or apparent) interface depth z,, while the true diffuyivitj coirespoitds to the corrected (or actual) iiiterface depth (zl - Az). Their development assumes t h a t PA^ is zero a t both z = 0 and z = Az, aiid that P A = POA a t z = z,. This 1a.t :issumption of equilibrium a t the interface has recently heeii reaffirmed by Alan (1970). 1-sing the qecoiid equality in eq 2, Lee and Wilke (1954) obtained
=
=
DABP piRT
[lt('Ao) In
P -POA
dt1
-
(z,
- Az)
(5)
where y = zi - 21, the total drop in interface depth. Equation 5 assumes the temperature is constant but allows for the fact t h a t both P and PAO may be functions of time. The product DABP is pressure independent for a dilute vapor. Although pressure variations are not strictly allowed by this development, slow, small changes, made smaller by obtaining the logarithm, can be tolerated. A similar approach has been applied successfully t o predict total pressure changes in a closed system where solvent vapor accumulates external t o t h e Stefan cell (Pommersheim, 1971). Let
Then eq 5 becomes
(7) A plot of y/2 against T ( t ) / y predicts a straight line whose slope is proportional to DABP and whose intercept will yield
Az. If the total pressure and solvent partial preqsure a t t h e cell top remain constant, eq 7 becomes
and a plot of y/2 against t / y linearizes this equation. The use of a mean pressure P in eq 8 is justified for small to moderInd. Eng. Chem. Fundom., Vol. 12, No. 2, 1973
247
17
not affect the results. For the same solvent-inert pair an attempt was made to measure interfacial temperature gradients a t a n interface depth of 8 cm. Within =t0.loC no gradients were detected as the iiiterface descended across a fine thermocouple immersed in the liquid phase (refer to Figure 1). Lee and Wilke (1954) made more precise measurements and found (for a number of systems diffusing into air) that interfacial cooling was always less than 0.2OC.
PENTANE- NITROGEN
10.
8 -
s
Second Run
T = 31.4%
3/9/71 F = 7 6 7 rnm
ZI = 2.75cm
, INERT
-__
FLOW.44 mlfiln
PRESSURE UNCORRECTED _CORRECTED (dot0 not a h a n )
S64 -
Results and Discussion
2-
0
2
4
6
&
10
12
14
16
18
x i 6 3 (stm)
Figure 2. Linearized plot of experimental data
ate changes 111 barometric pressure. Equation 8 reduces to eg 1 i n the fortuitous case when Az = 0. Experimental Section
The basic open system used is shown in Figure 1. The constant-diameter well (0.80 cm i d . ) of a Pyres glass T n-as initially filled with Spectrograde pentane t,o a known depth and immersed in an 18 in. deep constant,-teml)erat,urebath kept a t 31.40 = 0.05'C. Vsing a pycnometer the liquid molar density of pentane a t 31.4'C was determined to be 0.008543 molel'cm3. Sit'rogen (dried, high purity) was passed a t a constant fiow rate over the top of the well and allowed to vent into the atmosphere. The gas was brought to the bath temperature by passing it through a copper coil preheater P i ? in. 0.d. X 14 f t long) immersed in the bath. [-sing valves H aiid IC the system was first closed off and allowed to reach thermal and saturative equilibrium, while stored under inert gas. During this operat,ion valve H was opened to the atmosphere. Valve K was then cracked slightly to remove any overpressure, arid soon afterward both valves were opened and inert gas was passed over t h e cell a t the desired flow rate and total pressure. At this point, with the initial well depth recorded, data collection began. Cathetomet'er readings (+0.01 cm) of the liquid-interface depth were taken every I/? hr, and later at 1-hr or longer intervals as evaporation rates decreased. Bath temperature, cell temperature, and system and barometric pressure were all periodically recorded. System pressure was measured by connecting port J to a differential water manometer. For most runs an att'empt was made to maintain system pressure constant' near 760 mm by choking valve K. However, system pressure was found to drift with changes iii barometric pressure. I n order to assess the effect of inert flow rate, data were collected a t 16, 44, 83, and 130 ml of Sz/min. Runs lasted between 4 and 5 dags. Xpprosimately 50 data points were collected for each run. Initial studies were concerned with the effects of cell diameter and of interfacial temperature gradients. I n a test made a t 25°C: using helium as the inert gas (initial studies only) cells having internal diameters of 5, 8, and 13 mm were filled to within 3 cm of the top wit'h pentane and connected in parallel. Inert flow over each cell was maintained a t 150 ml/ miri! sufficient to ensure PAO 0. Interface levels were found to descend (within sighting error) a t the same rate. This confirmed the independence of interface level with cross-sectional area S predicted by eq 7 and also indicated that small changes in S with zi (caused by nonconstant tubing bore) probably did
-
248
Ind. Eng. Chem. Fundarn., Vol. 12,
No. 2, 1973
Interface level-time data were reduced using both eq 7 and 8. The values of PDAB and Az were determined from the best least-squares slope and intercept, respectively. L-siiig an analysis similar t'o that presented by Lee and Wilke (1954), estimates were obtained of the unsteady-state period, during which the quasi-steady-state assumption would not be valid. For all of the runs quasi-steady state was estimated to be reached wit,hin 1 min. I n all cases the drop in interface depth occurring during this interval was negligible. I n addition, the assumption that the flus of inert was zero a t quasisteady state was checked. It was found t'hat the increase in inert flus was always less than 0.1% of the average solvent flux. I n eq 8 the system total and partial pressures were assumed constant a t their mean values. Data from a typical run are plotted in Figure 2. For clarity, only representative points are shown. In general, for all runs, the linearity obtaiiied was good, but' it' was observed t,hat' data above the line were oft,en collected in periods of relatively high system pressure while values below the line were often collected in periods of low system pressure. The high vapor pressure of pent,ane a t 31.4OC (646 mm) makes the logarithmic term moderately sensitive to changes in total pressure, so that the use of a mean value of P may not be valid. At this temperature, near at'niospheric pressure, a mean pressure in error by 10% causes a diffusivity in error by 3%. -41~0,a t the relatively low inert flow rates used in the present study, the solvent part'ial pressure a t the cell top Pa0 was appreciable, especially at early times. Initial mole fract'ions a t t'he top of the cell varied between 2 and 10% pentane. These facts suggested that data aiialysis might be improved by accounting for the continuous change in total and partial pressures. ,Iccordingly, the data were reanalyzed using eq 7 . First the total pressure-time data (40 to 60 point's) were fitted to a n mth-order polynominal P,(t) of minimum variance m
P,(t)
ail'
= i=O
(9)
vi was '7 or less for all runs, with a maximum absolute deviation of 1 2 mm noted. P A Omas obtained by assuming that t'he solvent vapor mixed homogeneously with the inert gas at' the cell top. T ( t ) was then determined numerically from eq 7 using P,(t) and Simpson's rule. Iteration was used to obtain PAOa t each time. Values of T ( t ) were confirmed by their independence with grid size Af. I n Figure 2 is shown (dashed) the best leastsquare line obtained. For the run shown in Figure 2 , the average value of PDABin a t m cm2/sec mas 0.0913 + 0.0008 ( 5 ) for the pressure-uncorrect,ed data and 0.0869 f 0.0008 for the pressure-corrected data. (.ill f values denote 9570 confidence limits.) With pressure correction, data points (not shown) tended to move in toward the line in regions of high aiid low system pressure. However, the net effect for all runs was t'o produce approximately equal variances for pressure-correcbed as for pressure-uncorrected data. The percentage rela-
10
- EXPERIMENTAL - - _- THEORETICAL T G 8
-
s . Tee
Hc & B S 6
P
-
PVERAGE
+o.2 t
PRESSURE CORRECTED
m
UNCORRECTED
viscosity
L ,I
et ai. (1966)
.
nirshfelder el 01. (1954) SherwOod et al. (1964)
B
:
viriol Coefficient
0
P -
m
0
8 0
20
40
€0 EO 100 FLOW RATE ( m l / m i n )
120
140
Figure 3. Variation of diffusivity pressure product with inert flow rate
t i r e error of the individual diffusivities ayeraged 0.92 for the corrected data, as against 0.88 for the uncorrected data. Evidently the regular data were somewhat unsettled by the polynomial tot,al pressure fit'. For all runs the mean value of PDAB in a t m cm2/:sec was 0.0914 + 0.0018 for the pressure-uncorrected daia and 0.0864 0.0019 for the pressure-corrected dat,a. Because t'he confidence bands of the two data analyses do iiot encompass one another (even on the 99.9% confidence level), it can safely be assumed that' the two means are statistically different (Nickley, et al., 1957). Figure 3 shows the independence of PDAB with inert flow rate for both lwessure-uncorrected and pressure-corrected data,. Seit,lier method of data analysis showed any relation betiwen the inert flow rate and the diffusivity as indicated by the lack of significance i i i the calculated correlation coefficients. This justifies using an average value for PDABand suggests that similar studies could be conducted over any convenient range of inert flow rates. Slio\vn as dashed liiies in Figure 3 are some theoretical diffueivit'ies obtained from the literature. The diffusivit'y of 0.0895 cm?/:sec was calculated a t system temperature and 1 a t m pressure using those combined viscosity-virial coefficient force constants for the Lennard-Joiies 12-6 potential presented by Tee, et al. (1966), and subsequently recommended by Reid (1968). Calculation based on the force constants which they derived from viscosity data alone yielded a value of 0.0853 cm2,:gec, while calculations based on viscosity force const'aiits given by Hirslifelder, e l al. (1954), gave a diffusivity of 0.0871 cm2/sec. Collision int,egrals were evaluated using the iioniograpli presented by 13rokaiv (1969), as checked against t'he Cheii and O t h e r (1972) correlation. X value of 0.0928 was obtained using the method of Slat'tery and Bird (1958). Xccordiiig to the analysis of Tee, et al. (1966), LennardJones force cotistaiits calculated usiiig only viscosit,y data are less subject to variability for longer linear molecules like pelitalic arid Iieptaiie t'lian those calculated using virial coefficient, ( B ) data. For the iiresent . ten1 the diffusivity calculated from virial coefficient force constants was 0.0556 c m / see. This is well below the other theoretical values aiid comliletely outside the esperimental range. The theoretical diffusivity was also calculated using tables of collision integrals for the Kihara potential developed by
*
Figure 4. Variation of end length correction with inert flow rate
O'Connell and Prausnitz (1965), based on virial coefficient force constants originally presented by Sherwood and Prausnitz (1964). -1value of 0.0872 cm2jsec was obtained. This is also in good agreement with the esperimental diff u s i d i e s for the pressure-corrected data. The close correspondence with the Lennard-Jones viscosity value suggest,sthat the use of the Kihara pot,eiitial may be justified for estimating diffusivities for virial coefficient data alone. Certainly its use for long linear molecules where appreciable spherical cores exist is justified on theoretical grounds. S o t e from Figure 3 that the pressure-corrected average agrees more closely viith the theoretical values t'haii does the uncorrected average. In fact, nolie of the theoretical diffusivities are between the 95% confidence limits for the uncorrected data, n-hile three of the four are near the pressurecorrected average. Based on this correspondence it' appears that t'he pressure-corrected average is preferable. It' x a s rioted for both pressure-corrected aiid uncorrected data that two of the data points lay outside the 99% confidence band. .Lccordiiigly, the data \yere reanalyzed with tn-o fewer data points. The mean value of PDAB in a t m cm2/sec was 0.0863 i 0.0013 for the pressure-corrected data and 0.0916 + 0.0018 for the pressure-uncorrected data. Somewhat loner standard deviations are iioted in this case. but the diffusivities are essentially the same. and no additional conclusions can be d r a n . ~ ~ . The effect of fion rate on the eiid length correction factor, Az, was also itivestigatecl. 1 2 \vas found for both methods of analysis from the best value of the intercept. Figure 4 shows the effect for the 1,ressure-corrected data with all data poiitts iiicluded. The iio-flon. asymptote of - d 8 is shown as a dashed line. 1-ertical lilies denote the 95% coiifideiice limits on each Az. Az increases from negative values with increasing flow rate, as espected qualitatively from theory. -1significant linear correlation coefficient esists here between the meaii values of the correction factor xiid the flon rate. S o t e : though, that the intercepts plot imprecisely, n-itli 110 hint of the form of the exact relation. Results similar to those slio\vn iri Figures 3 and 1 were obtained for the 1,ressure-uncorrected data. 130th diffusivity independeiice and eiid length correction factor increase were ~ I n additioii the correcobserved with iiicreased iiiert f l o rate. tion factor-flon- rate correlation \vas somewhat better for t,hese data. Lee and Kilke (1954) could find iio correlation between flow rate aiid end leiigth correction. ,Iceording to eq 3, errors Ind. Eng. Chern. Fundarn., Vol. 12, No. 2, 1973
249
in Az would arise from both errors in the intercept and the slope. This is because both bhe slope and intercept involve DAB, which itself is subject to error. A small number of data points collected in a differential cell would compound the error. In addition, this analysis gives disproportionate weight t o data collected a t early times, as previously discussed. The present analysis re-confirms that Az is not an easily defined term. Fortunately, diffusivities found are independent of the value of the end length correction factor. 13ecause the slope and intercept involve D A B and Az separately, errors in one do riot influence errors in the other. I n addition, dat’a collected a t short times are not given undue weight and a very large number of data points can conveniently be collected from a single experiment. For single experiment’s relat’ive errors averrged 01113- 127G of corresponding values reported by Altshuller and Colieii (1960) using n-decane.
DAB = MA = N A ~= P = P, P’A
PA^
P
R S t T T(t)
y zi
z1
Az
GREEKLETTERS pi
= =
u
=
Conclusions
Revised methods are presented for the collection and treatment of data obtained in a finite Stefan cell. These methods are preferable to standard methods for the determination of diffusivities of evaporation solvents. Withiri experimental error diffusivities were fouild to be i~idependentof inert flow rates, suggesting that similar studies can be conducted at any conveiiient flon- rate or over a range of such flon- rateq. Elid length correctioii factors m r e found to increase with inert flow rate, as espect’ed qualitatively from theory. The present study showed that variations iii system and partial pressure could be treated analytically by a suitable curve-fitting integrdtion technique. Significant differences were found hetveeii pressure-corrected and pressure-uncorrected diffusivities. The best, pressure-corrected value of P D A Bfor the system pentane-nitrogen a t 31.4OC IYas found to be 0.0864 + 0.0019 atm cm2/sec. This gives good agreement n-ith values predicted from the Lennard-Jones potential using force const’ants liased upon viscosity data as \Tell as Ivith values from the Kihara poteiitial using virial coefficient force constants. Nomenclature
A B C d D,
=
= = = =
evaporating, liquid inert gas; virial coefficient total concentration, moles/cm3 inside diameter of diffusion cell apparent value of DAB
250 Ind. Eng. Chem. Fundam., Vol. 12, No. 2, 1973
= = = = = = = = = = = = =
molecular diffusivity of A in B, cm2/sec molecular weight of A molar flux of A in z direction, moles/cm2 see total pressure, a t m total pressure fitted by polynomial (eq 9) vapor pressure of A partial pressure of A a t cell top mean total pressure universal gas constant transport cross-section, om2 time, see temperature, O K integral defined by eq 6 zI - zI, total drop in interface depth, em interface depth, cm initial interface depth, ern end length correction factor, ern
liquid molar density of A, moles/cm3 viscosity standard deviation
literature Cited
Altshuller, A. P., Cohen, I. R., Anal. Chem. 32, 802 (1960). Andrew, S.P. S., Chem. Eng. Sci. 4, 269 (1955). Arnold, J. H., Trans. A.I.Ch.E. 40, 361 (1944). Bennett, O., Myers, J. E., “Momentum, Heat, and Mass Transfer, p 444, McGraw-Hill, New York, K. Y., 1962. Brokaw, It. S.,Ind. Eng. Chem., Process Des. Develop. 8 , 242 flS69).
Chen, N. H., Othmei, D. F., J . Chem. Ens. Datu 7, 37 (1962). Hirshfelder, J. C., Curtiss, C. F., Bird, R. B., “Molecular Theory of Gases and Liquids,” pp 1111-1112, Wiley, Kew York, N. Y . , 19.54.
Larson, E. AI,, M.S. Thesis in Chemical Engineering, Oregon State University, 1964. Lee. C. Y.. Wilke. C. It.. Ind. Ena. Chem. 46. 2381 f1954) Ma;, J. It,’, 1x0. ENG.CHIX., F&DIM.9, 283 (19707. AIcKelvey, J. X, Hoelscher, H. E., Anal. Chem. 29, 123 (1957). Mickley, H. S., Sherwood, T. K., Reed, C. E., “Applied Mathematics in Chemical Engineering,” DD 70-71, AIcGraw-Hill. New York, X. Y., 19;i7.O’Connell. J. P.. Prausnitz. J. 11..Advan. Thermonhiis. Prov. Estremk Temp,’Pressure 19 (19651. Pommersheim, J. 11.,IXD. ENG.CHEM.,F U N D ~10, M134 . (1971). Reid, 11. C., Chem. Enq. Progr. Monogr. Ser. 64 ( 5 ) , (1968). Schwertz, F. A,, Brow, J. E., J . Chem. Phys. 19, 640 (1951). Sherwood, A. E., Prausnitz, J. ll.,J . Chem. Phys. 41,429 (1964). Slattery, J. C., Bird, It. B., A.I.Ch.E. J . 4, 137 (1958). Tee, S. T., Gotoh, S., Stewart, W. E., IND. ENG.CHEM.,FUNDAM. 5 . 3.56 11966). Weity, J. It., Wicks, C. E., Wilson, R. E., “Fundamentals of LIomentum, Heat, and llass Transfer, p 489, Wiley, New York, 5 . Y., 1969. - I
I
__
‘
Y
RECEIYED for review August 14, 1972 ACCEPTED January 15, 1973
