-
EXPERIMENTAL TECHNIQUES
Studies of the Loschmidt Diffusion Experiment. II. An Improved Interferometric Method Sukehiro Gotoh, Morten Manner,' Jan P. S@rensen,and Warren E. Stewart* Department of Chemical Engineering, University of Wisconsin, Madison, Wis. 53706
A new interferometric method i s given for measuring gaseous binary diffusivities. The theory of part I
is
used for accurate analysis of the data. The method i s demonstrated by determining DABaccurately for nitrogen-n-butane at 25, 105 and 165°C and atmospheric pressure. The Chapman-Enskog kinetic theory, with a Lennard-Jones 12-6 potential, fits the results within 0.2%.
I n part I of this work, Stewart, et al. (1973), have given a theoretiral analysis of the Loschmidt diffusion experiment, with variable Auid properties and absorption b y the seals of the cell. T h e solutions were done by a perturbation method, for convenience in analyzing experimental data. T h e theory is applied here to experiments performed in our laboratory with a recording optical interferometer. The experimental technique and data analysis are described and demonstrated by measurements of DABfor the nitrogen-nbutane system. Less detailed theories are shown to be inadequate for analysis of the experiments. Experimental Apparatus
T h e apparatus is shown schematically in Figure 1. T h e main components are t h e diffusion cell and the interferometer, rigidly mounted in a large constant-temperat'ureair bath. Figure 2 shows t h e details of the diffusion cell. It consists of two symmetric rectangular stainless steel chambers with cross sections 10.172 X 1.591 em. The half-cell height, L , is 15.225 + 0.003 em a t 20°C. Each chamber is bolted on a large lapped stainless steel plate, and t,he lower plate slides along t h e upper one to start or terminat'e the diffusion. X thin layer of fluorosilicone grease, less than 0.001 em thick, is applied between the sliding plat.es to keep the cell vacuumtight. Two quartz windows are provided a t t'he top of the cell for refractive index sensing. These windows are sealed with silicone rubber and Viton O-rings. One window is masked to trim the entering light beam to a height of 3 mm. T h e center of the trimmed light beam is a t a height { = 0.97327, relative to the half-cell height. Two chromel-constantan thermocouples of 0.001-in. xire are mounted in each half-cell a t 1 and 2 in. from the midplane. They are used to monitor changes in gas temperature caused by the diffusion-therm0 (Dufour) effect. T h e interferometer, designed and built' for this study, is shown in Figure 3. Light from a low-pressure mercury lamp (filtered t o isolat'e the 5461-A line) enters horizontally by reflection a t point a on tmhemirror md6.It splits into t x o beams Present address, Norsk Hydro A. S.-Porsgrunn Fabrikker, Porsgrunn, Norway.
a t the middle mirror, rn,. One beam comes back through the diffusion cell; the other goes on through a reference path in air. T h e beams are rejoined on the right-hand side of m, to form interference fringes of 0.17-em spacing; this corresponds to an angle of 14 see. The changes in refractive irides caused by the diffusion are detected as movemeiits of the interference fringes across the photomultiplier aperture (see Figure 1). The photomultiplier output is plotted continuously to give a permanent record of the interference fringes. Gases Used
The gases used in this investigation were nitrogen of prepurified grade (99.998 minimum mole %) from Air Products and Chemicals, Inc., and n-butane of instrument grade from the Matheson Co. (99.5 minimum mole 70) Experimental Procedure
Each diffusion test was done in the thermostat,ed bath cont>rolledwithin 0.01"C. Before each test, the diffusion cell vas filled with nitrogen and left for several hours to ensure desorption of an)- gases remaining from the previous test. T h e lower chamber was then evacuated, tested for leakage, purged twice with the heavier gas, and filled 1Yit.hthis gas to a pressure slightly above atmospheric. Nest, the same was done for the upper chamber with t,he lighter gas. T h e pressure changes of each half-cell and its connecting lines were measured over a 0.5-hr or longer interval to see if absorption was occurring. The two filling lines were then joined t,o equalize the pressures of the t'n-o half-cells. The cell pressure was further adjusted to at,mospheric by opening the valve V6 (see Figure 1) for about 2 min. This pressure was taken as the initial and reference pressure for the experiment. T h e valves V91, T'92, V81, and V82, were then closed, and the system was left for a few minutes for thermal equilibration before the cell was aligiied to start the diffusion. T h e diffusion was monitored by recording the interference fringes continuously. The small t,eniperature changes caused by the Lhfour effect were also recorded and fouad to decay to less than 0.01"Cafter t'he first 2 or 3 min. The barometric pressure was read a t regular intervals from a precision baromet'er. The test was terminated a t about 99% complet'ion of the Ind. Eng. Chem. Fundam., Vol. 12, No. 1, 1973
119
(GAS
Figure 1. Low-density binary diffusion apparatus: V, valve; F, filter
diffusion by sliding the cell back to the filling position and measuring the final pressure of each half-cell on the mercury manometer. The procedures given here are those of Gotoh (1971). They are refinements of those developed by Manner (1967). listing of Observations
From the continuous fringe record of each test, about 100 points (time and fringe number) were read. The readings were made a t intervals of several fringes in the densely spaced portion of the record and a t shorter intervals down to a quarter of a fringe in the less dense portions. Most of the readings were taken midway between the top and bottom of a fringe, where the slope of the record is steepest. This choice maximizes the sensitivity of the subsequent data analysis to SAB. A typical data set consists of about 100 points (ti, Yi)of time and fringe number, and one final point (fN, Ap) calculated from the cell pressure readings. The time and fringe numbers are measured from the instant when the motion to align the chambers for diffusion is half completed.
For a binary mixture, R M may be written in terms of the mole fraction X A
+
RM = XARA
where n, A, and IT’ are the refractive index of the diffusing gas mixture a t time t, the wavelength of the monochromatic light beam, and the geometrical path length of the measuring beam in the cell, respectively. The Lorent,z-Lorenz equation relates the refract,ive indes to t,hemolar refractivity of the gas misture:
120 Ind. Eng. Chem. Fundam., Vol. 12, No. 1, 1973
=
RA
+ (I - XA)(RB- RA)
(3)
and c and X A may be predicted by the following equations from part I C
-= 1 co
+ eyolFo +
Yol(Y01
XA
€
XAO
-
- Fo
+
e 2 [ ~ o l ( ~ o ~Aol)Fls
AOI) FIDf YozFo2
+
+ (Yaz -
Yo12)p201
KIT^" + ( e k A h A -
+ AodFls + KIGO+ KzH -
+
€[(yo1
(YOI
f
e k ~ h ~ ) I (4) p
- AodFloI
+
€ ~ A ~ A . ~ A o~
+
) ~ B A B ~ (B5 O
applied at the interferometer beam position, { = 0.97327. , ~A10 included in part I The quantity (3 In c a ) ~ ~ln/ bP ) ~ , = has been neglected here, since it is extremely small for experiments a t atmospheric pressure. For the cell pressure change, the theory in part I gives Ap = p
Mathematical Model
A mathematical model for predicting the fringe shift up to time t can be derived by taking two contributions into account: the refractive index change caused by changes of composition and pressure in the light path through the cell and that caused by barometric pressure changes in a n equal net path length through the surrounding air. The predicted fringe shift P i s then
XB@B
- po
=
pan
(6)
where
(7) This result is a function of time, but not of position. The functions Fo, F l s , F l D , P2n1GO, H , I,, fao, and SBOappearing in eq 4, 5, and 7 have been described in part I. They all depend on the dimensionless time
and all but P2o and I , depend on the position f = z / L . Tables of these functions up to di = 2.0 for { = 0.97327 (the location of our interferometer beam) are available from the American Chemical Society (Business Operations Department; document No. FUND-73-114) as part of the results of part I. These tables have been recalculated and supersede those given by Gotoh (1971).
"'r
Cf
a
b 0 1
@ Figure 2. The diffusion cell. A, Overall view. B, Details of the cell with a cross-sectional view of the upper half: a, diffusion chamber; b, optical window; c, fine-gauge thermocouples; d, filling hole for lower chamber; e, stainless steel filters; f, upper sliding plate; g, lower sliding plate; h, rack drive for cell alignment; i, springs; i, support frame plate; k, support rail for springs; I, stop plates; m, thermocouple terminals and wires; n, thermopile wells; 0, cell flanges; p, cell head; q, plug. C. Window details: a, fusedquartz window; b, O-rings; c, positioning mask; d, window packing gland
Several constants in eq 1-8 have to be found from the diffusion experiment. The diffusivity DAB)^ obviously falls in this category. Similar treatment is required for the initial fringe number Y I arid the molar refraction difference (@B @ A ) , which are not known with sufficient accuracy, and for ~ , are unt h e absorption constants Kl, Kz, and k ~ h which known. The parameter k ~ . ican ~ be adequately represented in terms of k B A B as described below. The other constants in the mathematical model are measurable or predictable with adequate accuracy; see Table I. The molar density c and its derivatives y t j are obtained from the virial equation of state, with second virial coefficients calculated from bhe Kihara parameters of Tee, et al. (1966b), and the combining rules of Eisenman and Stiel (1971). T h e coefficient Ao1 = ( b 111 C D A B / ~ Z A ) ~ is , Tcalculated from the Chapman-Enskog second approximation (Mason, 1957) with Lennard-Jones parameters obtained from viscosity (Tee, et al., 1966a). The absorption of nitrogen in the window seals should be very slight, but for completeness the following estimate of relative absorption is used.
1 0'
I
I
incher
@ Figure 3. The interferometer. A. Light beam paths. 8. Top view with the diffusion cell mounted in the front half: m45, 45' mirror; mf, front mirror (total reflection); Wf,Wr, front and rear window plates; mm, middle mirror and beam splitter; Cf,Cr, front and rear compensating plates; m, rear mirror (total reflection)
This is obtained by using Raoult's law and the correlation of Scheibel (1954) to predict the relative solubilities and diffusivities of A and B in the window seals. Calculation of Diffusivity
T h e diffusivity was calculated by fitting eq 1-8 to the observed fringe numbers Yo, . . . YN-l and observed pressure change Y N = A p . The weighted sum of squares, wi. ( Y I - P t ) 2 ,was minimized with respect to the parameters (DA& Y I , (RB - @A), 1
