One other point must be considered. The equations have been derived under t h e stipulation t h a t the charging rate throughout t h e period of charging is the rate corresponding t o zero particle charge. If t h e amount of charge placed on any particle is small this requirement ail1 be satisfied. The best means of ensuring this is t o charge for such a short time t h a t only a small fraction of the particles acquire even a single electronic charge. In t h e calculations done here, the charging time mas set to provide a single electronic charge on every tenth particle. The chance of a particle achieving two charges is then of the order of 170,and any error due to deviation from the zero charge rate is of t h e same order. The charging times required are generally of the order of a fraction of a milliqecond, and the corona length a t reafonable velocities is a fraction of a millimeter. The current a t the wire is of the order of 10-9 or 10-lO -1.None of these quantities is critical, hon-ever, and a. single meter could be constructed
Modified Stefan
to measure accurately over two orders of magnitude in particle concentration or more. Quantitative experimental results have not yet been obtained. Qiialitgtive experiments on smoke show that the effect is readi\y observed. Acknowledgment
The authors wish to thank D. T. Wasan for his helpful suggestions. Literature Cited
Dunham. S. B.. U. S.Patent 3.114.877 (1963). Grindel1,’D. H.’,Proc. Inst. El&. Ekg., Part A 107, 353 (Jan 1960). Grindell, 1).H., AEI Eng. 2,229 (1962). White, H. J., “Industrial Electrostatic Precipitation,” AddisonWesley, Reading, Mass., 1963. RECEIVED for review January 24, 1972 ACCEPTEDNovember 8, 1972
Cell for Gas Diffusion Measurements
Albert C. Frost”’ and Erwin H. Amick2 Department of Chemical Engineering and Applied Chemistry, Columbia Cnicersity, S e w York, S.Y . 10027
An unsteady-state diffusion cell is described. One end is closed; the other end is exposed to a purge gas stream whose composition is continuously monitored. Diffusion coefficients are calculated from this composition history and the pertinent cell dimensions. Results for the He-A and He-N systems compared favorably with those of other investigators. Helium-paraffin (methane, ethane, propane, and n-butane) and heliumolefin (ethylene, propylene, and 1 -butene) diffusion coefficients were determined for temperatures up to 767°K. These runs demonstrated that the cell can easily and quickly yield precise diffusion coefficients over a wide temperature range.
V a r i o u s methods have lieen used for the experimental determination of gaseous diffusion coefficients. One of t’he earliest of these was developed by Stefan (1871), who filled a closed end tube with one gas and allowed a second gas to pass over its open end. Equal molar couiiterdiffusion took place between the gas origiiially in the tube arid the purge gas. *kt the end of the run the t’ube \vas closed and its mean composition was determined. The diffusivity could then be calculated from t,his concentration, the duration of t,he run! and the dimensioiis of the tube. 111 order to simplify this technique, Rhodes (1956) determined diffusivities from the continuously changing composition of the effluent purge gas stream. However, these measured diff usivities varied inversely with t’he purge gas flow rates. Succeeding investigators (Chang, 1956; Epstein, 1958; Kaufmnuii, 1961) using this technique required several runs a t different flow rates to differentiate the diffusivity from this effect. Further modificatioii of‘ both the cell itself and the mathematical model describing it has permitted a flow rate-inPresent address, Union Carbide Corporation, TarrytoFn Technical Center, Tarrytown, 3.Y. 10591. To whom correspondence should be sent. Deceased.
variant diffusivity to be determined directly from the purge gas effluent (Frost, 1967). This diffusion cell has no moving parts, is quick and easy to use even a t high temperatures, and permits precise measurements. Diffusion Cell
The stainless st’eel diffusion cell (Figure 1) consists of a mixing volume, TIA, and a diffusion volume, VB. These two sections are bolted together and sealed tvith a copper gasket’. The influent purge gas is distributed to volume 17.4 through six tangential 0.03-in. i d . jets wit’h sufficient kinetic energy to ensure thorough mixing. The effluent purge leaves bhrough a single central outlet. The diffusion volume V g is filled mith 0.0274-in. i.d. stainless steel capillary tube of length L to confine gas mixing to volume I;*. tem and a Gow-Mac The cell was used with a manifold tungsten hot-wire thermal conductivity detect,or (Figure 2). The manifold system permitted the cell to be alternatively filled with the initial gas and purged with the second gas. Flows t,o t,he diffusion cell (above 50 mlimin to induce adequate misiiig in V.4) aiid the gas detector were controlled with needle valves and monitored nit11 bubble meters. Both the diffusion cell and the gas detector had separat,e thermoInd. Eng. Chem. Fundam., Vol. 12,
No. 1, 1973
129
I N I T I A L GAS: N I T R O G E N
3
i
STEP PURGE GAS: H E L I U M
i
i
TOP V I E W
1,2 INLET PURGE GAS 3 W T L E T PURGE GAS 4
1
6
MIXED VOLUME,
6
DIFFUSION VOLUME, Vg
3
4
5
6
7
8
0 , SECONDS
VA
Figure 3. Response of the detector system to a step change in concentration
(FILLED WITH CAPILLARY TUBES1
7
2
INLET PURGE GAS JETS
COPPER GASKET
If a perfectly mixed boundary condition is assumed for the well mixed volume, VA, a t the top of the diffusion cell, a material balance for gas A around that volume yields the equation
SIDE VIEW
Figure 1. Diffusion cell
CONDUCTIVITY CELL
r-
-
VACUUM
DIFFUSION CELL
NZ
HE
uu
d B U B B L E METER
Figure 2. Schematic of complete apparatus
stated temperature baths with sufficiently long preheat coils to ensure isothermal operation. For high-temperature runs the diffusion cell was placed in a furnace designed to reduce radiation effects. The pertinent dimensions of the diffusion cell are VA = 2.31 f 0.04 cma, Vg = 25.94 f 0.09 cma, and L = 13.180 i 0.006 cm. Diffusion Model
The mathematical model for the Stefan cell is based on Fick's second law
where CA is the concentration of one of the gases in the binary gas system, x is the distance along the axis of the cell measured from its bottom, 6 is time, and D is the constant diffusion coefficient. Initially, the cell is filled with one of the gases, gas A, of the binary system so that
CA
=
CO
.f
sin 22, n=lz,+---+-2
-Cout =
co
(sin 2 ~ ~ ) e - ~ n ~ ~ ~ ' ~ ~ D V A Z , ~ DV~z,'sin 22, FL2 2
FLa
(5) where each of the values of Z , is a root of the transcendental equation
Cout leaves the diffusion cell and passes through a thermal conductivity cell. The overall response of this detector, including its preheat coil and recorder output, to a step change in input concentration is shown in Figure 3 to approximate a first-order system, whose time constant is l/X. With the detector in series with the effluent of the diffusion cell, the recorder output, C*outbecomes
( 2)
for all values of 2 P t time = 0. Since there is no molar flux through the closed end of the tube the boundary condition a t this point is
(3) 130 Ind. Eng. Chem. Fundom., Vol. 12, No. 1, 1973
where A is equal to the free cross-sectional area of the diffusion cell, Ci, is equal to the concentration of gas A in the inlet purge gas stream, Gout is the concentration of gas A in the outlet purge gas stream just as it leaves VA a t a particular instant, and F is the volumetric purge gas flow rate. Gout would also be the concentration of gas A just a t the borderline separating the perfectly mixed volume from the diffusion cell, Le., Gout = CA a t x = L. The solution to Fick's second law with these boundary and initial conditions was presented by Schumann (1931), who solved the analogous heat transfer problem. Replacement of Schumann's heat transfer constants with the corresponding diffusion equation constants describes Gout as a function of time
where
A,
=
sin 22, 2
sin 22, DVAZn'
z,+-----+- FL2
D V A Z , ~sin 22,
FL22
(8)
Table 1. Summary of Random Errors System
He-A HeN2 HeN2 (trace)
No. of runs
-
Std dev
Range
10
0.43% 0.38%
1.16% 1.63%
0.11 0.25
15
0.89%
2.28%
0.015
7
CA/CO
L
diffusion cell. An error analysis for these changes shows that a diffusivity measured a t temperature T should be increased by
(13) where a is the coefficient of thermal expansion. These increases were as much as 2.6% a t 763°K. Experimental Errors
RUN TIME, MINUTES
Figure 4. Concentration tracings for different purge gas
When a measured diffusivity is proportional to T61a/P, where P is the pressure, a n error analysis on eq 6 and 9 will yield, a t the usual operating conditions
flow rates
B,
=
Zn2D/L2
dD - 6 d L + -5 -d + T - -4dB1 + - - + 3- d P D 3 L 3 T 3B1 3 p
(9)
A study of eq 7 with typical values for A,, B,, and M shows that the equation soon converges to
(14) The maximum individual systematic errors for each of these measurements were estimated as dL/L = =to.05~0,d T / T = k O . O l % , dBi/Bi = +0.3%, d P / P = *0.2%, d F / F = *0.3%, and dVB/VB = k0.4%. Substitution of these values into eq 14 yields an overall systematic error of +1.0 to -O.6Y0. This range does not include the systematic error present from any convective mixing that may take place between V A and V B .Although this effect was not measured a t the prevailing conditions, it was measured for an increased degree of turbulence in V A .When the 0.030-in. i d . tangential inlet nozzles were replaced with 0.007-in. nozzles, causing a n 18-fold increase in the inlet momentum to V A ,the measured diff usivity increased by only o.35yO. Table I shows the random errors found for the He-A, He-N2, and He-N2 (trace) runs made a t 303°K. The high standard deviation for the latter run reflects the large dB1/B1 error experienced with low concentration measurements.
Figure 4 shows the logarithm of the recorder reading, found to be directly proportional to C*,,/CO, plotted against time for various flow rates. Each curve converges to a straight line plot, from which its slope, -B1 is measured. -B1 is then used in conjunction with eq 6 and 9 to obtain Z1and D. Although a lower flow rate run has a smaller -B1, Z1 also becomes smaller. These two changes are compensatory so that D remains constant, regardless of the flow rate. Concentration and Temperature Effects
Since gas diffusion coefficients can vary with concentration, a mean concentration, (?A, present in the cell during the course of a run should be defined. A reasonable definition is
-6.4_ CO
irIL +
( 2 sin 2,)(cos Z1z/L) (e-21208/L2 )dedz ~ ~ 221 sin 2Z1 D V A Z I ~ D V A Z sin 21 -- _ _ _ 2 FLZ FL2 2 ~
(0,
+
- OB) (L)
where OB and eE are the times corresponding to the beginning and end of the time interval over which D is being measured. Evaluation of the integral in eq 11 yields
I n most of our runs ~ A / varied C ~ between 0.10 and 0.30. Lower average concentrations required Cout/Co to be less than 0.01, a t which range precision fell off markedly. High-temperature diffusion coefficients were corrected to account for the dimensional changes that had occurred in the
TB)
1 d F+ A 3(F
~
Results and Discussion
Table I1 compares the diffusion coefficients described in Table I with the corresponding 303°K values obtained from the diffusion coefficient-temperature relationships of other selected investigators. Most of their work was done with a Ney and Armistead (1947) two-bulb apparatus or a LoSchmidt (1870, 1871) cell which used larger than trace quantities of both gases. The He-Nz (trace) diffusion coefficient was compared to those obtained from the gas chromatographic technique of Giddings and Seager (1960) and the pointsource method of Walker and Westenberg (1958a). In all of these systems the diffusion coefficients found in this work agreed to within less than 1% with the averages of the other results. Figure 5 shows helium-paraffin (methane, ethane, propane, and n-butane) and helium-olefin (ethylene, propylene, and Ind. Eng. Chem. Fundam., Vol. 12, No. 1, 1973
131
5.0
/
3.0 -
Table 111. Temperature Dependence of Helium-Hydrocarbon Diffusion Coefficients
4.0
2.0
-
1.5
-
?I
Y 5
Stondord
error of System
Helium-methane Helium-ethane Helium-propane Helium-butane Helium-ethylene Helium-propylene Helium-1-butene
I=
5 0 ki0
Equation
5.923 X 4.566 X 3.776 X 3.653 X 5.468 X 3.934 X 3,582 X
10-5(T)1.6426 10-5(T)'.6332 10-b(T)1.631b 10-5(T)1.6070 10-5(T)1.6177 10-5(T)1,63327 10-5(T)1.6194
estimate
1.1% 1.2% 1.1%
1.5y0 1.3% l,O% 0.8%
1.0.9
-
.a-
V
z 2
runs were not made because of the poor reproducibility of the detector a t low concentrations. Needless to say, a more sensitive detection system such as a flame-ionization cell could yield these trace hydrocarbon concentration diffusion coefficients.
.7-
2
.6-
LL
LL
.5
0
Nomenclature
.4
1
A
I
300
1
400
I
500
I
I
I
600 700 800900
TEMPERATURE,
e*/
OK
Figure 5. Helium-hydrocarbon data ~
~
~~
Table 11. Comparison of Measured Diffusion Coefficients
System
He-A
Hex2
Hex2 (trace)
Meosured diffusion coefficients, cm2/sec at 1 otm, 303'K This work Other works
0 773 0 0 0 0 0 0 0 0 0 0 734 0 0 0 0
792 (Kosov and Bogatyrev, 1969) 778 (Schrieider and Schafer, 1969) 771 (Van Heijningen, et al., 1968) 768 (Ivakin and Suetin, 1964) 735 (Holsen and Strunk, 1964) 773 (Saxena and lIason, 1959) 760 (Srivastava, 1959) 760 iStrehlow. 1953) 767 Mean 735 (Ellis and Holseii, 1969) 719 (Ivakin and Suetin, 1964) 728 (Rumpel, 1955) 727 Mean
0 704 0 720 (Chang and Kobayashi, 1967) 0 691 iseager. et al.. 1963) 0.709 (Walker and Westeiiberg, 1958b) 0 707 Mean
1-butene) diffusion coefficients for temperatures up to 767°K. The He-1-buteue system showed signs of thermal decomposition above 522°K. Each data point represents t'he average of three or more rutis. Table I11 s h o w that the standard errors of estimate for these straight line plots are less than 1.57,. The 373°K helium-methane diffusion coefficient found by Fuller arid Gidding (1965) was 1,294 higher than the correspondiiig interpolated value from t,his work. These hydrocarbon ruiis were made with e,/Ca ranging bet'ween 0.13 and 0.31. Low 6*/C0 data would haye been valuable in obtaining interaction energy parameters. Such 132
Ind. Eng. Chem. Fundom., Vol. 12, No. 1, 1973
the free cross-sectional area of the diffusion cell, cm2 = a value defined by eq 8, dimensionless A, = the exponential term Z,2D/L2, sec-I B, = the point concentration of gas component A, CA nioles/cm3 Ca = an average mole fraction in the cell bet\\-een OB and BE, dimensionless = the coiicent'ration of gas component' .A in the diffuco sion cell a t the start,of a diffusion run, moles/cm3 = the Concentration of gas h leaving the diffusion C*O"t cell as recorded by the measuring instrument, mole./ cn13 c,, = the coilcentration of gas X in the purge gas stream before it enters the diffusion cell, moles/cm3 C0"t = the concentration of gas h in the purge gas stream as i t leaves the diffusion cell, moles/cm3 = diffusivity, or diffusion coefficient, cm2/sec D = the purge gas flow rate through the diffusion cell, F cni 3ise c L = the length of the diffusion cell, cm = the reciprocal of the time constant for the instruJI nierit used to measure and record the concentration in the stream leaving the diffusion cell, 3ec-l P = the absolute cell pressure, psia T = t,hecell temperature, O K = the volume a t the top of the diffusion cell, cni3 J'A = the free volume of the diffusion cell, cm3 VB 5 = axial distance, em = the roots of a t'ranscetideiital equation, radians 2, =
GREEKLETTERS a
=
e eB
= =
eE
=
the mean coefficieiit of thermal expansion for stainless steel. "C-1 time, see the beginning of the t h e interval over which the diffusion coefficient is meabured, sec the end of the time interval over which the diffusion coefficieiit is measured, sec
literature Cited
Chang, G. T., Kobayashi, R., Ind. C h i n . Belge 32,337 (1967). Chang, N.L., 1123. Thesis, Columbia University, Sew York, N. Y., 1936. Ellis, C. S.,Holsen, J. N., IND.EZG. CHLM.,F G N D ~8, M787 . ( 1969). Epstein, M,,A1.S. Thesis, Columbia University, New York, N . Y., 1938. Frost. il. C. doctoral dissertation. Columbia University, New Yoik, N. Y., 1967. Fuller, E. N., Giddings, J. C., J . Gas Chromatogr. 3 ( 7 ) , 227 (1965).
Giddings, J. C., Seager, S. L., J . Chem. Phys. 33, 1579 (1960). Holsen, J . N., Strunk, &I. R.,ISD.ENG.CHEM.,FUNDAM. 3, 143 (1964). Ivakin, B. A., Sue&, P. IC., Zh. Tekhn. Fiz. 34 (6), 1115 (1964). Kaufmann, T., 3I.S. Thesis, Columbia University, New York, N. Y., 1961. Kosov, N. D., Bogatyrev, .4.F., Vop. Obshch. Prikl. Fiz., Tr. R ~ s p u bKonf., . 1st 1967 134 (1969). Loschmidt, J., Sitzber. A k a d . Wiss. Wien 61, 367 (1870). Loschmidt, J., Sitzber. Akad. V i s s . Wien 6 3 , 468 (1871). Sey, E. P., Armistead, P. C., Phys. Rev.71,14 (1947). Rhodes, 13. P., 31,s. Thesis, Columbia University, New York, N . l%X. - .Y - ., -Rumpel, University of Wisconsin Naval Research Lab Rept. CAI-851 (1953). Saxena, S.C., Mason, E. A,, X o l . Phys. 2,379 (1959).
Schneider, M., Schafer, K., Ber. Bunsenges. Phys. Chem. 73 (7), io2 fIR6Ri. Schumann,-T. E. W., Phys. Rev. 37, 1508 (1931). Seager, S. L., Geertson, L. R., Giddings, J. C., J . Chem. Eng. D a t a 8 , 168 (1963). Srivastava, K. P., Physica 25,571 (1959). Stefan, J., Sitzber. Akad. Wiss. Wien 63, 63 (1871). Strehlow, It. A . , J . Chem. Phys. 21,2101 (1953). Van Heijningen, R. J. J., Harpe, J. P., Beenakker, J. J . AI., Physica38, 1 (1968). Walker, It. E., Westenberg, A. A., J . Chem. Phys. 29, 1139 (1958a). Walker, R. E., Westenberg, A. A., J . Chem. Phys. 29, 1148 (195Sb). RECEIVED for review February 22, 1972 ACCEPTED September 20, 1972
Sorption Apparatus for Diffusion Studies with Molten Polymers J. Larry Duda,' Grant K. Kimmerly, Wilmer L. Sigelko, and James S. Vrentas*2 The Dow Chemical Company, JIidland, Jlich. 48640
A sorption apparatus and an experimental procedure for diffusion studies with molten polymers are described. It is possible to examine the concentration dependence of penetrant diffusion into polymers whose glass-transition temperatures are significantly above room temperature. Diffusivity and solubility data are presented for the ethylbenzene-polystyrene system over the temperature interval 1 60-1 78°C and for weight fractions of ethylbenzene ranging from 0 to 0.1 5.
M e a s u r e m e n t of the diffusion of relatively low molecular weight penetrants in high polymers provides a convenient method of deterniiiiiiig the nature of the kinetic agitation of polymer molecules. I n addition, such diffusion measurements are needed for the design of polymer processing equipment for molten polymers and for the evaluation of the barrier properties of solid polymers. For example, the diffusion of monomers, catalysts, diluenta, or by-products can have a sigiiificant influence 011 the behavior of polymerization reactors. I n addition, a processirig step involving the removal of residual components from polymers often succeeds these reactors. The distribution of blowiiig agents, plasticizers, and other additives in molten polynit,rs is often strongly dependent on molecular diffusion. Finally, the utilization of polymers in packaging or as protective coatings is usually based on how well the polymers resist the diffusion of penetrants. Most studies of penetrant diffusion i t i high polymers have been concerned with two regions of the penetrant coiicent'ratioii-temperature diagram for the polymer-diluent system (Duda atid Vreritas, 1970). One region involves diffusion a t low penetrant concentrations below the glass-transition temperature where the diluerit and polymer do riot significantly interact during t'he time scale of the diffusion process. Experiments for this region involve measuring the diffusioii Present address, Department of Chemical Ihgineering, The Pennsylvania State University, University Park, Pa. 16502. Present address, Ilepartment of Chemical Fhgineering, 11linois Institute of Technology, Chicago, Ill. 60616.
of relatively small inert penetrants with the st'andard permeation technique (Stannett, 1968). The other region is characterized by diffusion of lorn t o moderate amounts of penetrants at temperatures sufficiently above the effective glass-transition temperature so that relaxation of the polymer molecules is very fast compared to the time scale of the diffusion process. Data in this region have been taken for simple gases and organic vapors using both sorption and permeation techniques (Fujita, 1968). As Fujita observes, it is difficult to perform either sorption or permeation experiments a t temperatures significantly above room temperature. Coiisequeiitly, not surprisingly, most investigations in the second region have been concerned n-ith the study of penetrant diffusion into the so-called soft polymers whose glass-transition temperatures are near or below room temperature. Therefore, there exist very few data on the diffusion of penetrants into industrially important amorphous polymers, such as polystyrene, the glass-transition temperatures of which are significantly above room temperature. The design of experiments for the study of penetrant diffusion above the effective glass-transitioii teniperature for polymers with high glass-transitioii temperatures must take into accouiit the fluid-like or niolteii 1)ropertiea of the polynier and the strong concentration dependence of the diffusion coeficient for such systems. The diffusivity is typically an exponential fuiictioii of the penetrant coricentrat,ion a t low conceiitrations, and it is not uncommon to observe increases of the diffusion coefficient by a factor of 1000 with the addiInd. Eng. Chem. Fundam., Vol. 12, No. 1, 1973
133
