cont'rol. The optinial flow rate policy in each case had a terminal singular arc corresponding t o the new steady value of
49 Nomenclature
d B
dimensionless inlet concentration change dimensionless inlet temperature change G(0) = sivit'ching function K = proportional gain 31 = number of control values in direct search n = reaction order P = performance index p ( 0 , n ) = adjoint variable T(0,n) = dimensionless temperat,ure 40) = dimensionless velocity x ( 6 , ~ )= dimensionles concentratioii = =
GREIX LETTLRS Q ~ , C Y Z = constants p = dimensionless reaction group = dimensionless activation energy (: . ) = Dirac delta function ?I = dimeiisioiiless spatial variable 0 = dimensionless time 4 = dimensionless frequency factor 4 = dimensionless constant W = dimensionless heat generat,ioii constant,
SUPERSCRIPTS d = desired * = maximum SUBSCRIPT * = minimum literature Cited
Hwang, ill., 31.S. report, Depart'ment of Chemical Engineering, California 1nst.itute of Technology, June 1968. Katz, S.,J . Elect. Conlrol 16, 189 (1964). Koppel, L. B., IXD.EXG.CHEM.FUKDAJI. 5 , 403 (1966a). kroppel, L. B., ISD. EXG.CHEM. F U S D A h f . 5, 413 (1966b). Eioppel, L. 13., ISD.ESG. CHEM.FUXDAJf. 6, 299 (1967). Koppel, L. B., Shih, Y. P., Coughanow, D . R.,ISD. ESG. CHEM.FUXDAM. 7,286 (1968). Ogunye, A. F., Ray, W, H., A.1.Ch.E. J . , in press, 1970. Seinfeld, J. H., Lapidus, L., Chem. Eng. Sci. 23 (12), 1461 (1968a). Seinfeld, J. H., Lapidus, L., Chem. Eng. Sci. 23 (12), 1486 (196813).
RI:CEIVED for review February 28, 1969 ACCEPTED July 21, 1970 Work supported in part by Kational Science Foundation Grant GK-3342.
EXPERIMENTAL TECHNIQUES
Modified Ney-Armistead Cell for Gas Diffusion Measurements Arvo Lannus' and Elihu D. Grossmann Department of Chemical Engineering, Drexel Cniversity, Philadelphia, Pa. 191 04
A two-bulb gas diffusion cell i s described which i s suitable for measurements over a wide range of temperature and pressure, and i s completely and easily accessible for cleaning and inspection. The new cell equation was tested experimentally using the carbon dioxide-nitrogen system.
N e y and iirniistead's gas diffusion apparatus ( ~ e yand Arniistead, 1947) is capable of providing more accurate diffu5ion data than any of the other devices reported in the literature. It consists of two bulbs separated by a narrow capillary fitted in the center with a stopcock or valve by which the bulbs may be isolated or connected. While the cell is usable over a wider range of temperature and pressure than the older Loschmidt (1870) apparatus, it is often difficult to avoid a distorted diffusion path because of the stopcock design. The distortion can be excessive if the stopcock has been designed for hermetic sealing over a mide range of temperature. A recent two-bulb apparatus (Van Heijiiiiigeii et al., 1968) has Pre-eiit addre-$ Department of Chemical Engineering, The Coopei Uniori, New York, X. E'.10003. To Thorn correspondence should be sent.
eliminated the stopcock altoget,her, but is usable only a t very low pressures. Our modification places the valve a t the entrance to the capillary t'uhe, thus involving no distortion of the diffusion path. The design a l l o w easy access to both chambers and the tube, and maint'aiiis simple cylindrical geometry for absolute measurenients. Diffusion Cell
The diffusion cell (Figure 1) consists of a top chamber, A , of volume TIA, bottom chamber, B , of volume V B ,and a center block, C, through which has been bored a capillary of length L and area of cross section A t . The upper end of the capillary may be closed by valve E. l'alve D controls the inlet tube, F. Sections d and B are modified 6-inch-0.d. =\S1150-pound welding neck flanges of 304 stainless steel. The faces are grooved to fit the machined lip on block C. Lead gaskets provide the best seals for both pressure arid vacuum service. The capillary sealing valve, E , is a noiirotating stem high pressure Ind. Eng. Chem. Fundom., Vol. 9, No. 4, 1970
655
B , and its concentration in chamber A a t any time thereafter is given by cI*(t), a material balance on this component entering chamber A may be written as
- Di2A t(dCi/dz)z=L
VAdci,/dt
(11
where z is the axial distance in the tube measured upward fromz = Otoz = L . If we assume the main resistaiice to diffusion to be the capillary tube, so that the gases in chambers A and B are always perfectly mixed, then, after an initial transient, the concentration gradient in the tube will be linear, or
(dci/dz)z=L
= (CIA
- clB)/L
(2)
Elimiiiatiiig cIB from Equation 2 by using an over-all material balance of the component originally in B
Figure 1, A.
+
Diffusion cell
B. C. D. E. F.
+
where V = Va V B A I L , and applying the initial condition clA(0) = 0, the cell equation becomes, after integration of Equation 1,
Top chamber Bottom chamber Tube block and capillary tube Inlet valve Capillary valve Filling line
DIZ= -111[1 - c i ~ / c i ~ l / ( f l t )
(4)
The cell constant, p, for our apparatus is (1.148 =t 0.005) X IO+ a t 297.7 OK (tin seconds). T o test Equation 4, the diffusion coefficient of the carbon dioxide-nitrogen system was determined over a temperature range of 282' to 399°K using our apparatus. The gases were analyzed by gas chromatography. The results are plotted in Figure 2, together nith data found in the literature. Pressure and Concentration Effects
Measurements a t elevated pressures may be corrected to a reference condition (usually the ideal gas density a t 1 atni) by the formula
(PDidO
30 0
250 Figure 2. 0
350 T, O K
400
Carbon dioxide-nitrogen data This work Fristrom and Westenberg ( 1 9 6 5 ) Boordman and Wild (1937) Chapman and Cowling ( 1 939) Boyd et al. ( 1 95 1 ) Pakurar and Ferron ( 1 9 6 6 )
V
type, modified under its packing nut with a n additional Teflon ring to provide vacuum sealing. The regular valve disk has been replaced by a Teflon plug which seals the capillary. The total motion of the plug is 5 mm. The gas inlet valve, D , is a commercial bellows type. The over-all cell is 6 inches 0.d. and 8 inches high and fits coiiveiiieiitly into laboratory thermostats. The vital dimensions for establishing the cell constant are
V A = 100.5 =t 0.5 ema V B = 83.7 =t 0.4 em3 At
L
= =
0.0198 + 0.0002 em2 3.77 =t 0.01 em
Cell Model
Derivation of the cell equation follow closely that of S e y and airmistead (1947). If cl(z,t) is the concentration in the capillary of that component which is iiiitially all in chamber 656
Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
=
(PDIZ)
(5)
where p is the deiisit'y of the diffusing mixture. To account for gas imperfection, the density niay be calculated from t'he truncated virial equation p =
pX/[RT(l
+ B'p)]
(6)
where J4 is t,he average molecular weight and B' is the second virial coefficient in the pressure expansion. dlteriiatively, Enskog's theory of the transport coefficients of dense gases (Hirschfelder et al., 1964) may be used with similar results. It is harder to correct for concentration effects. A known disadvantage of this and other static methods is that the concentration of the measurement is somewhat indeterminate. Nevertheless, we niay proceed as follows: According to the Chapman-Enskog kinetic theory, the mth-order approximatioii to the biliary diffusion coefficient, [DI2],,is given by
[Di2lm =
[~i?]i.f~'"'(~~)
(7)
The first approximation, [D1?Il,depeiids only on the temperature and unlike-molecule interaction parameters, whereas the correction for higher order terms, .fD(m), depeiids on the composition of the diffusing mixt,ure as well as bemperature and like and unlike molecule interaction constants. Kihara's formula for .fO@),too complicated to be given here, has been discussed by hIason (1957). Except where the molecular weights of the components differ widely, f D @ ) is only slightly different from unity. Severt,heless, if the inclusion of concenmay be calculated from tration depeiidence is desired, [012]1
Equation 7 by taking as [Dl2I2the experimentally det'ermiiied value and approximating the composition as z1 G xi,. In t,his manner, [Dl.I2 at any other concetitratioii may be estimated, once [Di2l1is known. A\ccording to n'alker and 11-esteiiberg (1958), the maximum value of f D f 2 )for C02-X2is 1.004 in the temperature range of this work. K e have therefore taken jD@) G 1. However, the theory generally underestimates the magnitude of the correction (Van Heijtiingen et al., 1968). Experimental Errors
At least for measurements a t low and moderate densities, it is customary to correct diffusion coefficients to atmospheric pressure. Thus, Equation 4 becomes
Di?patm = - l n ( l
- ~i~/xi~)p/Pt
(8)
where p is the pressure of the measurement' and xiAaiid xImare the mole fractions. The root-mean-square absolute error in the diffusion coefficient is then defined as
Table I.
Summary of Measurement Errors
Std. deviation of C 0 2 - S 2data of this work, Reproducibility, Yo RlIS absolute error, %
yo
0.841 1.11 1.53
engineering purposes. (We have already obtained good diffusion data on two additional systems over a nominal temperat'ure range of 241" to 474OK with 3ur apparatus and will report the results in a subsequent publication.) Our apparatus is limited principally by the maximum service temperat'ure of the plastic (Teflon) components, and, a t lower temperatures, bl- the possible condensation of one or both components being measured. The maximum pressure is about 100 atni a t ambient temperature. The theoretical limit,ation of indeterminat,e composition has been mentioned. Nomenclature
cross-sectional area of capillary tube, em2 mixture second virial coefficient, atm-1 concentration of component 1 in capillary a t c1, C l m time t and t + a ,g mole/cm3 = concentration of component 1 in cells -4 and B , CIA, C ~ B g mole/cm3 = binary diffusion coefficient, cm2/':ec D12 = mth Sonine approximation to kilietic theory [Dnlm binary diffusion coefficient, crn2, sec = mth order correction to [DlPll j d m ] L = length of capillary tube, em IIf = molecular weight of mixture, g/g mole P = pressure, a t m t = time, see T', VA, V B = yolume of cell. volume of chambers -1 and B , cm3 21: 21, = mole fraction of component 1 in capillary at' time t and t + ~0 rl.4 = mole fraction of component 1 i n chamber .I 2 = displacement, em P = cell constant, cmF2 = density of mixture, g/cm3 P -4 t
=
B'
where the differential coefficients are given by
bDiz/bxla
=
P / [Pt(ri=-
I
21~)
b D i z ' b ~ - = - p ( ~ i ~ / x i m ) /[Pt(zi, dD12/dp = -1nV bDi2lbP dDi2/bt
=
ln[l
-~
-Z~~/T~,I/P~
1
~
1
1
(10)
- XIA/~I,IP,'P~~
= 111 [I - Z ~ A 21,]p/Pt2 /
The individual measurement errors foi a typical run were AxiA = 0.00042, AXI, = 0.0046, A p = 1.14 X attn, $3 = cni-', and At = 10.0 seconds. The temperature of 5x I sumour experiments was controlled to within 10.05 "C. . mary of the experimental errors and the standard deviation of our CO2-X2data from giaphically smoothed values is given in Table I. The reproducibility mas estimated from the two measurements a t 313OK shonn in Figure 1. Results and Discussion
Equation 4 was tested by measuring the diffusion coefficient of C 0 2 - S 2from 282" to 399'K (see Figure 2). X summary of the measurement errors is given in Table I. Our results for the C02-N2 system are in good agreement with previously reported values from the literature. I n a recent paper describing diffusion measurements (Van Heijniiigen et al., 1968) it is estimated that the scatter in diffusion data obtained by the two-bulb or Key-Armistead method is around 2 t'o 5%. Our chromatographic analyses are routinely between 0.5 and 1.5% average deviation. From Table I it is seen that the reproducibility aiid data scatter are of the order of 1% and the (maximum) theoretical error is as low as 1.5y0. It is concluded t'hat this cell is capable of yielding diffusion data of sufficient precision and accuracy for most
= =
literature Cited
Boardman. L. E.. Wild, S . W., Proc. Rou. SOC.A-162. 511 11937). Boyd, C. A'., Stein, N., hteingr "I\lathemat,ical Theory of Konuniform Gases," p. 252, Cambridge University Press, London, 14x4 *""".
Fristrom, R. Westenberg, A . .L, "Flame Structure,'! p. 265, IIcGraw-Hill, Kew York, 1965. Hirschfelder, J. O., Curtiss, C. F., Bird, R. B., "llolecular Theory of Gases and Liquids," p. 646, Kiley, Kern York, 1964. Loschmidt, J., A k a d . TT-iss. TT7z'en61, 367; 62, 468 (1870). lIason, E. A,, J . Chem. P h y s . 27, 782 (1957). Sey, E. P., Armistead, P. C., Phys. Rev. 71, 14 (1947). Pakurar. T. .4..Ferron. J. It.. ISD.ESG.CHEU.Fvs~.\ar.5 . 553 (1966): Van Heijiiingen, R . J. J., Harpe, J. P., Beenakker, J. J. X, Physica 38, 1 (1968). Walker, R . E., We.;teiiberg, A. A,, J . Chem. Phys. 29, 1147 (1958). I~ECCIVED for review October 3, 1969 ACCEPTEDAugust 26, 1970
Ind. Eng. Chem. Fundom., Vol. 9, No.
4, 1970 657
