n = concentration of species, molecules/cm3 N = Avogadro's number q = Bjerrum radius of bounds ions rc = critical distance of repulsion, cm rm = mean distance of approach of dissociated ion pairs, cm R = gasconstant R , = reaction rate for water dissociation T = absolute temperature u = v%, quantity in eq 17 x = distance from edge of dilute b.1. toward membrane, cm x' = distance from edge of concentrated b.1. toward membrane, cm z = valence of ion Greek Letters 6 = boundary layer thickness c = electric permittivity, C2/erg cm 0, = critical angle of repulsion u = selectivity coefficient, dimensionless cp = J,lzJ/RT,dimensionless electric potential @ = ratio of dissociation constants, with and without electric field J, = electric potential, V w = mobility of ion Subscripts and Superscripts b = bulk solution
b.1. = boundary layer i = ith ion; 1,2,3,4, respectively, K+, C1-, H+, OHlim = limiting r.z. = repulsion zone - - membrane phase
Literature Cited Bethe, A., Toropoff, T., Z.Phys. Chem., 88, 686 (1914). Bjerrum, Kgl., Danske Vld. Selskab. Math-fys. rnedd., 7, 9 (1926). Bragg, W. l.,Proc. Roy. Soc. london, Ser. A 89, 468 (1914). Cowan, D. A., Brown, J. H., lnd. Eng. Chem., 51, 1445 (1959). Frank, H. S., Evans, M. W., J. Chem. Phys., 13, 507 (1945). Gregor, H. P., Miller, I. F., J. Am. Chem. Soc., 86, 5689 (1964). Gregor, H. P., Peterson, M. A., J. Phys. Chem., 68, 2201 (1964). Ghosh, J. C., J. Chem. Soc., 113, 449(1918). Harned, H. S., Owen, B. B., "The Physical Chemistry of Electrolytic Solutions," 3rd ed,Reinhold, New York, N.Y. 1957. Kressman, T. R . E.,Tye, F. L., Discuss. Faraday Soc., 21, 185 (1969). Lang, K. C., Ph.D. Thesis, Polytechnic Institute of New York, 1974. Lange, N. A., "Handbook of Chemistry," 10th ed,McGfaw-Hill, New York, N.Y., 1961. Manegold, E., Kalauch, K.,Kolioid b,86, 3 13 (1939). Onsager, L., J. Chem. Phys., 2, 599 (1934). Peers, A. M., Discuss. Faraday SOC., 21, 124 (1956). Robinson, R. A., Stokes, R. H., "Electrolytic Solutions." 2nd ed, Butterworths, London, 1959. Rosenberg, N. W., Tirrell, C. E., lnd. Eng. Chem., 49, 780 (1957). Uchino, T., Nakaoka, S., Hani, H., Yawataye, T., J. Elecfrochem. SOC., Jpn., 26, 366 (1958).
Received f o r reoiew September 16, 1976 Accepted April 20,1977
Oscillatory Motion in the Cone and Plate. Perturbation Analysis of Secondary Flow Effects Raffi M. Turian' Department of Chemical Engineering and Materials Science, Syracuse University, Syracuse, New York 13210
An analysis of the flow in the cone and plate system aimed at establishing the working equations for interpreting oscillatory measurements using the rheogoniometer is carried out. The three-dimensional flow analysis yields explicit analytical expressions for the time-dependent velocity and pressure fields, and these are used to establish simple analytical expressions for the stress, the normal force, and the power, clearly depicting the influences of the secondary flows as well as those of the frequency and the cone angle. The analysis predictsthe occurrence of higher harmonics associated with the secondary flows, and consequently also establishes the experimental conditions needed to promote or suppress them. The significance of these results to the practitioner derives from the fact that they are simple, straightforward, and analytical, and also from the fact that they pertain to a range of the variables adequate enough for and often exceeding that of most available rheogoniometers.
The problem considered here concerns the transient motion of the fluid contained between the surfaces of a cone and a plate (Figure 1)when either surface is forced to execute a simple harmonic oscillation. The flow patterns resulting from such oscillatory motion, and the associated torque and normal force responses, form the basis for dynamic rheological measurements with what is perhaps the most widely used rheometer: the Weissenberg rheogoniometer. It is recognized (though not widely it seems) that it is not an easy matter to obtain meaningful oscillatory data with this instrument. This is principally due to the fact that the gap width does not all Division of Engineering, National Science Foundation, Washington, D.C. 20550.
348
Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977
ways remain unchanged against the normal forces generated by the flow, and also due to the fact that no adequate theory exists providing a definitive assessment of the influence of the secondary motions. The former can be resolved by modification in design, which entails stiffening the instrument frame to restrain axial motion of cone or plate and replacing the normal force sensing spring system by an array of pressure sensitive strain gages distributed across the plate surface (see Christiansen and Leppard, 1974). The latter problem must be resolved through analysis, and this is the purpose of this paper. While the primary motivation for this work is to establish the theoretical framework needed for unambiguous interpretation of rheometric data, the problem is also intrinsically
Figure 1. Cone and plate system.
interesting because it belongs to a small class of problems concerned with analytical treatment of unsteady three-dimensional flows. Of course there is an extensive literature on oscillatory flows in general, but the main published works dealing specifically with the geometry examined here are by Maude and Walters (1964), Nally (1965), Walters and Kemp (1968), and MacDonald et al. (1969). All these studies share a common heritage: they derive from what will, for brevity, henceforth be designated as the one-dimensional theory, Le., the theory which views the flow to be purely tangential uninfluenced by the secondary motions. The distinctions among them, however, merely derive from the varying levels of approximation with which the problem is addressed within the bounds of the one-dimensional theory. I t will be simpler to establish a ranking for each of these studies when we have defined the framework of the more comprehensive threedimensional theory which is the basis of the present analysis. Accordingly we will recall these publications a t an appropriate later stage. The motion in the cone and plate is inherently three-dimensional. The systematic analysis developed here admits secondary motions, and is based upon an ordering scheme in which a suitable Reynolds number (defined in terms of the characteristic amplitude and frequency) plays a central role in ascertaining the rank of each result in the hierarchy of approximations. The present analysis reveals some interesting properties of the flow and yields results which are beyond the reach of one-dimensional theory. The steady rotation counterpart of this problem appeared in this journal earlier (Turian, 1972),and the present study claims direct descent from it. Both are concerned virtually exclusively with the small cone angle problem because they are principally undertaken in response to the need of experimentalists to interpret data. Except for possibly facilitating flow visualization, or also for permitting measurements with systems comprising suspended solids (which are more appropriately handled in other types of instruments anyway), the large cone angle (0 > loo) flow would appear to be a hydrodynamic curiosity. Indeed the exaggerated influence of the edge (which may under these circumstances need to be enclosed to contain the test fluid) transforms the large cone angle flow to a substantively different problem. Furthermore, in this system the asymptotic behavior, as the angle becomes vanishingly small, is that of uniform shear across the gap and this is a singular attraction of the cone and plate geometry. Therefore any modification which undermines this feature must come with significant compensating advantages. I t might therefore appear curious that Walters (1975) prefers the parallel plate geometry (which does not enjoy this feature) on the grounds that the governing theory is simpler, but the assessment is made a t the level of approximation of one-dimensional theory. When secondary motions are admitted, as they must be in either case, this apparent advantage becomes quite a bit thinner. Nevertheless the present analysis can easily be applied to the parallel plate geometry. Indeed the present results can be adapted to the narrow gap parallel plate system by a straightforward limiting process under the appropriate coordinate transformation.
Formulation of the Problem We consider the system shown in Figure 1and take ur,ug, and u4 to be the components of the velocity vector nondimensionalized with respect to (RR), in which R is the cone radius and R is a characteristic “rotational speed” comprised of a suitable combination of the amplitude and frequency of the oscillation. The pressure p is nondimensionalized with respect to /IR. I t is convenient to use the complement of the angular coordinate 0 given by p = ( ~ / 2 )- 0. Reference to Figure 1then shows that the surface of the plate corresponds to /3 = 0 and that of the cone @ = PO. We define a stream function by
+
and set W
u+ = r cos p In the small cone-angle analysis considered here we use the transformation {=-=sin p sin p (3) sin PO t in which t = sin POsin2r rR2pRReo Reo 40 168
(
354
Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977
M4 +-cos2 105
T
+.
T
-
[1 + -" ( 3 P - 2)
M4r4 + 6 t 2 p + O(c4)] cos + [(15p - 3 0 p + 7) 360 T
+ O(c2)]sin + . . . + Re2r4 [(350p - 315p 25 200 - 280c3 + 83) + O(c2)]sin3 + . . . T
T
(67)
The special expressions for the velocity field corresponding to the present case can also very easily be obtained from the results already established. As an illustration we merely present that corresponding to woo given by eq 53, which yields woo = r2
sinh ar{ sinh ar
3
sin ar sin ar
-+ X I * -
(XI
(68)
where XI = (sin 7 - i cos 7)/2 in the present situation, while the small -ar representation of eq 68 reduces to woo = r2{ sin T
+ M2 -r4( ? - () 6
cos T
--M 4 r 6 ( 3 p - l o p + 7 0 sin T + . . . (69)
360 Clearly one can depict, more explictly if necessary, through use of the trigonometric identities, such as sin2 T = M (1- cos 2 T ) etc., the detailed nature of the higher harmonics occurring in eq 64-66. It would perhaps be instructive to indicate the magnitudes of the various correction terms in the equations developed above for prescribed conditions of operation. Accordingly we present in Table I1 ratios of the absolute maximum value of the various correction terms to that of the term O(1) for eq 64, 65, and 66. Three cases are considered, the first pertaining to high frequency-small amplitude and small cone angle, and the second and third pertaining to moderate and low frequencies, respectively, but both with large amplitude and cone angle. The column headings in Table I1 are defined as f 0110ws. R(Re2'c 2jM2')
I maximum value of term 0 (I)I (64)
(66)
The term O(M4) in eq 66 is obtained from eq 62. Equations 56 and 57a through 57g for ragreduce, in the present case, to
= Jmaximum value of term O(Rezie2jMzL) I
+ ...
+ 2739Re2 (1 - r6) sin4 + . . . 079 000
>
+ . ..
3 +Re2 sin4 + . . 4900
erog =
It is clear from the first integral of eq 59a that the righthand side of eq 60 is independent of {, as decreed by eq 61. As for the power, the expansion in eq 54 permits calculation of the terms corresponding to +OO(O), @oo('), and @ o o ( ~ of ) the dissipation function given by eq 49. Accordingly terms of O(l), O(Reocoa2),and O(Re0c0a4)in the expansion for the power can be calculated. The first two have already been determined, of course, and appear in eq 50, the term O(Reocoa2)was found to be zero. The term O ( a 4 )is given by
TCOS T
(70)
Thus, for example, the term R(c2M2)for eq 66 is given by
in which the M factor is inserted because of the fact that it is the maximum value of cos 7 sin T .
Table 11. Ratio of Maximum Absolute Value of Term O(Re2ic2iM2’) to Term 0(1)
R( c2M 2 )
K(c4)
R(M4)
R(Re2)
1.55 x 2.32 x 10-8 7.42 X lob2 1.28 X 3.06 x 10-4 1.65 x 10-3 4.22 x 10-7 ( 3 ) ~ 5.72 x 10-3 Normal force (1)” 1.15 x 10-4 2.07 x 3.48 X 9.90 x 10-2 4.59 x 10-4 Eq 65 ( 2 ) b 1.65 x 2.20 x 10-3 6.33 x 10-7 ( 3 ) ~ 7.32 x 10-3 Power (1)” 5.08 x 10-5 1.23 x 10-3 6.0 x 10-9 9.0 x 10-10 0 4.05 x 10-7 Eq 66 (2)b 7.28 x 10-3 1.86 x 10-5 0 2.78 X 2.81 X 7.91 X 10-5 1.39 x 10-5 1.09 x 10-7 3.68 X 0 2.76 x 10-6 (3Ic 3.24 X ” (1) R = 5 cm; w = 607 rad/s; A1 = 00 = 1°/2; u = 1cm2 s-1; M2 = 0.3589; Re* = 9.807 X 10-6. (2) R = 5 cm; w = 2 7 rad/s; Al = 2p0 = 2(6O);u = 1 cm2 s-l; M 2 = 1.7163; Re2 = 0.12921: c (3)R = 5 cm; w = 0 . 1 radls; ~ A1 = 5po = 5(4O); u = 1cm2 s-l; M2 = 0.03822; Re2 = 1.780 X 10-4.
Pressure Eq 64 with r = 0
( i ) ~ 8.94 x 10-5
(2)b
Conclusions Comprehensive and detailed oscillatory flow data using Newtonian fluids are apparently unavailable. Data from Christiansen’s laboratory are for highly viscoelastic solutions (see Christiansen and Leppard, 1974; Leppard and Christiansen, 1975; and also Leppard’s (1975) thesis). These data, while unsuitable for detailed quantitative comparisons, are in accord with the present results insofar as they confirm that to a first approximation the torsional response has the same frequency as the input oscillation, eq 57a, while the normal force and the pressure distribution have twice the fundamental frequency, eq 39 and 47. No oscillatory flow data taken under sufficiently severe conditions to test the higher order correction terms, particularly those of O(Re2),are known to be available a t the present time. However, the present oscillatory flow results can be shown to reduce appropriately to our steady rotation results (Turian, 1972), and these were shown to be in excellent agreement with available torque and pressure distribution data. The results established in this work lead to the following additional conclusions. (1) The one-dimensional theory for the rheogoniometer is incapable of providing even the most primitive level of approximation to the pressure and associated normal force. Secondary flows have a primary influence on the pressure distribution and must be accommodated if an adequate theory is to be developed. This is true in steady as well as in oscillatory rotation. The ability to measure the pressure distribution extends the instrument’s capability to provide accurate firstand second-normal stress difference functions. (2) The pertinent “Reynolds number” used as a gauge of the importance of the secondary flow effects is comprised of the product of the characteristic amplitude and the frequency of the oscillatory mot ion. Designations such as “large-amplitude” or “small-amplitude” rotation, in common use among practitioners, are therefore indefinite and vague. The flow is governed by an amplitude-frequency based Reynolds number, Re2, a frequency parameter, M2, and a cone-angle parameter, t2. The Reynolds number gives a measure of the relative importance of secondary flows, which clearly demonstrates that these may be promoted or suppressed by selecting a suitable combination of operating conditions. The asymptotic results established here, like the previous steady rotation calculations, have a range adequate for, and often exceeding that of, most available rheogoniometers. (3) The occurrence and nature of higher harmonics in the response to the forced simple harmonic input is rooted in the nonlinearities associated with the secondary motions. The systematic analysis presented here, aside from predicting these, establishes a logical ordering scheme which ties these to the set of gauge functions (Rezn).Again this suggests op-
erating strategies for promoting or suppressing these higher harmonics as desired. (4) The frequency can, depending upon conditions, have a strong influence on the flow even when the “Reynolds number” is small enough to render secondary flow effects imperceptible. Acknowledgments The interest and valuable comments of R. B. Bird, E. B. Christiansen, M. M. Denn, W. R. Leppard, A. B. Metzner, and M. C. Williams are very much appreciated. Nomenclature Ao, A1 = amplitudes of angular displacement of plate and cone, respectively, radians ao, a1 = dimensionless real angular velocity amplitudes of plate and cone, respectively (eq 9) bo, bl = phase angles of plate and cone, respectively, radians E2 = [ ( l / r 2 ) ( d 2 1 d p + ) ] c2[(d2/dr2) - ( F l r 2 ) ( d 2 / d P )dimen], sionless second-order operator M = v ‘ m , dimensionless frequency parameter No = normal force on the plate defined by eq 44, dyn P = power required to maintain the flow, dyn-cmls p = pressure nondimensionalized with reference to MQ pa = ambient pressure nondimensionalized with reference to p R p i , ( i ) = term in asymptotic expansion forp of O(Re21c2~a2‘), eq 38 R = coneradius, cm Reo = R 2 R p / p , first Reynolds number Re = R e d , second Reynolds number r = FIR dimensionless radial coordinate 5 = radial coordinate, cm t = time, s t = ut/R2,dimensionless time u, = r component of velocity nondimensionalized with reference to RR u~ = 8 component of velocity nondimensionalized with reference to R Q u4 = d~ component of velocity nondimensionalized with reference to R O w = r(cosP)u4 wi = term in asymptotic expansion for w , eq 16 w i j ( ’ ) = term in asymptotic expansion for w of 0(Re2’ 2 i , 21) Greek Letters = vZPcomplex frequency parameter flo = cone angle, radians fl = ( ~ / 2 ) 8, angular variable 6 = sin Po { = sin fllsinpo = sin f l / c KO = a (&)! eLbo,eq 10 ~1 = (a1/2) eibl, eq 10 (Y
Ind. Eng. Chem., Fundam., Vol. 16,No. 3, 1977
355
XO(S) = K O e I r , eq 26 X,(r) = K l e t r , eq 26 A i = A 1 - Xo p = viscosity of liquid, P u = p / p , kinematic viscosity, cm2/s K,) = the total stress tensor nondimensionalized with ref-
= w A 1 , peak angular velocity amplitude of cone, eq 9, radls u/2r = frequency of forced oscillation, Hz * = asterisk designates the complex conjugate
erence to p R p = liquid density, g/cm3 T = u t , dimensionless time x = $/Reot3, dimensionless stream function xI = term in asymptotic expansion for x,eq 17 x l , ( l ) = term in asymptotic expansion for x of O ( ReZLt 2/~21) $ = dimensionless stream function defined by eq 1 Q = l Q o l + lRll Ro = wAo, peak angular velocity amplitude of plate, eq 9, radls
Christiansen. E. B., Leppard, W. R., Trans. SOC.Rheol., 18, 65 (1974). Leppard. W. R.. Christiansen,E. B.,A./.Ch.€. J., 21, 999 (1975). Leppard,W. R.. Ph.D.Thesis, University of Utah, 1975. Maude, A. D., Walters, K., Nature(London), 201, 913 (1964). MacDonald,I. F., Marsh, B. D., Ashare, E., Chem. Eng. Sci., 24, 1615 (1969). Nally, M. C., Brit. J. Appl. Phys., 16, 1023 (1965). Turian, R. M., lnd. Eng. Chem., Fundam.. 11, 361 (1972). Walters, K., “Rheometry”, Wiley, New York, N.Y., 1975. Walters, K . , Kemp, R. A,, Rheol. Acta, 7, l(1968).
R1
Literature Cited
Received for review October 21,1976 Accepted March 28,1977
Intrapellet Diffusivities from Integral Reactor Models and Experiments L. Louis Hegedus’ and James C. Cavendlsh GeneralMotors Research Laboratories, Warren, Michigan 48090
For pore mouth poisoning,the activity of the partially poisoned catalyst pellets depends on the diffusivity characteristics of the poisoned shell. Since this quantity is not easily amenable to direct experimental measurement, we set out to construct a mathematical model which, upon comparison with reactivity measurements, allows one to estimate the effective diffusivity of the reactant across the poisoned shell. The utility of the above technique has been demonstrated for propylene and propane oxidation over a partially poisoned automobile exhaust catalyst. The paper also discusses the Rltz-Galerkin method for solving reaction-diffusion equations in catalyst pellets with stepwise variable material and chemical properties along their radius.
Introduction Earlier work (Hegedus and Baron, 1975) on poisoned automotive catalysts indicated that poisonous P b and P containing species tend to deposit primarily in the micropores of the catalyst support and leave the macropores essentially unobstructed. Electron microprobe experiments showed that both phosphorus (Hegedus and Summers, 1975) and lead (Schlatter, 1975) cover the support’s inner surfaces in a monolayer-equivalent thickness which explains why the effective diffusivity of the poisoned catalyst remains essentially unchanged with respect to the fresh catalyst. While this “chemical” poisoning did a reasonable service to explain the performance of most moderately poisoned catalysts, it failed to explain the performance of certain heavily deactivated samples. Scanning electron microscopic and electron microprobe experiments on these heavily deactivated catalysts showed that in addition to the “chemically” poisoned layer they were coated with a 3-10 p thin, apparently partially impervious layer of poison deposits over their outer surface. It is not intended here to discuss the chemistry of these surface deposits. Instead, we will restrict our attention to a mathematical description of their effect on the conversion performance of an integral, isothermal reactor. The schematics of the cross section of a typical, partially deactivated catalyst pellet is depicted in Figure 1. It is the 356
Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977
result of electron microprobe and scanning electron microscopic observations which indicated that the boundaries of the various zones within the pellet are sufficiently sharp, such that they can be approximated by the idealization shown in Figure 1. The catalyst is impregnated by the noble metals through zones 2 and 3. The poisons penetrate the catalyst pellet in the form of a sharp zone (zone 3). The interior of the pellets (zone 1)is not active. Zone 4 represents the poisonous deposits over the catalyst’s outer surface. It is quite logical to hypothesize that the diffusivity of the reactants across zone 4 must be reduced because chemical poisoning alone cannot explain the activity losses associated with its presence. However, attempts to selectively determine the diffusivity of this thin outer crust by steady-state counterdiffusion experiments failed due to the extreme thinness of zone 4 with respect to the radius of the pellet. For this reason, we determined the diffusivity of zone 4 by diffusion influenced reactor experiments as described below. In this paper we construct a mathematical model in which D4,the diffusivity in zone 4, is an adjustable parameter. Comparison of the model’s predictions with experimental rate measurements, then, promises to be suitable for evaluating D4.This will establish a relationship between the catalyst’s activity and D4,and thus the model will allow us a quantitative characterization of the effect of the obstructive deposit on a reactor’s performance.
