4001
NOTES for pressures >1 Torr, the accuracy of measurement of the end point is a function of the photomultiplier sensitivity. The “extinction-point” of the afterglow may be defined as the minimum value of (O)(NO) which will produce a detectable photocurrent. For the detection system used this product may be estimated to be of the order of 5 X g mol2cm-6 for pressures of about 5 Torr, Therefore the minimum atomic oxygen concentration required t o produce a detectable photocurrent is given by (0) = 5 X 10-21/(NO) g mol eme3, and (0) calculated in this manner may be called the 0-atom detection limit. The detection limits for conditions simulated in Figure 1 are shown on the diagram. The detection limits for the cases illustrated by curves 3 and 4 in Figure 1 are 6 X loy6 and 5 X lo-’, respectively. Now the indicated atom mole fractions at the photomultiplier slit, 22 cm downstream from the NO2 addition point, are low5and 5 X lo-’, respectively, both close to the detection limit. I n these cases NO2 addition equivalent to 90% of atomic oxygen produced bring about extinction of the glow at the photomultiplier position, and the degree of undertitration due to removal of 0 atoms by reaction 3 is 10%. Hence, in general, if the detection limit is accurately known and with a knowledge of the decay reactions involved, the minimum flow of added NO2 which would extinguish the glow a t the photomultiplier can be calculated and the degree of undertitration, and hence the real titration value, assessed. I n the case of curves 1 and 2 in Figure 1, the undertitration errors would have been negligible since the 0 atom concentration at the detection point are well above the limit in each case and the flow rate of KO2 would have been increased until luminescence ceased. The rapid onset of exponential oxygen atom decay a short distance from the point of NO2 admixture with the gas stream is explained by the very fast nature of reaction 1, which is substantially complete a short distance from the jet. For such a rapid reaction the efficiency of mixing of the gaseous reactants is important and in experiments in this laboratory NO, was introduced into the oxygen stream through a “pepper-pot” nozzle in which the titrating gas was fed through a network of fine holes located on the rounded tip of the feed tube. The photomultiplier slit must be positioned at a point downstream from the region where this fast reaction is taking place, where the rate of 0 atom decay is controlled by reaction 3. It may be concluded that the NO2 titration is reasonably accurate for determining atomic oxygen flow rates for pressures of up to 10 Torr if precautions are taken to ensure good mixing at the point of NO2 admixture and t o position the photomultiplier close t o the NO2 injection jet, bearing in mind the volume required for the completion of reaction 1. I n addition, if low 0 atom flow rates are to be measured the accuracy will increase still further because the importance of reaction 3 is then much reduced.
Lifetime of a Soluble Sphere of Arbitrary Density’ by Daniel E. Rosner2 and Michael Epstein Department of Engineering and Applied Science, Yale University,
New Haven, Connecticut 06690 and AeroChem Research Laboratories Incorporated, Division of Sybron Corporation, Princeton, New Jersey 08640 (Received April 9,1970)
Despite the importance of transport-controlled dissolution of spheres (particles, or bubbles) in industrial and research applications, previous theoretical work has not yielded sufficiently extensive or general results for the total lifetime of a soluble sphere of arbitrary density (compared to the surrounding solvent d e n ~ i t y ) . ~ However, Rosner4has recently shown that, in the dense sphere limit, a simple closed-form result can be obtained for the total particle lifetime which simultaneously accounts forstheeffects of appreciable solubility and interface kinetic limitations. I n the present note we (i) present the first quantitative results for the diffusion-controlled lifetime of spheres of arbitrary density and solubility in a universal form suitable for correlating future “exact” computer solutions and relevant experimental data; (ii) define the accuracy of the “mixed control’’ lifetime correlation developed in ref 4 and suggest the form of its extension to arbitrary solvent/ sphere density ratio; and (iii) illustrate one way in which the versatile integral (profile) method5 may be applied to this important class of nonlinear moving boundary problems.
Physicochemical Model With the exception that we here confine our attention to the frequently encountered limiting case of diffusion-controlled di~solution,~” our continuum model and nomenclature are identical with that of ref 4, vix. (1) Supported by the Propulsion and Energy Conversion Division of the U. S. Air Force Office of Scientific Research under Contracts AF 49(638)-1654 and F 44620-70-C-0026. (2) Associate Professor, Chemical Engineering Group; to whom inquiries concerning this paper should be addressed. (3) Numerical (finite difference) calculations for specific cases have been reported (see, e.g., D. E. Readey and A. R. Cooper, Chem. Eng. Sci., 21, 917 (1966) and M. Cable and D. J. Evans, J . A p p l . Phys., 38, 2899 (1967)); however, we have found several such lifetime results t o be higher than an expected upper bound (71,ie = 1) discussed in footnote 14. Similar discrepancies have been independently noticed by J. L. Duda and J. S. Vrentas (A.1.Ch.E. J., 15, 351 (1969)) in a mathematically interesting paper devoted to the radius-time relation for dissolving bubbles. (4) D. E. Rosner, J. Phys. Chem., 73, 382 (1969). [Author’s note: The reader should correct an obvious printing error in the location of the subscript zero in the denominator of eq 13 of this reference.] (5) T . R . Goodman in “Advances in Heat Transfer,” Vol. I, T. F. Irvine and J. P. Hartnett, Ed., Academic Press, New York, N. Y . , 1964, pp 51-122. (6) Thus, departures from solute concentration equilibrium at the phase boundary are neglected compared t o concentration differences across the outer “long-range” diffusion ”boundary layer.” In ref 4 it is shown that this limit (c, = csat) is attained provided ( k R o / D )* (1 - csat) >> 1, where k is the first-order rate constant for dissolution. The Journal of Physical Chemistry, Vol. 74, No. B, 1970
4002
NIXES
together with the obvious conditions
Figure 1. D-lution of an isolated sphere controlled by long-range solute diRusion; configuration and notation.
we consider an isolated sphere of initial rad.ius Ro d i s solving into an isothermal, constant property, otherwise quiescent solvent (see Figure 1 and the Nomenclature). As before,’ the assumption of spherical symmetry implies that the Reynolds number (based on terminal settling or rising velocity) and Grashof number (based on solute concentration-induced density differences) must both be small.‘ Our results will therefore be immediately relevant to “captive” spheres (constrained to be motionless) but can also provide useful (upper) limit values for the dissolution lifetime of isolated spheres free to move under the influence of gravity and/or other forces. Clearly, this “stagnant” limit will be approached for a sphere of any size or density provided the solvent (continuous phase) is sufliciently viscous.’
e(R,t)= 1 e(m,l) = 0
(2)
e(r,O) = 0
(4)
(3)
The convective (second) term in eq 1, evaluated here for the case of an external solution of constant density, is seen to depend on the pure solute/solution density ratio, p s / p , and the instantaneous dissolution rate -R. The latter quantity is, however, related to the instantaneous value of @13/&r),-~through a solute balance at the solute/solution interface; vir.
PSR= DpB(be/br),,(,)
(5)
These equations, and the “initial” condition R(0) = Ro, are in principle sufficient to simultaneously determine e(r,l)and R(1), from which the lifetime would follow from the definition R(t,ite) = 0. Rather than numerically dealing with this nonlinear boundary value problem in its present form, we adopt the following method. 1. A reasonable functional form for e(r,t) is postulated, involving two undetermined, timedependent functions, viz., a “shape” function 0 and a boundary layer thickness 6. 2. These latter functions (0, 8) are evaluated by imposing the conditions that e(r,t)satisfy (i) an inbqraled conservation relation and (ii) a “curvature” condition a t r = R (derived from eq 1). 3. The lifetime is then calculated from an integrated dimensionless form of eq 5, viz.
Analysis To obtain quantitative expressions and numerical predictions for the time, kit., necessary to completely dissolve the sphere, applicable for arbitrary solvent/ sphere density ratio and solubility, we reduce the nonlinear part,ial differential equation-boundary-value problem to a computationally simpler one involving an ordinary differential equation for R(1) using a variant of the integral method: We thereby obtain an expres sion for the sphere lifetime in the form of a readily calculated quadrature. Experience with this method in other nonlinear transient transport processes with spherical symmet$ suggests that its accuracy should be sufficient for design purposes. Moreover, in addition to guiding future numerical calculations and gaining experience with such approximate methods, the procedures outlined below could be readily generalied to compute the simultaneous effects of finite interfacial kinetics. Consider first the normalized concentration variable e(r,t), [defined by (c - cm)/(c.,, - c,)] which must satisfy the local solute conservation relation.’ Tha . l o u d o/ Phvsicol Chnnblru, Vol. 74.No. e#, 1870
(7) For relevant eonelstion formula8 8ee. eo., R. B. Bird. W. E. Stewart. and E. N. Lightfoot, “Tran8port Phenomena,” Wiley. New York. N. Y..1960. If the terminal Reynolds number (based on sphere diameter) is in the Stokes range (50.3) and the corRsponding Peelet group R c . ( v / D ) k also small (say 50.3). then the relative change in instantaneous mtransfer coeffioient (for B solid or vismus sphere of fired radius) can be shown to be I ( d p ) 11. @R’)/(gD.). For B dense sphere. the initial value of this g r o u p ing should provide an uppet bound to the eRwt of gravitational settling on mi. sinoe riifs depends upon some timeaverale of the effective mtransfer cwffieient. (8) See. for example, G. Pwta. In!. J . H d Mara Tronnler. 5. 625
-
(1862). (8) See for example, L. E. Seriven. C h m . Ew. M.. 10, 1 (1859).* who first demonstrated a class of self-similar sdutions to eq 1 for the special eof sphere wmdh from zero initial radius. While similarity solutions do not e r k t for the dissolution of finite radius spheres. we have ueed Seriven’s exact solutions to show that the integral method exploited below, when combined with the assump tion of a thin concentration boundary layer, adegustely predicts the transwrt limited growth of bubbles in the parameter range: 10 < ( d p . 4 . IS1 < 10’. 4 x lo1 < P / P S < ID. E.Rosner and M. Epatein. in preparation]. However. we expect the integral method to be put to a more severe tcst in the e&% of the p-nt d b d i r l i o n predictions. the accuracy of which will remain to be systematically studied a8 more reliable finita diRerence reaulta become available.
-
4003
NOTES For 0 we choose the transcendental profilelo
1
DIFFUSION LIMITED SPHERE DISSOLUTION
B
I, ( p / p s I . 8 * O , 2
Dud0 and Vrenla8 lNumorico1,B
