Ind. Eng. Chem. Res. 1994,33, 3078-3085
3078
Diffusion-Controlled Instantaneous Chemical Reaction in Single Drops Anurag Mehra’ and B. V. Venugopal Department of Chemical Engineering, Indian Institute
of
Technology, Powai, Bombay 400 076, India
The problem of mass transfer into a rigid, spherical, liquid drop, accompanied by a n instantaneous reaction between the absorbing solute and a diffusing nonvolatile reactant, within the drop, has been investigated. Experiments on a gas-liquid system were carried out with drops suspended from a buret where a n acid-base reaction using phenolphthalein a s an indicator was utilized to track the moving reaction boundary. A model, for the motion of this reaction front, using a quasi steady state approximation based on the notion of a n “undisturbed” core, and inspired by the shrinking core model, has been proposed. The experimentally determined position of the reaction front shows a fair degree of agreement with the theory, and some deviation is observed. The observed time for conversion of the complete drop agrees reasonably well with the theoretical predictions. A comparison of the proposed model with the unsteady state, numerical solutions shows that the proposed model is a reasonable analytical approximation for certain combinations of the concentration and diffusivity ratios of the two reacting species.
Introduction
h
Diffusion accompanied by a chemical reaction within drops is of considerable importance in multiphase reaction systems where the locale of the reaction is the dispersed phase. Gas absorption into dispersed liquid droplets, such as in spray columns and aerosol reactors, liquid-liquid systems where the reaction proceeds within the dispersed liquid phase, and even slurry reactors where one of the reagents lies in the suspended solid are typical examples of such contact. The rates of reaction in these systems are determined by the diffusion of the reactant, originally resident in the continuous phase, into the dispersed constituents where reaction and diffusion proceed simultaneously. It is therefore important to quantitatively describe the diffisionreaction behavior in a single dispersed drop in order to evaluate the performance of the total system. Figure 1shows schematic sketches for the concentration profiles of the two reacting species, A and B, within a spherical, rigid (noncirculating) drop, for various relative rates of reaction to diffusion. Species A is originally resident in the continuous phase and diffuses into the drop containing reagent B which can also diffuse (exclusively confined t o within the drop); the reactants are simultaneously consumed by an irreversible reaction (A zB products). Usually, A is likely to have a small solubility in the drop phase (which is the justification for having a multiphase contact in the first place). For the case shown in Figure l a , the reaction is very slow compared to the possible diffusional rates so that the drop is simply consumed at the kinetic rate, i.e., R l = (R&) = V ~ Z C A ~(mol/s). C B ~ The situation shown in Figure l b represents the pseudo first order scenario where there is no spatial depletion of B due to reaction in the drop; here the concentration of B is uniform everywhere within the drop, a t any given instant, and the diffusion of B is not of any consequence. The form of the kinetic term, to appear in the diffusion equation where the value for A, simply becomes k z C & ~ = klC~, of k1 declines with time as B gets consumed. The mathematical analysis for this case is identical to the
+
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* Author to whom correspondence should be addressed.
r + R r = O r:R
C
d
9
CAO
r:
front
Figure 1. Concentration profiles for reactants A and B within a liquid drop for various relative rates of diffision and reaction: (a) slow reaction (kinetically controlled); (b) fast reaction (pseudo first order); (c) very fast reaction (depletion); (d) instantaneous reaction. q = CB~(ZCAJ; Thiele modulus #t = dotted line in (d) shows the profile of B for quasi steady state model.
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one used for calculating effectiveness factors in catalyst pellets (Fogler, 1992). Figure ICshows the case for very fast reactions where the concentrations of A and B vary along the radial coordinate and in the limit for an instantaneous reaction fall to zero at a reaction shell which is depicted in Figure Id. The rigorous, unsteady state models for the last two cases require numerical solutions, usually complicated by stiffness of the species balances and, in the limit for instantaneous reactions, by the presence of a moving reaction front. The above cases have been constructed by analogy with the regimes outlined in the theory of mass transfer with chemical reaction (Doraiswamy and Sharma, 1984). Tyroler et al. (1971) have studied the problem of
Q888-5885f94l2633-3Q78~Q4.50f0 0 1994 American Chemical Society
Ind. Eng. Chem. Res., Vol. 33,No. 12, 1994 3079 diffusion accompanied by a fast chemical reaction in stagnant and circulating drops in a liquid-liquid system, using acid-base reactions (acetic acid in cyclohexanol, continuous; sodium hydroxide in water, dispersed). Phenolphthalein was used as an indicator, and the reaction front was tracked photographically. The position of the reaction front was computed theoretically from a numerical solution of the unsteady state, coupled diffusion-reaction equations for A and B and found t o be in good agreement with the experimental measurements. These experiments are, however, only for a single (small) concentration ratio and of greater relevance t o liquid-liquid systems (in gas-liquid systems drop circulation may be unimportant). This study falls in the regime depicted in Figure IC. The limiting problem, assuming a truly instantaneous reaction occurring at a well-defined moving front within a spherical, rigid drop, has been theoretically investigated by Dutta et al. (1988), who used transformed variables to convert the moving boundary problem to a fixed boundary one and subsequently solved it numerically. However, they have not compared the solutions to any experimental data. A similar methodology was used by Basu et al. (1986) to theoretically analyze gas absorption accompanied by an instantaneous reaction in a laminar falling film. Various experimental studies have been reported in the literature, within the context of gas-aerosol reactions, on the interaction of a gaseous solute with single aerosol droplets (in the micron size range) bearing a reactive species. The most notable work in this area has emanated from Rubel and Gentry (1984,1986,1987) and from Davis and co-workers. These single-drop studies have used the principle of electrodynamic levitation in order to suspend single drops in gaseous environments. This pioneering use of the electrodynamic balance has been attributed to Davis (1992); this excellent review also provides details on the various aspects of microchemical engineering. Even though the predominant focus in most of these investigations has been on obtaining (solid) precipitated products within the aerosol drop, a variety of features have been investigated, such as the influence of the external gas phase mass transfer, simultaneous reaction and intradrop diffision of the reactants, the product concentration fields, and the solid product layer thickness. For instance, Li et al. (1994) have dealt with the formation and properties of TiO2, coated microspheres produced by water-alkoxide reactions. Here, lowering of the water vapor partial pressure in the gas phase allows larger depths of the solid coating to be produced on account of the greater extent of diffusion before the appearance of a solid retards diffusionheaction. Similarly, Foss and Davis (1994) have provided the theoretical basis for the analysis of such systems and have reinterpreted the data of Rubel and Gentry (1984), whose experimental work involved the production of ammonium phosphate from ammonia absorption into drops of phosphoric acid. The theory reported by Foss and Davis (1994) uses the unsteady state diffusionreaction species balance for the reactants as well as the dissolved product (in spherical coordinates) where the reaction has been treated as second order (numerical solution) or pseudo first order in the dissolved gas species (analytical solutions). Thus, in order to examine the diffusion-reaction effects, in the absence of all other factors, it was thought desirable to carry out gas absorption experiments into
Table 1. Physicochemical Properties of the Carbon Dioxide-Sodium Hydroxide System Used in This S t u d y property value source DA (water) 1.97 x m2/s Danckwerts (1970) DB(water) 3.35 x m2/s Danckwerts (1970) CA,,(water) 2.85 x km0Vm3 Perry and Chilton (1973) Danckwerts (1970) kz 1 x lo4 m3/(kmol.s) Temperature = 28 "C, pressure = 1 atm. ,*blocked
region''
oction
IDEAL DROP
ACTUAL DROP
e
DIMENSIONS
CORRECTION FACTOR
Figure 2. Schematic details about suspended drops including "blockage" at point of attachment and correction to be applied to measured front positions due to refraction; n = 0.746.
single drops, for the limiting case of instantaneous reactions since these are convenient to track. For this situation, the developmentof a quasi steady state model has also been attempted in this work. Comparisons between the numerical solutions obtained from Dutta et al.'s unsteady state model and the proposed quasi steady state model (hereafter referred to as the qss model) have also been carried out in order to determine the region of validity of the proposed model.
Experimental Section The system adopted for this study was the absorption of carbon dioxide into single, suspended drops of an aqueous solution of sodium hydroxide. Table 1 shows the physicochemical property data for this system. Drops of aqueous sodium hydroxide, of a chosen strength, were suspended from the tip of a buret. The radius of the drop was measured by means of a traveling microscope. All stable drops that were created were nearly of the same size, and not much size variation could be obtained. A similar experiment was tried with a syringe, but these drops did not remain suspended. The use of a buret allowed the drop t o remain in the suspended state but at the expense of a larger area of attachment of the drop to the buret tip. This area of attachment was about 10% of the total surface area of the drop, and the volume "lost" was of the order of 3% (see Figure 2 for schematic details of ideal and actual drops). A Perspex box with provision for carbon dioxide inlet and outlet was used as a chamber around the suspended drop in order to avoid the effect of ambient draft currents and to maintain a pure carbon dioxide atmosphere in the vicinity of the drop. This box was provided with a hole on the top face for the insertion of the buret which could be fitted snugly. A very low flow rate of carbon dioxide was maintained, during a run, into and
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