Experimental study of mass transfer limited reaction Part II: Existence of cross-over phenomenon

Chemical Engineering Science 60 (2005) 4147 4156 www.elsevier.com/locate/ces Experimental study of mass transfer limited reaction Part II: Existence of cross-over phenomenon Kiran B. Deshpande, William B. Zimmerman Department of Chemical and Process Engineering, University of Sheffield, Newcastle Street, Sheffield S1 3JD, UK Received 9 November 2004; received in revised form 18 January 2005; accepted 20 January 2005 Available online 15 April 2005 Abstract In Part II, we extend the inverse methodology which is discussed in Part I to various experiments performed in a tubular reactor to infer mass transfer coefficients. Mass transfer coefficients, inferred from the concentrations of the reactants which are extracted from the absorbance of the reaction mixture using multicomponent spectrum analysis, are then used to solve the convection diffusion reaction equations for the concentrations of the reactants in the bulk phase and in the dispersed phase, to explore the possibility of cross-over for a mass transfer limited reaction. Experiments are performed incorporating the asymmetry in the transport rates of the premixed reactants, which is the potential reason for the existence of cross-over. The different mass transfer coefficients of the two premixed reactants indeed indicate a switch in the concentration of the reactants in the dispersed phase, which is termed as cross-over. The experimental results are further analysed by validating the theoretical criterion proposed by Mchedlov-Petrossyan et al. (2003, Chem. Eng. Sci. 58, 3005 3023 & 2691 2703) to obtain the parametric space for the existence of cross-over, in order to optimize the length of the tubular reactor. 2005 Elsevier Ltd. All rights reserved. Keywords: Reaction engineering; Mass transfer; Drop 1. Introduction Transport limited heterogeneous reactions, such as ionic reactions or simultaneous absorption of two gases and subsequent reaction in a liquid phase, are often used in the chemical industry. Most of the research on such reactions is restricted to modelling or theoretical analysis, due to instantaneous nature of the reaction and is focused on initially separated reactants. Mass transfer with instantaneous chemical reaction occurring in a single droplet has been studied theoretically by various researchers (Dutta et al., 1988; Noh et al., 1995a,b; Warnecke et al., 1999). They considered a system where a non-circulating liquid droplet contains reactant B and another reactant, A, diffuses from the surrounding continuous phase into the dispersed phase where A and B react instantaneously. The governing equations for A and B are comprised of a time dependent Corresponding author. Tel.: +44 114 222 7517; fax: +44 114 222 7501. E-mail address: w.zimmerman@shef.ac.uk (W.B. Zimmerman). concentration term and a diffusion term. The moving boundary problem was converted to a fixed boundary by using transformed variables. The reaction front was considered as a plane where transformed concentrations of both A and B were zero. The propagation of the reaction front was found to be governed by the diffusion rates of A and B and the relative amount of B present in the dispersed phase. A larger concentration of B inside the droplet and the higher diffusivity of B have a decelerating effect on the movement of the reaction front. Mehra et al. (1994,1997) have reported a similar result for the experiments performed using the carbon dioxide absorption reaction where the position of the reaction front wasmeasured using a travelling microscope. Most studies of instantaneous reaction are focused on initially separated reactants. In the present work, however, we are interested in studying transport limited characteristics of the premixed reactants, particularly with asymmetric transport rates. We are dealing with a system where the two reactants transport from the continuous phase to the dispersed phase and react in the dispersed phase. We have 0009-2509/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2005.01.034

4148 K.B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 60 (2005) 4147 4156 discussed various potential reactions in our earlier communication (Deshpande and Zimmerman, 2005). Simultaneous absorption of the two gases with chemical reaction in the liquid phase is one of the potential reactions which occurs very commonly in the chemical industry, e.g., simultaneous absorption of CO 2 and NH 3 into water. The above reaction with the premixed reactants has been theoretically studied by a number of researchers (Roper et al., 1962; Ramachandran and Sharma, 1971; Chaudhari and Doraiswamy, 1974; Juvekar, 1974; Hikita et al., 1977; Zarzycki et al., 1981). They found that chemical reaction increases the absorption rate of the two reactants by an amount that depends on the ratio of the solubilities, the reaction rate constant and the diffusion coefficients. In most of the above mentioned research work, the concentrations of the gaseous reactants had been arbitrarily fixed, which is not satisfactory. Bhattacharya and Chaudhari (1972) proposed an exact analytical solution which requires no arbitrary choice of bulk liquid phase concentrations of the gaseous reactants. It is indeed more logical to determine the bulk liquid phase concentrations of the gaseous reactants through stoichiometric coefficients, ratios of solubilities and diffusivities. There has been very little attention paid to experimental study of mass transfer limited reaction. There have been articles reporting an experimental approach to evaluate mass transfer flux (Petera and Weatherley, 2001; Mao et al., 2001; Li et al., 2003), but they considered transport of a reactant from the continuous phase to the dispersed phase without any chemical reaction. In this work, we are interested in mass transfer of the premixed reactants with instantaneous chemical reaction. Also, the formulation proposed in the above mentioned work, to evaluate the mass transfer coefficient of a single reactant, assumes the surface concentration of the reactant to be equal to zero. The above mentioned formulation cannot be extended to evaluate mass transfer coefficients of two reactants with asymmetric transport rates because of the possibility of cross-over, where there is a switch in the concentration of the reactants in the dispersed phase, from one limiting reactant to the other (Mchedlov-Petrossyan et al., 2003b; Zimmerman et al., 1999). Mchedlov-Petrossyan et al. (2003a) studied transport limited heterogeneous reaction for premixed reactants with unequal transport rates (termed kinetic asymmetry) and proposed the parametric space for the existence of crossover, even for unequal diffusion fluxes of the reactants. The existence of cross-over is indeed due to kinetic asymmetry in the transport rates of the reactants, as it also occurs for the case of a single droplet (Deshpande, 2004). We first briefly discuss the theory proposed to depict the cross-over phenomenon for mass transfer limited reactions. The experimental procedure and results are discussed in the following section. If the experimentally inferred mass transfer coefficients of the two species are different, the existence of cross-over can be theoretically computed. The theoretical criterion proposed to estimate the parametric space for the existence of cross-over is experimentally verified by performing experiments for different operating conditions and by evaluating mass transfer coefficients for those conditions. 2. Theory The axial transport of reactants A and B and product C can be captured by solving the convection diffusion reaction equations which are represented as U C A U C B U C C = D A = D B = D C 2 C A 2 κ A a(c A C A,s ), (1) 2 C B 2 κ B a(c B C B,s ), (2) 2 C C 2 κ C a(c C C C,s ), (3) κ A a(c A C A,s ) = κ B a(c B C B,s ), (4) κ A a(c A C A,s ) = κ C a(c C C C,s ), (5) C A,s C B,s = KC C,s. (6) D A, D B, D C are the coefficients of turbulent diffusion, a is the active catalyst surface area per unit volume of the reactor, and j A, j B and j C (terms proportional to κ A, κ B and κ C, respectively) are the fluxes representing disappearance of the species from the bulk phase to or from the surface of randomly distributed catalyst particles (dispersed phase) due to mass transfer. The concentration difference of the reactants or product between the bulk phase and the dispersed phase is the driving force for mass transfer and hence the fluxes are represented in terms of mass transfer coefficients κ A, κ B and κ C. The above system of equations has been numerically solved by Deshpande (2004) which showed an interesting switch in the surface concentrations of the two premixed reactants, as shown in Fig. 1. The switch in surface concentrations of the reactants, which was termed cross-over phenomenon by Mchedlov-Petrossyan et al. (2003a), is a good measure for optimizing the design of the reactor, as the molecular efficiency is greatest at the point of cross-over. The greatest molecular efficiency at the time of cross-over is achieved in the sense that all reactant molecules arriving in the dispersed phase react simultaneously giving maximum product formation per unit reactant mass the molecules are in stoichiometric balance at that point in the dispersed phase. The cross-over length is a good indicator of the scale for optimal reactor length as conversion level was found to be 99% at a distance five times longer than that required for crossover to occur for a tubular reactor (Mchedlov-Petrossyan et al., 2003a). This result was consistent across the variation of the operating parameters. A similar result is reported for a batch reactor (Mchedlov-Petrossyan et al., 2003b). The

K.B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 60 (2005) 4147 4156 4149 It can be clearly seen that the cross-over phenomenon is not possible for the first two cases, since either of the reactants is in excess over the entire domain of the reactor. The crossover phenomenon is possible for cases 3 and 4. We focus on case 3 in the present analysis, since cases 3 and 4 are exactly the same except for the fact that the concentrations of two reactants are interchanged. The theoretical criterion for the existence of the cross-over phenomenon for case 3 can be written as, 1 A 2 < σ < 1 ν 1, (7) Fig. 1. Change in surface concentration of reactants A and B along the length of a one-dimensional tubular reactor. cross-over phenomenon which is applied for a two-phase system in the present work can be well exploited for a solid catalysed reaction as there are no unreacted reactants present in the dispersed phase, and hence there are least separation related issues.the optimum length (where cross-over occurs) is considered in this work to be a trade-off between the greater conversion and the least operational difficulties such as separation. The concept of cross-over was first applied by Mchedlov-Petrossyan (1998) in a solid state mixture for the precipitation reaction and was further extended by Zimmerman et al. (1999) to study mass transfer limited reactions for initially separated reactants and by Mchedlov-Petrossyan et al. (2003b) for premixed reactants in a batch reactor. Deshpande (2004) also reported a modified approach to the above system of equations in order to bring out the existence of a boundary layer surrounding the dispersed droplets, which will be used later in the inverse methodology to infer mass transfer coefficients of the reactants. Mchedlov-Petrossyan et al. (2003a) reported an analytical solution for the above system of equations for an irreversible reaction. They considered the following four possible cases: (1) Reactant A is in excess of reactant B over the entire domain. (2) Reactant B is in excess of reactant A over the entire domain. (3) Both the reactants are depleted due to chemical reaction, but the surface concentration of reactant A is more depleted at the entrance of the reactor, whereas reactant B is more depleted at the other end. (4) Both reactants are depleted due to chemical reaction, but the surface concentration of reactant B is more depleted at the entrance of the reactor, whereas reactant A is more depleted at the other end. where, ( ν 1 A 2 = α 2 (α 2 + P), α 2 = DB U ak P = B, ν 1 = D A, D B D B P θ 2 1 2 ) 2 + ν 1 θ 2 1 P θ2 1 2, θ 1 = D A D B and σ = C A 0 C B0. The above theoretical criterion is very useful in demarcating the parameter space where the existence of cross-over is possible. We perform various experiments in order to explore the possibility of existence of cross-over for mass transfer limited reaction and to obtain the parameter space for the existence of the same using the above theoretical criterion. 3. Experimental We have developed a calibration technique to evaluate the concentration of the individual reactants from the absorbance of the reaction mixture, which is discussed in detail in Deshpande and Zimmerman (2005). We have also outlined a methodology to evaluate mass transfer coefficients from experimentally measured concentrations of the reactants in the same article. We apply those techniques in this section for the various experiments performed maintaining uniform operating conditions. We chose the reaction mixture where nicotinic acid and bromophenol blue were dissolved in 1-chlorobutane (solvent) as it satisfies all the criteria required to study mass transfer limited characteristics for the premixed reactants. We performed the experiments in the same experimental rig, which is discussed in detail in our earlier communication (Deshpande and Zimmerman, 2005). The experimental procedure followed to measure the concentration of the reactants is discussed in the next section. 3.1. Experimental procedure (1) The tubular reactor is initially filled with the reaction mixture consisting of nicotinic acid and bromophenol blue dissolved in the organic phase, 1-chlorobutane.

4150 K.B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 60 (2005) 4147 4156 (2) The absorbance spectrum of the initial reaction mixture is obtained using a fibre optic spectrometer. The concentration of each individual reactant is evaluated from the absorbance spectrum of the reaction mixture, which is discussed in detail in the multicomponent spectrum analysis section of our earlier communication (Deshpande and Zimmerman, 2005). (3) Droplets of the acidic aqueous phase (0.004 M nicotinic acid) are introduced into the reaction mixture through the distributor. A constant pressure head, maintained at the topof the distributor, ensures that the droplets generated are approximately uniform in size. (4) The reaction mixture and the acidic aqueous phase are fed into the reactor at a constant flow rate and the product and the unreacted reactants are removed at a particular flow rate in order to maintain a steady state operation. (5) The experiment is run for 15 s. All the inlet and outlet streams are closed immediately after the experimental run. The aqueous phase present in the reactor is allowed to settle down. (6) The absorbance spectrum of the unreacted reaction mixture is obtained at various locations along the length of the reactor to ensure that the concentration of the unreacted reactants is the same in the bulk phase. (7) The total volume of the reaction mixture and the aqueous phase collected over the experimental run is measured to evaluate the average velocity of the reaction mixture and the aqueous phase. (8) The volume of the aqueous phase present in the reactor is measured to evaluate the average phase fraction of the dispersed phase in the reactor, which is used to evaluate the average surface area of the dispersed phase per unit volume of the reactor. (9) The mass transfer coefficients can be evaluated for the two reactants from the initial concentrations and the concentrations of the unreacted reactants, using inverse methodology, as discussed in our earlier communication (Deshpande and Zimmerman, 2005). It should be noted that the mass transfer coefficients evaluated in the present work are overall mass transfer coefficients. The local mass transfer coefficients cannot be inferred because of the experimental difficulties in measuring the local concentration of the reactants in the presence of the moving droplets, which is further elaborated in the next section. (10) The mass transfer coefficients evaluated experimentally are then used in the theoretical model to explore the existence of the cross-over phenomenon. 3.2. Experimental results Firstly, we performed experiments to obtain the absorbance spectrum in the presence of moving droplets. The droplets were continuously introduced at the top of the reactor and were moving from the topto the bottom of the Absorbance 3.5 3 2.5 2 1.5 1 0.5 0 Initial Position1 Position2 Position3 Position4 Position5 Position6 Position7 Position8 Position9 Position10-0.5 240 260 280 300 320 340 360 380 400 Wavelength, nm Fig. 2. Absorbance spectrum for nicotinic acid and bromophenol blue dissolved in 1-chlorobutane in the presence of moving droplets, in an attempt to gather information about concentration of the reactants in the dispersed phase. reactor due to density difference. The absorbance spectra were obtained at various positions along the length of the reactor by moving a collar on which a light source and a detector of the spectrometer are mounted. Experimental results, as shown in Fig. 2, indicate that the presence of droplets of the acidic aqueous phase in the optical path does not provide any meaningful information, except indicating a vertical shift in the absorbance spectrum along with the base line. This could be attributed to the time spent by the droplets in the optical path, which is very small (fraction of a second), and possible diffraction of light in the presence of moving droplets. We cannot reduce the velocity or the phase fraction of the moving droplets present in the reactor, because we do not get a substantial change in the concentration of the reactants at lower phase fraction and lower velocity of the droplets. Since the reaction mixture is continuously fed at the top of reactor, the change in the concentration of the reactants across the length of the reactor is going to be uniform if the droplet velocity is constant. As we cannot measure the absorbance spectrum in the presence of moving droplets, we trigger the above system by running the experiment in a continuous mode for 15 s and then shutting down all the inlet and outlet streams. The aqueous phase present in the reactor is allowed to settle down and the settling period is found to be 15 s. We measure the average velocity of the moving droplets for the continuous operation and for the settling period by measuring the volume of the aqueous phase over a known period of time. The average velocity is found to be approximately uniform for the continuous operation and for the settling period. Thus, the above mechanism of triggering can be well justified. We performed experiments for different operating conditions, which are summarized in Table 1. All the experiments

K.B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 60 (2005) 4147 4156 4151 Table 1 Experiments performed for different operating conditions maintaining approximately uniform hydrodynamic conditions. The absorbance values measured at the different wavelengths for nicotinic acid bromophenol blue reaction mixture are also reported Run Aqueous Organic Total Aqueous Initial Initial Final Final phase phase volume phase in A 269 A 297.6 A 269 A 297.6 collected collected (cm 3 ) the reactor (cm 3 ) (cm 3 ) (cm 3 ) Run1 80 215 295 140 1.3798 0.3224 0.9557 0.2862 Run2 120 220 340 135 1.5018 0.3686 1.1026 0.3278 Run3 85 235 320 145 1.635 0.3462 1.0947 0.2744 Run4 70 245 315 160 1.5156 0.3337 0.9237 0.2625 Run5 80 230 310 145 1.7957 0.5953 1.2731 0.5045 Run6 90 240 330 150 1.3602 0.4183 0.9266 0.3538 were run for a period of 15 s and the average droplet diameter, scaled during the experimental run, is found to be approximately 2 mm. The total volume collected is reported by measuring the organic phase and the aqueous phase, during the experimental run. The aqueous phase present in the reactor is also reported, which is useful in evaluating the phase fraction of the dispersed phase and the average surface area of the dispersed phase per unit volume of the reactor, and also to ensure that the velocity of the droplets is constant in the continuous mode of operation and during the settling period. The absorbance of the reaction mixture at wavelengths equal to 269 and 297.6 nm, which correspond to the peak wavelength of the reactants nicotinic acid and bromophenol blue, respectively, is also reported. We apply the non-linear calibration technique, which is well illustrated in our earlier communication (Deshpande and Zimmerman, 2005), to evaluate the concentration of nicotinic acid and bromophenol blue from the measured absorbance spectrum of the reaction mixture, and are reported in Table 2. The initial concentration of nicotinic acid and bromophenol blue in the organic mixture and the final concentration of the reactants after droplets pass through the tubular reactor are obtained, and hence we can estimate the change in the concentration of the reactants along the length of the reactor. We discuss the inverse methodology used to evaluate mass transfer coefficients from the change in concentration of the reactants, in the next section. 3.3. Evaluation of mass transfer coefficients using inverse methodology The experimental results shown in Tables 1 and 2 are analysed using inverse methodology to evaluate mass transfer coefficients. The governing equations (Eqs. (8 11)) are applied to extract information about mass transfer coefficients from the experimentally measured initial and final concentrations of the reactants: U C A = D A 2 C A 2 κ A a(c A C A,s ), (8) U C B U C A,s U C B,s = D B 2 C B 2 κ B a(c B C B,s ), (9) = κ A a(c A C A,s ) kc A,s C B,s, (10) = κ B a(c B C B,s ) kc A,s C B,s, (11) The average superficial velocity, which is calculated from the experimentally measured volumetric flow rate, is used in the above model system to represent the convection of reactants. Diffusion coefficients of nicotinic acid and bromophenol blue are evaluated using the Wilke Chang equation (Reid, 1977). The average surface area of the dispersed phase per unit volume of the reactor is calculated using the phase fraction of the dispersed phase. A high reaction rate constant k = 10 6 M 1 s 1 is used to capture mass transfer limited characteristics. The dependence of the reaction rate constant on the mass transfer coefficients evaluated using the inverse methodology will be discussed later in this section. The initial concentrations and the final concentrations of the reactants extracted from the absorbance of the reaction mixture are used to solve the above set of equations and to infer the mass transfer coefficients of the reactants. The inverse methodology, as outlined by Deshpande and Zimmerman (2005), is used to infer mass transfer coefficients for the known set of concentrations with minimum error. The accuracy of the predicted mass transfer coefficients, for the different experimental runs, is checked by running numerical experiments for different initial guesses of the mass transfer coefficients, and the results obtained using the inverse methodology are represented in Table 3. The error calculated in predicting mass transfer coefficients is very small (of the order of 10 4 ) for all the initial guesses used in the analysis. Also, the mass transfer coefficients obtained for different initial guesses are exactly the same, indicating that there exists a unique solution for the present formulation involving convection diffusion reaction equations. The predicted mass transfer coefficients of nicotinic acid and bromophenol blue are found to be of the same order

4152 K.B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 60 (2005) 4147 4156 Table 2 Concentrations of nicotinic acid and bromophenol blue extracted from the absorbance of the reaction mixture using the principle of non-linear additivity Run Initial Initial Final Final Initial Initial Final Final A 269 A 297.6 A 269 A 297.6 C NA (M) C BP (M) C NA (M) C BP (M) Run1 1.3798 0.3224 0.9557 0.2862 0.001560 0.004072 0.0009964 0.003960 Run2 1.5018 0.3686 1.1026 0.3278 0.001613 0.004956 0.001121 0.00463 Run3 1.635 0.3462 1.0947 0.2744 0.001889 0.004201 0.001243 0.003468 Run4 1.5156 0.3337 0.9237 0.2625 0.001740 0.004106 0.001002 0.003504 Run5 1.7957 0.5953 1.2731 0.5045 0.001445 0.00965 0.0009811 0.008112 Run6 1.3602 0.4183 0.9266 0.3538 0.00128 0.006248 0.0008085 0.00541 Table 3 Mass transfer coefficients of nicotinic acid and bromophenol blue are evaluated using the inverse methodology for the various experiments performed maintaining approximately uniform hydrodynamic conditions. The error calculated for the various initial guesses is also reported Run Guessed Guessed Predicted Predicted Error κ BP κ NA κ BP κ NA (cm s 1 ) (cm s 1 ) (cm s 1 ) (cm s 1 ) Run1 0.001 0.001 0.008 0.166 8.0167e 5 0.01 0.05 0.008 0.166 7.7475e 5 Run2 0.001 0.001 0.023 0.134 1.7153e 4 0.01 0.05 0.023 0.134 6.7783e 5 Absorbance 1.4 1.2 1 0.8 0.6 0.4 0.2 20 sec 40 sec 60 sec 80 sec 100 sec 300 sec 600 sec Run3 0.001 0.001 0.057 0.125 8.4386e 5 0.01 0.05 0.057 0.125 2.6466e 4 Run4 0.001 0.001 0.041 0.159 2.7013e 4 0.01 0.05 0.041 0.159 7.1513e 5 Run5 0.001 0.001 0.053 0.11 1.4443e 4 0.01 0.05 0.053 0.11 6.5863e 5 Run6 0.001 0.001 0.047 0.145 1.6329e 4 0.01 0.05 0.047 0.145 9.7441e 5 of magnitude, except for Run1, where the change in the concentration of bromophenol blue is found to be not very significant. Since the hydrodynamical conditions for the various experimental runs are approximately uniformly maintained, the predicted mass transfer coefficients are of the same order of magnitude. This also confirms the accuracy of the inverse methodology in evaluating the mass transfer coefficients. 3.3.1. Dependence of reaction kinetics on the evaluation of mass transfer coefficients The reaction that we have chosen for the present experimental study, between nicotinic acid and bromophenol blue, is not well studied, and hence the reaction rate constant for the reaction is not available in literature. We attempted to evaluate the reaction rate constant experimentally by mea- 0 300 350 400 450 500 550 600 650 700 Wavelength, nm Fig. 3. Absorbance spectra of nicotinic acid and bromophenol blue dissolved in the aqueous phase for various time steps, to evaluate the reaction rate constant. suring the concentration of nicotinic acid and bromophenol blue in the aqueous phase. The absorbance spectrum of the reaction mixture in the aqueous phase is obtained for various time steps, as shown in Fig. 3, and it is observed that the absorbance spectrum of the reaction mixture does not change. This indicates that the reaction is very fast in nature. Also, the reaction between nicotinic acid and bromophenol blue in the aqueous phase is a protonation reaction, which requires just an exchange of proton and such reactions are instantaneous in nature (Astarita, 1967). Since we are dealing with a transport limited reaction in the present work, the reaction is controlled by transport of reactants and it should not depend on the reaction kinetics. To check the dependence of the reaction rate constant on the dynamics of the system, we perform numerical experiments for the experimental run, Run1, and evaluate mass transfer coefficients by varying the reaction rate constant k, ranging from 10 6 to 10 2 M 1 s 1, and the results are shown in Table 4. The mass transfer coefficients of bromophenol blue and nicotinic acid are found to be exactly the same for the lower values of the reaction rate constant. However, at the higher values of the reaction rate constant, the mass transfer

K.B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 60 (2005) 4147 4156 4153 Table 4 The dependence of the reaction rate constant in evaluating the mass transfer coefficients of nicotinic acid and bromophenol blue is checked for a wide range of values of the reaction rate constant Reaction rate κ BP κ NA constant, k (cm s 1 ) (cm s 1 ) (M 1 s 1 ) 10 6 0.008 0.166 10 5 0.008 0.168 10 4 0.008 0.177 10 3 0.008 0.199 10 2 0.008 0.207 10 0.008 0.208 1 0.008 0.208 10 1 0.008 0.208 10 2 0.008 0.208 coefficients of bromophenol blue are constant and those of nicotinic acid are found to be only slightly different. The mass transfer coefficients evaluated using the inverse methodology for various experiments performed under roughly uniform conditions, which are fairly uniform, confirm the robustness of the methodology, as the mass transfer coefficients depend only on the hydrodynamic conditions and not on the reaction kinetics. The mass transfer coefficients are also evaluated for all the experimental runs for different reaction rate constant, k = 1M 1 s 1, and the predicted mass transfer coefficients are found to be of the same order of magnitude as those obtained for k = 10 6 M 1 s 1 (results not shown here). This illustrates that the evaluated mass transfer coefficients are indeed independent of the reaction rate constant. Thus, the reaction rate constant, k = 10 6 M 1 s 1, considered in the present methodology to evaluate mass transfer coefficients for transport limited reactions can be well justified as representative of very fast reactions. Although mass transfer coefficients inferred using the inverse methodology are not dependent on the reaction rate constant, the cross-over phenomenon which potentially occurs in transport limited reactions (instantaneous reactions) is found to be dependent on the rate of disappearance of the reactants in the dispersed phase, and is discussed in a later section. 4. Investigation of the existence of cross-over phenomenon We investigate the possibility of the existence of the crossover phenomenon for transport limited heterogeneous reactions in this section. Since there is kinetic asymmetry in the transport rates of the two reactants, a possibility of the existence of cross-over is investigated by changing the ratio of the initial concentrations of the reactants. The model system of equations (Eqs. (8 11)) is solved using experimentally evaluated mass transfer coefficients for different operating conditions as shown in Tables 1 and 2. We obtain the concentration profiles of the reactants in the bulk phase and the dispersed phase for known mass transfer coefficients and known operating conditions. Since there is a switch in the surface concentration (in dispersed phase) from one limiting reagent to the other in the cross-over phenomenon, we are interested in analysing the concentration profiles of the reactants in the dispersed phase. The surface concentration profiles of nicotinic acid and bromophenol are plotted along the length of the reactor for various operating conditions, as shown in Fig. 4. The dotted line and the solid line in these figures represent the evolution of concentration of nicotinic acid and bromophenol blue, respectively, in the dispersed phase along the length of the reactor. Since the mass transfer coefficient of nicotinic acid is considerably greater than that of bromophenol blue, we can potentially expect the occurrence of the cross-over phenomenon if the initial concentration of bromophenol blue is considerably greater than that of nicotinic acid. The ratio of the mass transfer coefficients of nicotinic acid and bromophenol blue is greater than the ratio of the initial concentrations of the two reactants for Run1, which results in cross-over at a distance equal to 17 cm, as shown in Fig. 4(a). The operating conditions are changed in order to validate the theoretical criterion (Eq. (12)) proposed for existence of the cross-over phenomenon which is later discussed in detail. The cross-over phenomenon is achieved within the length of the reactor for Run2 and Run4, as shown in Figs. 4(b) and (d), respectively. As the ratio of the mass transfer coefficients of the reactants approaches the ratio of the initial concentrations of the reactants, cross-over occurs almost at the entrance of the reactor, as shown in Fig. 4(c) for Run3. As the mass transfer coefficient ratio becomes lower than the ratio of the initial concentrations of the reactants, it can be seen that bromophenol blue is already in excess in the dispersed phase, indicating the cross-over will never happen for the experimental runs, Run5 and Run6, as shown in Figs. 4(e) and (f), respectively. A quantitative study of different ratios of the initial concentrations and the mass transfer coefficients of the two reactants is presented in the parametric study in a later section. 4.1. Dependence of reaction rate constant on cross-over The surface concentration profile of the reactants, as shown in Fig. 5, represents the imbalance between the transport of the reactants to the dispersed phase and the consumption of the reactants in the dispersed phase due to chemical reaction. If the transport of the reactant is in equilibrium with its consumption, surface concentration of that reactant is equal to zero. If the transport rate of the reactant is greater than its rate of consumption, the reactant is in excess in the dispersed phase. The switch in the excess concentration of one reactant to the other results in cross-over. Hence, the cross-over phenomenon is dependent

4154 K.B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 60 (2005) 4147 4156 Fig. 4. The possible existence of the cross-over phenomenon is verified by using experimentally evaluated mass transfer coefficients in the theoretical approach which was discussed earlier: (a) Run1, (b) Run2, (c) Run3, (d) Run4, (e) Run5, (f) Run6. on the reaction rate constant, unlike the evaluation of mass transfer coefficients. Since the reaction rate constant for the reaction between nicotinic acid and bromophenol blue in the aqueous phase is not available in literature and also cannot be easily determined experimentally because of its fast nature, we have performed various numerical experiments to determine the reaction rate constant that can be used in the numerical simulations. We have varied the reaction rate constant from 10 3 to 10 6 M 1 s 1 and the surface concentration profiles obtained are shown in Fig. 5. It can be seen that, for the reaction rate constant k = 10 3 M 1 s 1, the transport of both the reactants is greater than the consumption of the reactants and hence the switch in the concentration profiles of the reactants resulting in the cross-over is not obtained (Fig. 5(a)). Therefore, in order to increase the consumption of the reactants, we need to increase the reaction rate constant. It can be seen that, by increasing the reaction rate constant, the switch in the concentration profiles of the two reactants is obtained, as shown in Fig. 5(b), and with a further increase in the reaction rate constant no change in cross-over length is observed, as shown in Figs. 5(c) and (d).in the present experimental work, the reaction is very fast. Hence, the reaction rate constant k = 10 6 M 1 s 1 is used and can be well justified as representative of the fast reaction. 5. Analysis of parametric space for the existence of cross-over We revisit the criterion proposed by Mchedlov-Petrossyan et al. (2003a) for the existence of the cross-over phenomenon in their theoretical approach to study transport limited heterogeneous reactions, which can be rewritten for the present reaction mixture as 1 < σ < 1, (12) A 2 ν 1

K.B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 60 (2005) 4147 4156 4155 Fig. 5. The dependence of the reaction kinetics on the possible existence of the cross-over phenomenon is verified by performing numerical experiments for different reaction rate constants: (a) k = 10 3 M 1 s 1, (b) k = 10 4 M 1 s 1, (c) k = 10 5 M 1 s 1, (d) k = 10 6 M 1 s 1. where, ( ν 1 A 2 = α 2 (α 2 + P), α 2 = DNA P θ 2 1 2 U aκ P = NA, ν 1 = κ BP, θ 1 = D BP D NA κ NA D NA σ = C BP 0 C NA0. ) 2 + ν 1 θ 2 1 P θ2 1 2, and We have already validated the above criterion numerically using the level set simulations for a single moving droplet, as reported in Deshpande (2004). We now investigate the above criterion for the experimentally measured mass transfer coefficients of the two reactants in this section. We first investigate the right-hand-side inequality of the criterion (Eq. (12)) by varying the ratios of the mass transfer coefficients and the initial concentrations of the two reactants. We have calculated σ and ν 1 for various operating conditions and the results are represented in Table 5. It can be seen that there exists a cross-over when σ < 1/ν 1, which is illustrated by experimental runs: Run1, Run2 and Run4, as shown in Figs. 4(a), (b) and (d), respectively. For experimental run, Run3, σ is almost equal to 1/ν 1, and hence cross-over occurs at the entrance of the reactor, as shown in Fig. 4(c). For the parametric space where σ > 1/ν 1, the cross-over phenomenon will never happen and this is illustrated by Run5 and Run6, as shown in Figs. 4(e) and (f), respectively. Table 5 The ratio of the mass transfer coefficients of nicotinic acid and bromophenol blue, evaluated using the inverse methodology, is compared with the ratio of the initial concentrations of the reactants, for various experiments performed maintaining approximately uniform hydrodynamic conditions, to study the parametric space for the existence of cross-over. The criterion for the existence of cross-over is validated for the experimental results Run σ 1 ν1 P α 2 1 A2 Run1 2.61 20.36 974.61 5.04e 5 1.0 Run2 3.07 5.77 1257.05 1.38e 4 0.999 Run3 2.22 2.19 1182.49 3.86e 4 1.0 Run4 2.36 3.79 980.62 2.69e 4 1.0 Run5 6.68 2.095 1216.59 3.92e 4 1.0 Run6 4.88 3.07 1110.17 2.9341e 4 1.0 The left-hand-side inequality of Eq. (12) is checked by calculating P, α 2 and 1/A 2 for the various operating conditions. It is found that A 2 is constant and is lower than σ for all the experimental runs. Thus, the left-hand-side inequality is satisfied at all times and the possibility of existence of cross-over very much depends upon the ratios of the mass transfer coefficients and the initial concentrations of the two reactants. 6. Conclusion A technique developed using inverse methodology, as discussed by Deshpande and Zimmerman (2005), to infer the mass transfer coefficients of the premixed reactants from the

4156 K.B. Deshpande, W.B. Zimmerman / Chemical Engineering Science 60 (2005) 4147 4156 experimentally measured concentrations of the reactants, is extended in this article to various experiments performed to study mass transfer limited characteristics of a heterogeneous reaction. This unique technique is potentially important to infer asymmetric mass transfer coefficients. The mass transfer coefficients evaluated using the inverse methodology are of the same order of magnitude for the experiments performed under roughly uniform hydrodynamic conditions, which illustrates the robustness of the inverse methodology. The mass transfer coefficients are also evaluated using different initial guesses, which result in the same solution indicating that there exists a unique solution for the system of convection diffusion reaction equations. A system with asymmetric mass transfer coefficients potentially indicates the existence of a cross-over phenomenon (Mchedlov-Petrossyan et al., 2003a). The various experiments performed confirm the above hypothesis that crossover indeed exists for a system with kinetic asymmetry in the transport rates and the initial concentrations of the reactants. The theoretical criterion, proposed by Mchedlov-Petrossyan et al. (2003a) to demarcate the parameter space for the existence of the cross-over phenomenon, is validated by obtaining the concentration profiles of the two reactants in the dispersed phase using the experimentally evaluated mass transfer coefficients. Since the mass transfer coefficients of the two reactants are found to be different, experiments were performed for various initial concentration ratios of the two reactants to capture the parametric space for the existence of cross-over. The parametric study performed for different operating conditions experimentally satisfies the proposed theoretical criteria for the existence of cross-over and is potentially useful in optimizing the design of the reactor. 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