Theorem of Corresponding Hydrodynamic States for Estimation of

Sep 21, 2006 - A theorem of corresponding hydrodynamic states analogous to the theorem of corresponding states in thermodynamics is proposed. Accordin...
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Ind. Eng. Chem. Res. 2007, 46, 3095-3100

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Theorem of Corresponding Hydrodynamic States for Estimation of Transport Properties: Case Study of Mass Transfer Coefficient in Stirred Tank Fitted with Helical Coil Bhaurao P. Nikhade and Vishwas G. Pangarkar* Mumbai UniVersity Institute of Chemical Technology (formerly UDCT), Chemical Engineering DiVision, Matunga, Mumbai-400019, India

Application of concepts similar to those used in thermodynamics provides an alternate, simple, and scaleindependent approach to predict hydrodynamics-dependent transport properties in multiphase systems. A theorem of corresponding hydrodynamic states analogous to the theorem of corresponding states in thermodynamics is proposed. According to the former, systems in the same corresponding hydrodynamic state exhibit identical mass transfer coefficients. A posteriori proofs from experimental measurements bear testimony to the validity of this physical postulate. It is shown that this approach unifies data from widely different and distributed sources in terms of time as well as methodology and results in simple and adequately accurate correlations for mass transfer coefficients in stirred reactors, the most complex yet widely used process equipment. For this a posteriori proof based on literature data as well as new data on solid-liquid mass transfer coefficient (KSL) in a widely differing system, stirred tanks fitted with helical coils are used. It is concluded that KSL in a two-/three-phase stirred tank reactor (STR) can be correlated reliably over a wide range of system configurations, scales, and operating conditions by a two-parameter equation involving a reduced hydrodynamic state similar to the reduced thermodynamic state in thermodynamics. 1. Introduction An appropriate design model of solid-liquid and gasliquid-solid reactors requires the information of various transport, kinetic, and mixing parameters. The overall rate of the process depends either on the chemical reaction kinetics or on the physical gas-liquid and solid-liquid mass transfer processes. Therefore, to design reactors for such processes, it is necessary to have knowledge of the mass transfer parameters. Mechanically agitated three-phase reactors are widely used in the industry because of their reliability and flexibility in operation. Notable examples include traditional catalytic reactions, such as hydrogenation in many fine and bulk chemical production processes, hardening of unsaturated oils as well as aerobic fermentations, and waste treatment.1 Mechanically agitated reactors/contactors are preferred over sparged contactors in specific situations where the superficial gas velocities are relatively low (e10 mm/s), the range in which sparged contactors are inefficient. In these reactors, it is important that the solid particles, which may act as a catalyst or may undergo a reaction, are completely suspended to ensure that the maximum surface area is available. A perfect homogenization at further expenditure of energy is not desirable. Therefore, the knowledge of the speed at which all particles become suspended is a prerequisite in the design of mixing equipment. It has been well-established that the presence of internal parts such as detection probes, thermocouple wells, internal cooling coils, etc. and the location of these internal parts inside stirred tanks greatly affect the hydrodynamics of stirred tanks. It has been reported that even the presence of a gas sparger increases the critical impeller speed for solid suspension.2,3 In our earlier communication, it has been shown that the presence of cooling coils (which may be used in certain * To whom correspondence may be addressed. [email protected]. Tel./Fax: +91 253 2316093.

E-mail:

situations to cater to extra heat load) can seriously hamper solid suspension.4 The latter is an important requirement in all threephase stirred reactors with the solid acting as the catalyst. In the present communication, the hydrodynamics of a stirred tank reactor (STR) is revisited. A new concept of “corresponding hydrodynamic states” is introduced. This physical postulate is substantiated a posteriori using literature data on particle-liquid mass transfer coefficient, KSL, in STR and new data on KSL in STR fitted with helical coils generated using the same equipment as that used in our earlier communication.4 2. Background Transport phenomena are an integral part of engineering operations. These phenomena are complex in nature and many times defy rational analysis, particularly in the case of multiphase systems of importance in engineering applications. A new discipline, complexity science, is emerging to treat such complex phenomena. It has been proposed that an integrative approach can supplement the predominantly reductionist approach previously practiced.5 Such new approaches can help in wrestling with ongoing questions. For most practical engineering problems, a simple, reliable approach is desired but may prove elusive. The hydrodynamics in multiphase systems is often complicated by a mean flow combined with a high level of turbulence, which enhances the rate of interphase transport of one or more species. Interphase transport plays an important role in these processes, and therefore, knowledge of transport coefficients is essential for economic and efficient design of the multiphase contactors. Turbulent transport phenomena are being increasingly analyzed using nonequilibrium thermodynamics. These are essentially nonequilibrium phenomena occurring under a driving force. Nonequilibrium thermodynamics has been shown to be useful in obtaining closure of turbulence models.6,7 Thus, transport processes have a close relationship with nonequilibrium

10.1021/ie060385f CCC: $37.00 © 2007 American Chemical Society Published on Web 09/21/2006

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thermodynamics. The concept and scope of nonequilibrium thermodynamics have been discussed in detail elsewhere.7 Nonequilibrium thermodynamics provides a general framework for the macroscopic description of irreversible processes. All turbulent transport processes are macroscopically irreversible and, hence, can be analyzed using nonequilibrium thermodynamics. Sadiki and Hutter6 have shown that nonequilibrium thermodynamics can provide a better description of turbulence. In the theory of nonequilibrium thermodynamics, the state parameters used to model an irreversible process are treated as continuous variables of space and time. The equations contain quantities referring to a single point in space at one time. This approach is not useful for practical purposes such as design of reactors involving interphase transport. In such situations, global values of the transport coefficients are measured on a small/ medium scale and correlated with measurable operating and system variables to obtain simple, adequately accurate correlations for scale-up. This approach has been very widely used in the literature,8-16 yielding empirical/semiempirical equations for predicting mass transfer coefficients. However, it has been found unsuccessful since the correlations depend highly on the system geometry, scale, and operating conditions under which they have been developed and are not useful for scale-up. In contrast to these empirical/semiempirical and the nonequilibrium approaches, in the present communication, it is proposed that an approach, similar to that used in equilibrium thermodynamics, can be used to predict transport properties in multiphase systems. In equilibrium thermodynamics, the state variables are usually independent of the space coordinates and, thus, a global value can be defined. This simplified integrative approach based on a well-defined critical hydrodynamic state (discussed later) should be of interest to a broad range of researchers from physicists to chemical engineers dealing with transport in complex multiphase systems. A thorough understanding of the mass transfer process in multiphase systems requires knowledge of the particle (bubble/ drop/solid particle)-fluid relative velocity, uR, which determines the drag on the particulate phase in multiphase systems.12 uR is a function of space and time. A rigorous evaluation of uR through the solution of a modified form of Tchen’s equation of motion17 for a particle in a turbulent fluid requires extensive information on the turbulence structure for situations of interest to engineers, which is rarely available for such systems.18 Although the primary energy input in most multiphase reactors (MPRs) results in non-isotropic turbulence, the dissipation of the input energy by viscous dissipation through the smallest eddies yields isotropic turbulence. An STR has a wide distribution of eddy sizes having different energies, and hence, it is very difficult to model it. The most important parameter governing transport processes is the turbulence intensity defined as

u′ )

xV′x2 Vx

(1)

u′ can vary significantly in time and space in a given MPR. Its variation in time and space causes variations in the rates of heat and mass transfer. For a design engineer, the overall rate of transport and, hence, the overall intensity of turbulence is more important rather than these variations. It will be shown later that, indeed, an overall u′ can be obtained, though indirectly, through simple considerations.

3. Theorem of Corresponding States in Thermodynamics and Theorem of Corresponding Hydrodynamic States The theorem of corresponding states in thermodynamics is stated as follows: “All fluids, when compared at the same reduced temperature (Tr) and reduced pressure (Pr), have approximately the same compressibility factor and all deviate from ideal gas behavior to about the same extent”.19 This theorem is a relatively simple way of describing P-V-T behavior of fluids. It suggests that a simple cubic equation of state such as the van der Waals equation can be converted to a universal form, applicable to all fluids using the definition of the reduced state. The reference fluid properties used are its critical properties, critical temperature, Tc, and critical pressure, Pc. The latter defines the fluid in the sense of the theorem of corresponding states. The transport coefficient defined as the rate of transport per unit driving force is an intensive property that depends solely on the hydrodynamic state of the system defined by the mean flow and turbulence levels. In this sense, the transport coefficient is similar to the fluid density, which is a function of the thermodynamic state. The fluid density is quantified by the reduced conditions (Tr, Pr), which refer the actual T, P to the fluid Tc, Pc, respectively, as per the theorem of corresponding states. A reduced hydrodynamic state is similarly defined when the prevailing hydrodynamic state is referred to a critical hydrodynamic state carefully selected for the particular system/phenomenon. In the following, it is shown that this approach of a reduced hydrodynamic state can be used for obtaining universal, scale-independent correlations for mass transfer coefficients in a three-phase stirred reactor, by far the most complex, yet most widely used, process equipment in the chemical/biotechnology and allied industries. The transport coefficient is related to the drag experienced by the particulate (solid/liquid/bubble) phase. The overall hydrodynamic state prevailing characterizes the drag. There exists a base hydrodynamic state (such as the critical Rayleigh number) for the onset of convection,20 which, as discussed later, is similar to the critical state in thermodynamics. When the actual hydrodynamic state is referred to this base level, a reduced hydrodynamic state similar to the reduced thermodynamic state is obtained. Thus, analogously, a theorem of corresponding hydrodynamic states can be proposed as follows: “The mass transfer coefficient in a stirred reactor is the same for different geometrical and operating conditions as long as these conditions result in the same corresponding hydrodynamic state.” The above theorem is justified a posteriori like all physical postulates using previous literature data and new data generated in this work in which the hydrodynamics and the flow patterns in the STR used are greatly altered by the presence of helical coils. The raison d’etre of cooling coils in STRs has been explained in a previous paper.4 The above statement of the theorem refers to STR inasmuch as the presently available evidence in its support pertains to STR. It is however, quite possible that it can be extended to other equipments/phenomena involving similarity of hydrodynamic states when similar substantiating data are available. 4. Experimental Section The experimental setup used has been described elsewhere in detail.4 Three types of impellers, namely, six-bladed disk turbine (6DT), six-bladed pitched turbine downflow (6PTD), and six-bladed pitched turbine upflow (6PTU), with both pitched

Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007 3097 Table 1. Properties of Benzoic Acid Particles average screen diameter, dp′ × 106 m surface area, m2/kg shape factor, φ sphericity, ψ mean particle diameter, dp × 106 m

150 22.62 0.5 0.67 112

300 19.86 0.5 0.67 224

500 16.18 0.5 0.67 372

turbines having an angle of pitch of 45°, were used. The details of the impellers and helical coils used are given by Nikhade et al.4 The solids used were benzoic acid particles of mean diameter 112, 224, and 372 µm. The size range selected was relevant to industrial catalytic applications. Table 1 shows the properties of the benzoic acid particles. Two solid loadings (X), 0.1 and 0.4 wt %, were used. This relatively small solid loading was used to avoid saturation of the solution. Other details of measurements of impeller speed and power drawn by the impeller are described elsewhere.4 The reproducibility of the NSG measurements by visual observation was typically within (3% of the impeller speed. The experimental procedure used to determine KSL was the same as that used by Jadhav and Pangarkar.21 4.1. Determination of NSG. The critical impeller speed for just suspension of solid particles (NSG) was measured using the 1 s criterion proposed by Zwietering,22 in his pioneering work in this field. “Just suspension” is the most commonly encountered level of liquid agitation and is of practical significance since it often represents an economic optimum. Higher levels of agitation do not lead to significantly enhanced rate processes, while lower levels of agitation lead to the formation of permanent deposits of solids on the tank base, which in turn can lead to lower process rates and/or the formation of undesired byproducts.23 The experimental procedure used was the same as that reported by Nikhade et al.4 The only difference was that, instead of spherical glass beads used in the previous communication, benzoic acid particles of three different sizes were used in this work and the aqueous solution used was a saturated solution of benzoic acid. The latter replaced water in order to prevent a decrease in dp during the NSG measurements. 4.2. Determination of KSL. The dissolution of benzoic acid in water has been used extensively to determine KSL as benzoic acid is sparingly soluble in water.8,21,24,25 The rate of dissolution is given by

V

dC ) KSLAS(C* - C) dt

(2)

Integrating the above equation between initial (Ci) and final concentrations (Cf) yields

KSL )

(

C* - Ci V ln ASt C* - Cf

)

(3)

For the experiments carried out in the present work, the initial concentration of the solute was zero. Therefore, the above equation becomes

KSL )

(

C* V ln ASt C* - Cf

)

(4)

As dissolution continues, there is a decrease in particle size. This reduction in particle size, in other words, particle surface area, is accounted for by the following equation26

()

AS ) SfWf ) SiWi

Wf Wi

2/3

(5)

Figure 1. Effect of solid size on KSL.

The mass of undissolved particle can be calculated from the mass balance:

Wf ) Wi - VCf

(6)

The particle surface area was determined by the procedure described by Carman.21 Table 1 gives the particle size and specific surface area of the particles as measured by the method referred to above. The particle-liquid mass transfer rates were measured by conducting experiments at, below, and above NSG. 4.3. Analysis. The analytical method used has been described in detail by Jadhav and Pangarkar.21 5. Results and Discussion 5.1. Effect of Solid Size. Figure 1 shows the variation of KSL with solid size for the three impellers at 2% solid loading in the presence of coil 3 at NSG (i.e., at N/NSG ) 1). The effect of dp on KSL (under otherwise similar conditions) is very weak (KSL ∝ dp-0.027) and is within the limits of experimental errors and differences in properties which include particle shape and exact size, etc.). This observation indicates that, at N/NSG ) 1, a unique hydrodynamic state is achieved which eliminates the effects of other important variables. Previous workers21 had obtained a higher negative effect (KSL ∝ dp-0.5) at a fixed value of N (not necessarily at N/NSG ) 1 for the range of dp (5501100 µm) covered). This observation was mostly due to the fact that this dependence was not obtained for N/NSG ) 1. At constant N, the smaller particles experienced a much higher level of turbulence than the larger particles and, hence, yielded higher values of KSL. This effect of dp on KSL obtained previously is, however, eliminated in Figure 1 when the cause of the effect (the different degrees of suspension for different dp) itself is eliminated by plotting KSL vs dp at N/NSG ) 1. 5.2. Effect of Solid Loading (X). For all the impellers and coils, KSL was determined at two solid loadings, 0.1 and 0.4 wt %. This relatively low solid loading is typical of gas-liquidsolid three-phase reactions and also was dictated by the possibility of saturation of the solution at higher loadings. It was observed that the solid loading has a negligible effect on KSL for all the coils used when the comparison is made at NSG for all solid loadings. The plot of KSL vs X obtained was exactly identical to Figure 1 and is, therefore, not included. This observation renders further support to the argument that NSG represents a unique hydrodynamic state. 5.3. Effect of Impeller Type. The effect of impeller type on KSL is shown in Figure 2. The symbols shown by arrow indicate the value of KSL at N/NSG ) 1. These points again for different impellers and different coils are identical. This observation buttresses the importance of the hydrodynamic state at N/NSG

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suspension of the solid particles occurs. It was found that an approach based on the analogy between momentum and mass transfer could satisfactorily correlate data covering a wide variety of geometric and operating variables as well as system properties encompassing Newtonian liquids to non-Newtonian liquids. The corresponding measurement techniques range from physical dissolution, dissolution with reaction, and ion exchange, to solid-liquid mass transfer controlled gas-liquid-solid reaction.25 The conclusion drawn by Pangarkar et al.25 can be reinterpreted through the theorem of corresponding hydrodynamic states enunciated in Section 3 as follows: Jadhav and Pangarkar,21 based on their data for DT and PTU for low- and high-viscosity solutions in a two-phase STR, proposed the following correlation for KSL,

KSL ) 1.72 × 10-3

Figure 2. Effect of type of impeller on KSL. Table 2. Comparison of Different Impellers impeller

N/NSG

P0/V (W m-3)

KSL (m s-1)

PTD

0.52 1.05 1.5 0.49 1 1.52 0.5 1 1.51

134.85 483.57 962.46 356.15 998.46 2054.17 395.48 1345.47 2267.42

3.87 × 10-5 6.05 × 10-5 7.25 × 10-5 2.83 × 10-5 5.84 × 10-5 6.48 × 10-5 2.54 × 10-5 5.57 × 10-5 6.21 × 10-5

DT PTU

) 1. On the other hand, the behavior of KSL with variation in P0/V does not show any clear/unique dependence of KSL on P0/V, as would have been anticipated from Kolmogoroff’s theory. This observation clearly reflects the failure of the P0/V approach in correlating transport properties in multiphase systems. Among the three impellers, PTD gave the highest KSL at a given power input. This can be attributed to the fact that PTD can suspend the particles to a higher degree at low P0/V as compared to the other impellers. Thus, it can be argued that KSL is solely decided by the degree of suspension at a given Sc. Table 2 shows the KSL and P0/V values at various N/NSG for all the three impellers. It can be seen from Table 2 that, above the critical impeller speed, very high-energy input is necessary to obtain a relatively small increase in mass transfer coefficient. Therefore, it is advisable to operate STRs at NSG. 5.4. Approaches to Predict KSL. Studies available in the literature on particle liquid mass transfer coefficient in STR attempted to correlate KSL by one of the four approaches mentioned below: (1) dimensional approach, (2) slip-velocity-based approach, (3) Kolmogoroff’s theory-based approach, and (4) approach using analogy between momentum-mass transfer Pangarkar et al.25 have made a critical and systematic analysis of the four approaches based on the information available in the literature. 6. Correlations for KSL 6.1. Correlations from Previous Investigations. The correlation for solid-liquid mass transfer coefficient, KSL, in twophase and three-phase stirred reactors can be interpreted in terms of the theorem of corresponding hydrodynamic states proposed in Section 3. Different investigators have suggested different mechanisms of solid suspension. Generally, when the mean flow and its associated turbulence result in a sufficiently large upward force on the particles resting on the bottom, off-bottom

( ) N NS

1.16

Sc-0.53

(7)

The theoretical basis of the above equation is demonstrated elsewhere.25,27 The ratio N/NS represents the fraction of the solids suspended and experiencing the turbulence in the STR. It is only at N/NS ) 1 that the entire solid mass experiences the turbulence. Jadhav and Pangarkar21 observed that eq 7 correlates their data as well as the data of Harriott,9 Brain and Man,28 Nienow,29 Miller,30 Nienow and Miles,31 Boon-Long et al.,32 and Sicardi et al.33 satisfactorily. Dutta and Pangarkar,34 in their study on KSL in low- and highviscosity liquids and multiple impeller systems using DT and PTD for a two-phase STR, proposed the following correlation, which is similar to that of Jadhav and Pangarkar:21

KSL ) 1.34 × 10-3

( ) N NS

1.07

Sc-0.52

(8)

From the work of Nienow and Miles31 on a two-phase system, Chapman et al.35 speculated that, for a three-phase system at NSG (minimum impeller speed for complete suspension of the particles in a three-phase STR), the value of KSL should be the same irrespective of geometric configuration. Kushalkar and Pangarkar36,37 and Dutta and Pangarkar,34 from their experimental work on a three-phase STR, observed that, indeed, the value of KSL is approximately constant at NSG in both Newtonian and non-Newtonian liquids either agitated by a single impeller (DT, PTD, and PTU) or by a multiple-impeller system (DT and PTD). The correlations proposed by Kushalkar and Pangarkar,36,37 Dutta and Pangarkar,34 and Pangarkar et al.25 are given by eqs 9, 10, and 11, respectively.

KSL ) 1.19 × 10-3 KSL ) 1.02 × 10-3

( ) ( ) ( )

KSL ) 1.8 × 10-3

N NSG

1.15

Sc-0.47

(9)

N NSG

0.97

Sc-0.45

(10)

N Sc-0.53 NSG

(11)

Sc in these equations is the Schmidt number for mass transfer and accounts for varying system properties. It will be shown that equations of these types which have the basis of the theorem of corresponding hydrodynamic states fit extensive data for both two- and three-phase systems, different types of impellers, single/multiple impellers/reactor combinations and scales, etc. available in the literature from mid-20th century to Very recent times. On the other hand, empirical correlations based on dimensionless groups (mechanical similarity) proposed by

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different investigators are generally valid for a narrow range of geometric parameters/system variables, scales, etc. and are also dependent on the measurement technique used. Considering these facts, the application of the momentum transfer-mass transfer approach-based correlations,25 which correlate relative particle suspension (N/NSG) with the mass transfer coefficient KSL, seems to be simple and reliable for estimating KSL. 6.2. Correlation Based on Present Data. Considering the success of the analogy between momentum and mass transfer, the same approach was extended to the data obtained in the present work for KSL in an STR fitted with helical coils. As mentioned above and also discussed at length by Nikhade et al.,4 the helical coils seriously disturb the hydrodynamics and flow patterns in the STR. For the KSL data in the presence of helical coils, the following correlation was obtained:

KSL ) 1.92 × 10-3

( ) N NSG

0.90

Sc-0.53

(12)

It is clear from eqs 7-12 that they have a striking similarity to each other, although as mentioned earlier, the hydrodynamics and flow patterns in each case are vastly different. The success of these equations over a very wide range of conditions is an a posteriori proof of the proposed theorem of corresponding hydrodynamic states. NS and NSG represent a unique combination of mean flow and turbulence structure defined here as the base hydrodynamic state at which the solid particles are just suspended and also experience the same drag. Starting with the pioneering work of Zwietering,22 correlations are available to predict NSG for a wide variety of impeller types, geometric configurations, and operating conditions. 25 Thus, knowing NSG and Sc, it is possible to have a reliable estimate of KSL. (N/NSG) represents the reduced hydrodynamic state referred to the just suspended condition just as P/Pc, T/Tc represent the reduced pressure and temperature Pr and Tr, respectively, in thermodynamics. Equations 7-12 can be interpreted in conformity with the theorem of corresponding hydrodynamic states as follows: At the same reduced hydrodynamic state (N/NSG), systems having the same Sc yield the same value of KSL irrespective of types of impeller/ reactor combinations/dimensions, operating conditions, presence of helical coil, etc. Such system combinations are said to be in corresponding hydrodynamic states. 6.3. Overall Correlation Based on Present and Previous Data. All the above correlations (eqs 7-12) have different constants and indices on N/NSG and Sc since they were obtained by independent regression of the respective investigators’ data. In particular, the value of the dimensional constant varies from 1.02 × 10-3 to 1.92 × 10-3. This variation is large enough to cast doubts on the proposition that equations of these type are scale-independent. To test the proposition that, indeed, such an equation is unique and scale-independent, it was thought desirable to combine all the available literature data to obtain a single correlation. Thus, the data of Jadhav and Pangarkar,21 Kushalkar and Pangarkar,36,37 Dutta and Pangarkar,34 and Pangarkar et al.,25 other literature data quoted in the above investigations, and the data obtained in the present work were combined and regressed to obtain the following correlation.

KSL ) 1.46 × 10-3

( ) N NSG

1.04

Sc-0.53

(13)

A total of 850 data points were regressed within 95% confidence limit to obtain the above correlation. The standard deviation is 17%. It is to be noted that the constant (1.46 × 10-3) remains

unchanged with variations in N/NSG (from 0.5 to 1.5) and Sc (from 600 to 1000). Variations in NSG takes into account variations in geometric (different types of impellers and CI/TR and DI/TR combinations) and operating conditions (dp, N, ∆F, etc.) of different investigators, and variations in Sc take into account differences in physical properties. Thus, eq 13 holds for a range of geometries and operating conditions. It is, thus, evident that eq 13 is scale-independent. 6.3.1. Physical Significance and Unique Features of the Dimensional Constant in Equation 13. Equation 13 is not dimensionless and has a dimensional constant on the right-hand side, which has the units of KSL (m s-1). The physical significance of the dimensional constant can now be interpreted. If eq 13 holds universally for different combinations of STRs as discussed above, this constant represents the value of KSL for N/NSG and Sc ) 1. Alternatively, all types of combination STRs should yield KSL ) 1.46 × 10-3 m/s at N/NSG and Sc ) 1. To check the sensitivity of the constant on indices of N/NSG and Sc, the index on N/NSG was varied from 0.8 to 1.2 and that of Sc was varied from -0.45 to -0.60. These ranges were selected on the basis of the logic given by Pangarkar et al.25 Further, since the solid-liquid mass transfer process in a highly turbulent medium is governed by an unsteady-state process, the following approximate dependence should hold: KSL ∝ DA0.5 (or KSL ∝ (Sc)-0.5). It was observed that there was no change in the constant when the data were regressed using the above ranges of indices for (N/NSG) and Sc. Evidently, the constant has a unique value that lends further credence to the physical significance enunciated above and the theorem of corresponding hydrodynamic states. 6.4. Prognosis for Transport in Multiphase Systems. It is possible to extend the approach presented above to other situations if an appropriate critical hydrodynamic state can be identified. For instance, a very large number of empirical correlations based on dimensionless groups are available for gas-liquid mass transfer coefficient, kLa, in bubble columns. All these correlations give different values of kLa under similar conditions. If a critical hydrodynamic state similar to NSG is identified in this case, then the present approach can unify the various literature data in the form of a single correlation. It is realized that this is a much more onerous task because of the strong dependence of the turbulence levels on system properties and holdup structure besides a wide variation in the bubble size. Nonetheless, efforts in this direction can be highly rewarding, as is evident from the success of eq 13 in unifying KSL data from widely different and distributed sources in terms of time as well as methodology. 7. Conclusion A theorem of corresponding hydrodynamic states analogous to the theorem of corresponding states in thermodynamics is proposed, and it is shown that this approach involving a reduced hydrodynamic state can be used for obtaining universal, scaleindependent correlations for mass transfer coefficients in two-/ three-phase stirred reactors. Notations As ) particle surface area (m2) c ) clearance of the impeller from the tank bottom (m) C ) concentration of solute in the liquid at time t (kg m-3) C* ) concentration of solute in the liquid at saturation (kg m-3) Cf ) final concentration of solute in the liquid at time t ) 5 min (kg m-3)

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CI ) clearance of impeller from bottom of the vessel (m) Ci ) initial (t ) 0) concentration of solute in the liquid (kg m-3) DA ) diffusivity of solute A in the liquid phase (m2/s) DI ) impeller diameter (m) dP ) mean particle diameter (m) dp′ ) average screen diameter (m) KLa ) gas-liquid mass transfer coefficient (s-1) KSL ) solid-liquid mass transfer coefficient (m s-1) N ) impeller speed (s-1) NS ) critical impeller speed for complete suspension of the particles in two phase (solid-liquid) system (s-1) NSG ) critical impeller speed for complete suspension of the particles in three-phase (gas-liquid-solid) system (s-1) P ) pressure (N m-2) P0 ) power drawn by the impeller (W) Pc ) critical pressure (N m-2) Pr ) reduced pressure (N m-2) S ) specific surface area of particle (m2 kg-1) Sc ) Schmidt number, (µ/FDm) t ) time (s) T ) temperature (K) Tc ) critical temperature (K) Tr ) reduced temperature (K) TR ) tank diameter (m) u′ ) turbulence intensity (m s-1) V ) volume (m3) VG ) superficial gas velocity (m s-1) W ) weight of undissolved solid (kg) X ) solid loading (Wt. %) Greek Symbols F ) density (kg m-3) ∆F ) density difference between solid and liquid phase (kg m-3) φ ) shape factor ψ ) sphericity τ ) torque acting on the table (N m) µ ) viscosity (Pa s) Literature Cited (1) Fishwick, R. P.; Winterbottom, J. M.; Stitt, E. H. Effect of gassing rate on solid-liquid mass transfer coefficients and particle slip velocities in stirred tank reactors. Chem. Eng. Sci., 2003, 58, 1087. (2) Rewatkar, V. B.; Raghava Rao, K. S. M. S.; Joshi, J. B. Critical impeller speed for solid suspension in mechanically agitated three-phase reactors. 1. Experimental part. Ind. Eng. Chem. Res. 1991, 30 (8), 1770. (3) Rewatkar, V. B.; Raghava Rao, K. S. M. S.; Joshi, J. B. Critical impeller speed for solid suspension in mechanically agitated three-phase reactors. 2. Mathematical model. Ind. Eng. Chem. Res. 1991, 30 (8), 1784. (4) Nikhade, B. P.; Moulijn J. A.; Pangarkar, V. G. Critical impeller speed (NSG) for solid suspension in sparged stirred vessels fitted with helical coils. Ind. Eng. Chem. Res. 2005, 44, 4400. (5) Gallagher, R.; Appenzeller, T. Beyond reductionism. Science 1999, 284 (5411), 79. (6) Sadiki, A.; Hutter, K. On thermodynamic turbulence: Development of first-order closure models and critical evaluation of existing models. J. Non-Equilib. Thermodyn. 2000, 25, 131. (7) Huang, Y. N.; Durst, F. A. Note on thermodynamic restriction on turbulent modeling. Int. J. Heat Fluid Flow 2001, 22, 495-499. (8) Barker, J. J.; Treybal, R. E. Mass Transfer Coefficients for Solids Suspended in Agitated Liquids. AIChE J. 1960, 6, 289. (9) Harriott, P. Mass transfer to particles. Part I. Suspended in agitated vessels. AIChE J. 1962, 8, 93. (10) Levins, D. M.; Glastonbury, J. R. Application of Kolmogoroff’s theory to particle-liquid mass transfer in agitated vessels. Chem. Eng. Sci. 1972, 27, 537-543. (11) Aussenac, D.; Alran, C.; Couderc, J. P. Presented at the Proceedings of 4th European Conference on Mixing, Noordwijkerhout, The Netherlands,

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ReceiVed for reView March 28, 2006 ReVised manuscript receiVed June 14, 2006 Accepted August 16, 2006 IE060385F