Fast Competitive Reactions in Taylor−Couette Flow - American

The upper limit ReC for the reactor Reynolds number is ... and more probable reactor Reynolds numbers Re > ReC, the inertial time scale defined in ter...
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Ind. Eng. Chem. Res. 2005, 44, 7306-7312

Fast Competitive Reactions in Taylor-Couette Flow L. J. Forney,* Z. Ye, and A. Giorges School of Chemical and Biomolecular Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332

Similarity parameters are derived to correlate the selectivity of mixing controlled parallel reactions within Taylor-Couette reactors. For the common case of laminar reactant feed the Kolmogoroff time is shown to correlate selectivity in turbulent, Taylor-Couette flows provided the reactor Reynolds number Re < ReC. The upper limit ReC for the reactor Reynolds number is shown to be constrained by the volumetric flow ratio of vortex cell-to-reactant feed. For larger and more probable reactor Reynolds numbers Re > ReC, the inertial time scale defined in terms of a reactor feed length is shown to correlate selectivity. The latter scaling law is used to correlate the present data and for comparison with similar stirred tank data. The advantages of the Taylor-Couette geometry are discussed. Attempts to improve selectivity with rotor-generated cavitation, however, were unsuccessful. 1. Introduction Common reactor configurations for bulk chemical production such as specialty organic chemicals are stirred tanks1 or static mixers.2 Control of selectivity for mixing sensitive reactions normally favors turbulent flow in such devices with one or more laminar feed streams.3 The level of turbulence in these reactor designs is independently controlled by either the stirring rate for stirred tanks or the flow rate for the case of static mixers. However, control of the selectivity, particularly during scaleup, can be difficult because of the relative importance of a number of turbulent length and time scales and their relationship to both the flow rate (or size) of the reactant feed streams and the kinetic reaction times. For the common case of laminar feed tubes the choice of mixing times is either the viscous (Kolmogoroff) time scale or an inertial scale (proportional to the eddy dissipation time) for small or large feed rates, respectively. This concept was useful in attempts to correlate the product distribution in stirred tanks.4 The relationship between these inertial and viscous time scales and the important operating parameters such as reactor size, Reynolds number, and reactant feed rate are recently outlined for both stirred tanks5 and plug flow reactors.6,7 In the present paper a turbulent Taylor-Couette reactor is considered. Such designs can provide large values of the turbulent energy dissipation rate of ∼104 W/kg necessary to reduce unwanted byproducts, very large surface-to-volume ratios necessary to control temperature-sensitive reactions, and very small reactor volumes. Examples of previous work concerning practical applications of Taylor-Couette reactors include plug flow characteristics,8 electrochemical designs,9 catalytic chemical reactors,10 polymerization reactors,11 recovery of intermediate species,12 and UV-induced reactions.13 In the present paper the question of similitude is considered for mixing controlled parallel reactions in Taylor-Couette geometries. Necessary similarity parameters are derived to correlate the selectivity. Moreover, a physical interpretation in terms of a critical

Figure 1. Schematic of Taylor-Couette flow.

reactor Reynolds number is provided for the choice of time scale as done, for example, in a previous study of stirred tank reactors.5 2. Taylor-Couette Flow The flow pattern between two concentric cylinders with the inner cylinder rotating at moderate rates is shown schematically in Figure 1. For sufficiently large Taylor number Ta > 400, the vortices around the circumference are turbulent. The average turbulent energy dissipated per unit mass of fluid by the inner cylinder can be estimated by the torque coefficient Cm14 where

Cm )

M /2πFUr2Roh

1

(1)

Here, the torque M is defined by * Corresponding author. Tel.: 404-894-2825. Fax: 404-3852713. E-mail: [email protected].

M ) 2πτoRo2h

10.1021/ie050342j CCC: $30.25 © 2005 American Chemical Society Published on Web 08/16/2005

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Ind. Eng. Chem. Res., Vol. 44, No. 19, 2005 7307

sion that the fine scale structure of most nonisotropic but large Reynolds number, turbulent flows is almost isotropic (local isotropy). Thus, many practical problems in turbulent phenomena that are strongly influenced by the fine-scale structure can be solved to some approximation by applying the features of isotropic turbulence. In this case, the eddy spectrum in the equilibrium range depends only on the local eddy dissipation rate and the kinematic viscosity. Further discussion of local isotropy is provided by Brodkey,20 Davies,21 and Kresta.22 In the present paper, the reactor Reynolds number, where Re ) Urd/ν, is assumed to be sufficiently large such that the fine-scale structure of the turbulence is locally isotropic. The turbulent kinetic energy κ can thus be written in the form19

κ ∝ (u′)2 Figure 2. Turbulent energy dissipation rate due to TaylorCouette flow vs rotor frequency.

where τo is the wall shear stress on the inner cylinder. Thus, the wall shear stress can be expressed as

(6)

and the dissipation rate (power input per unit mass) is

 ∝ (u′)3/le

(7)

Since the average turbulent energy dissipated per unit mass for Taylor-Couette flow is

where le is the average size of the energy containing eddies and u′ is the root-mean-square turbulent velocity fluctuation. It is now convenient to write the eddy dissipation time (∝κ/) in terms of local turbulent properties or

t ) 2πτoRohUr/(2πFRohd)

td ∝ le/u′ ∝ (le2/)1/3

τo ) FUr2Cm/4

(3)

(4)

and the Kolmogoroff time

or substituting eq 3, one obtains

t ) Ur3Cm/4d

(8)

(5)

The frequency of rotation for the reactor in the present study varied from 20 e f e 90 Hz with a rotor radius of Ro ) 7.62 cm, rotor length of h ) 2.54 cm, and an annular gap of d ) 0.32 cm. For these conditions the dissipation rate due to Taylor-Couette flow within the annular gap was estimated to be 0.3 < t < 26 kW/ kg for the range of Taylor numbers considered 2500 < Ta < 11 000, where a value of Cm ≈ 0.004 was estimated from Schlichting.14 A plot of turbulent dissipation rate versus rotor frequency is shown in Figure 2. 3. Similitude Mixing and chemical reaction in fully turbulent liquids occurs within laminar shear layers of thickness lk ,the Kolmogoroff length, between contacting eddies.15,16 The selectivity of mixing sensitive parallel reactions in coaxial fully developed, turbulent tube and reactant feed flows from the work of Bolzern et al.17 was plotted as a universal curve of selectivity versus Kolmogoroff time.7 The latter data demonstrate that the reaction rates are independent of Schmidt number Sc (or molecular diffusion times) as shown by Li and Toor.18 In contrast, the controlling physics of mixing when the reactant feed is laminar may be limited by an inertial mixing time rather than the viscous or Kolmogoroff time. The choice in the latter case depends on the laminar feed flow rate. 3.1. Turbulence. Isotropic turbulence, although hypothetical, is a useful concept in practical flows since many conditions more or less approach the physics of isotropy.19 Experimental evidence supports the conclu-

tk ∝ (ν/)1/2

(9)

3.2. Micromixing Time Scale. When the reactant feed is laminar and the feed time is large (small flow rate), experimental evidence demonstrates that the reaction rates are limited by the micromixing or Kolmogoroff time scale.23 To properly correlate selectivity data within turbulent reactors, it is necessary to compare the Kolmogoroff time in different reaction zones for fixed fluid properties and reactor geometries. Since tk ∝ -1/2 from eq 9, it is convenient to write the Kolmogoroff time at any location “i” in reference to its value at the average dissipation rate t within the annular gap. Substituting i ) t(i/t)

tki ) tkt(t/i)1/2

(10)

where t ∝ Ur3/d, the Kolmogoroff time (sometimes called the viscous or micromixing time) becomes

tki ) d/Ur(1/Re1/2Cm1/2)(t/i)1/2

(11)

where Re ) Urd/ν is the reactor Reynolds number. 3.3. Mesomixing Time Scale. When the reactant feed is laminar and the feed time is small (large flow rate), experimental evidence demonstrates that the reaction rates are limited by a somewhat larger inertial or mesomixing time scale.4 Thus, in the present study we focus on the small feed tube in Figure 3 that is assumed to be laminar or the feed tube Reynolds number Ref < 2500,21 where

Ref ) 4Qf/(πndfν)

(12)

Here, n is the number of feed tubes and df is the

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Figure 3. Laminar feed tube and length scale.

diameter of each feed tube. The reactor feed scale is therefore defined by the expression

l ∝ (Qf/nUm)1/2

(13)

as done earlier3,7 from conservation of mass at the feed nozzle, where l represents the diameter of a thread of fluid traveling at the average fluid velocity Um in the annular gap of the reactor. The inertial time scale tm can now be defined from eq 8 by substituting l for le as suggested by Corrsin.24 To properly correlate selectivity data within turbulent reactors, it is necessary to compare the inertial time at different reaction zones for fixed fluid properties and reactor geometries. Since tm ∝ -1/3, it is convenient to write the inertial time scale at any location in the reactor in reference to its value at the average dissipation rate t ∝ Ur3/d in the annular gap or

tmi ) tmt(t/i)1/3

(14)

where tmt3 ∝ Qfd/(nUr4). If we now define the annular flow rate within a vortex cell of magnitude Qc ∝ Umd2, where the cell cross-section is nearly square14 and Um = Ur/2 in fully developed turbulent flow, the inertial time scale from eq 14 becomes

tmi ) [Qf/(nQc)]1/3(d/Ur)(1/Cm)1/3(t/i)1/3

(15)

3.4. Influence of Reynolds Number. For a laminar reactant feed, the selectivity of a parallel reaction scheme is correlated with the largest of either the viscous or inertial time scale as defined above. The criterion for the choice of either scale is described below in terms of the reactor Reynolds number. One assumes that when the reactor feed length l is smaller than the Kolmogoroff length where the latter is defined by

lk ) (ν3/)1/4

(16)

or l < lk, then the appropriate time scale for mixing is the viscous or Kolmogoroff time. In this case the reactor feed stream is entrained between ambient turbulent eddies. However, if l > l k, we assume that the time scale for mixing is the inertial scale. The choice of the mixing time is therefore determined by the magnitude of the ratio l/lk as described by Forney et al.7 It should be noted that the latter concept is consistent with the ordering of the time scales since tm/tk ∝ (l/lk)2/3. Substituting from eqs 13 and 16 with  ) t , one obtains a ratio of the feed-to- Kolmogoroff lengths of the form

l/lk ) [Qf/(nQc)]1/2(Urd/ν)3/4(i/t)1/4

Figure 4. Mixing time vs reactor Reynolds number in a TaylorCouette reactor (where, for example, 1.E-01 represents 1 × 10-1): Rec ) 20 000.

(17)

Since the reactor Reynolds number Re ) Urd/ν, setting l/lk ) 1 in eq 17 provides a transition reactor Reynolds number that is constrained by the volumetric flow ratio of vortex cell-to-reactant feed in the form

Rec ) c(nQc/Qf)2/3(1/Cm)1/3(t/i)1/3

(18)

where the universal constant c ∼ O(102) must be established from experimental data. The choice of time scale is therefore determined by the magnitude of the reactor Reynolds number. For Re < ReC the scale is the viscous or Kolmolgoroff time defined by eq 11, while for Re > ReC the mixing time is the inertial value defined by eq 15. A comparison of both scales is demonstrated in Figure 4 for a hypothetical Taylor-Couette reactor. The values of the volumetric flow ratio of vortex cell-to-reactant feed Qc/Qf indicate that an increase in rotor speed shifts the mixing scale from the Kolmogoroff (micromixing) to the inertial (mesomixing) value at the transition Reynolds number ReC ) 20 000. One wishes to remain, therefore, on either the left or right of the transition value ReC for accurate scaleup since the physics of mixing changes at ReC. 3.5. Micromixing (l/lk e 1). Similitude is assured with the micromixing time scale if the reactor Re < Rec, the critical value defined by eq 17. Equal selectivity and scaleup are therefore possible for equal kinetics and stoichiometry with constant Damkohler number Da ∝ kCotk for the feed tubes in Figure 3, where k is a rate constant. Here, Co is defined as the inlet feed stream concentration for the rate-limiting reactant in the slow competitive reaction, producing unwanted byproducts. Substituting for the reactor Reynolds number in eq 11, one obtains

Da ) kCo(νd/CmUr3)(t/i)1/2

(19)

where i refers to the location of the reaction zone. In the present study we have assumed that the jet momentum is large enough such that the reaction zone is within the fully developed turbulent core as shown in Figure 3. 3.6. Mesomixing (l/lk > 1). Similitude is assured with the inertial time scale if the tank Re > Rec, the critical value defined by eq 17. Equal selectivity and scaleup are therefore possible for equal kinetics and stoichiometry with constant Damkohler number Da ∝ kCotm for the feed tubes, where Co is the rate-limiting reactant concentration for the slow competitive reaction

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Figure 5. Schematic of annular gap including the reactant feed tube. Also shown is the rotor surface pressure Pr and the stator pressure Ps. The rotor hole geometry containing a cavitation layer of thickness Lcav at the fluid vapor pressure Pv is also shown. See the Appendix for a description of the cavitation. Figure 7. Outlet temperature increase above inlet values vs rotor frequency.

measure of the energy dissipation rate per unit mass due to cavitation since

 ) (1/τ)cp(Tout - Tin)

(21)

where τ ) V/Q is the fluid residence time. As shown in Figure 11 (see Appendix), the measured energy dissipation rates with cavitation are very large, exceeding 102 kW/kg, which represents an order of magnitude increase above that expected for rotor-stator devices.4 5. Reaction Selectivity

Figure 6. Measured pressure on rotor and stator vs rotor frequency.

and k is a rate constant. Substituting for the inertial time scale defined by eq 15, one obtains

Da ) kCo[Qf/(nQc)]1/3(d/Ur)(1/Cm)1/3(t/i)1/3 (20) where i refers to the location of the reaction zone. 4. Cavitation The rotating cylinder in the present reactor was altered to include two rows of holes 0.375 cm in diameter that penetrated to a depth of ∼3.8 cm along the rotor radius. The geometry was manufactured by Hydro Dynamics Inc. (Rome, GA). The fluid within the hollow volume of such recessed cavities is subjected to larger centrifugal forces with increasing rotation rates. Under these conditions at sufficiently high rotation rates the onset of cavitation occurs at a vapor liquid interface somewhat below the rotor surface, as shown in Figure 5. The modifications described in this section were included in an attempt to increase the energy dissipated per unit mass in the annular gap in order to promote mixing. A pressure gauge was installed on the stator wall, and the gage pressure Ps was recorded versus rotation rate as shown in Figure 6 for an outlet pressure fixed at 20 psig. The outlet temperature was also recorded as shown in Figure 7. The latter fluid temperature provided a

5.1. Reaction System. The reaction system used in the present study is the instantaneous acid (HCl), base (NaOH) reaction in parallel with the relatively slow acid (HCl) catalyzed decomposition of 2,2-dimethoxypropane (DMP) to acetone and methanol as shown below

NaOH + HCl f NaCl + H2O DMP + H2O f CH3COCH3 + 2CH3OH

(k1)

(22)

(k2) (23)

The solvent is a 25 wt % ethanol aqueous solution and salt (NaCl) at a concentration of 100 gmol/m3 as suggested by Walker.25 The kinetics of the reaction sequence are such that k2/k1 , 1 with k2 ) 700 m3/(kmol s) at 25 °C and a salt (NaCl) concentration of 100 gmol/ m3. The main feed contained 0.2 kmol/m3 of DMP and 0.21 kmol/m3 of NaOH which was combined with a side stream of 2.0 kmol/m3 of HCl. The total volumetric flow rate of Q(base)/Q(acid) ) 10 was fixed for all data with the side stream feed tube diameter df ) 0.001 m (1 mm). Moreover, a kinematic viscosity ν ) 2.6 × 10-6 m2/s was assumed for the reactant mixture. 5.2. Rate Constant. The process of cavitation near the rotor surface creates a temperature gradient across the annular fluid gap. Under these conditions turbulent heat transfer occurs in the radial direction, and fluctuations in the fluid temperature occur within the reaction zone. The influence of temporal variations in temperature on the reaction rate is introduced by defining a turbulent fluid temperature in the form

T ) To + T′(t)

(24)

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where To is the time-averaged fluid temperature and T′ is the turbulent fluctuating temperature. An Arrhenius expression for the instantaneous rate constant in eq 23 for the determination of reaction times is defined in the form

k2 ) z2 exp(-E2/RT)

(25)

Expanding k2 in a three-term Taylor series about the time-averaged fluid temperature and averaging over time provides

k2 ) ko2 +

( ) ( ) dk2 dT

T′ +

To

d2k2

T′2 + O(T′3) (26) dT To 2 2

where ko2 is the value of the rate constant at To and T′ ) 0. With substitution of eq 25 into eq 26, one obtains

[

k2 ) ko2 1 +

E2 3

RTo

(

)

]

E2 - 2 T′2 + O(T′4) RTo

Figure 8. Selectivity vs rotor frequency.

(27)

where T′2 is the mean-square turbulent fluctuating temperature. Values for the activation energy E2 ) 46.2 kJ/kmol and the constant z2 ) 7.32 × 1010 m3/(kmol s) were assumed in the present study.25 A correction for the influence of salt (NaCl) concentrations was also introduced in the form

log(z2′/z2) ) K2 + acs

(28)

for the values K2 ) 0.054, a ) 7.07 × 10-5m3/gmol, and salt concentration cs ) 100 gmol/m3. Finally, the average temperature in the reaction zone of the fully developed turbulent annular gap is To = Toutlet with turbulent fluctuations |T′| = Toutlet - Tinlet. 5.3. Selectivity. Two independent experiments were conducted over the range of rotor frequencies 20 e f e 90 Hz. The main stream flow rate was maintained at approximately 1.4 L/min such that the reactant residence time τ ) 0.84 s for a holdup volume within the annular gap of 20 cm3. The selectivity for both experiments was determined by the concentration of acetone (CH3COCH3) in the form

X)

C[acetone] Co[DMP]

(29)

The acetone concentration was measured with a HPLC, and the results are shown in Figure 8. The indicated increase in selectivity at rotor frequencies f > 80 Hz was probably due to the increase in the mainstream reactant temperature due to cavitation. The Damkohler number was computed for each data point averaged for each frequency and defined by the expression

Da ) k2Co[DMP]tmi

(30)

Here, tmi is the inertial time scale determined by eq 15, and the energy dissipation rate t is defined by eq 5. The data from the present experiment are compared with data in a turbulent stirred tank reactor from Forney,5 and the results are shown in Figure 9.

Figure 9. Selectivity vs Damkohler number for Taylor-Couette flow defined by the inertial time scale for the case Re > Rec.

5.4. Discussion. The data from the present study were compared with large Reynolds number data in a stirred tank because the properties of continuous TaylorCouette flow at large Taylor numbers approach that of a continuous stirred tank reactor.26 Calculation of both the Kolmogoroff scale lk and the reactor feed length l demonstrates that l . lk or that the inertial time is the correct mixing scale. In Figure 9 we have assumed a value for the product R(t/i)1/3 = 3.0 for the present data, where the dissipation rate in the reaction zone is less than the average value within the annular gap from the numerical computations of Forney et al.25 In the latter case i < t since  is a maximum at the solid walls of either the stator or rotor. It is also clear from the data plotted in Figure 9 that the correct time scale for the reaction is the inertial mixing time established by Taylor-Couette flow rather than the much smaller time scale associated with cavitation. The latter conclusion may be due to the distance between the side stream injection point and the cavitation layer shown in Figure 5. It is also possible that the fast reaction is rate-limited by the larger of the two inertial time scales, that is, ratelimited by the larger time scale established by TaylorCouette flow in the presence of the smaller cavitation scales.

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6. Conclusions Mixing controlled parallel reactions in turbulent, Taylor-Couette reactors with laminar feed streams are shown to be correlated with either a viscous or an inertial time scale. The choice between time scales is shown to depend on the magnitude of a transition reactor Reynolds number ReC that is constrained by the volumetric flow ratio of vortex cell-to-reactant feed. Since both the vortex cell flow and the reactant feed are ∼d2, the ratio of vortex cell-to-reactant feed is small and of order one for Taylor-Couette geometries. The latter implies that the transition Rec is small and that the inertial time scale is sufficient for the purpose of scaleup in most cases. For a reactor Reynolds number Re < ReC, the Damkohler number defined in terms of the viscous time scale (micromixing) is sufficient to scale selectivity. More probably if Re > ReC, the inertial time (mesomixing) is the appropriate scale. The selectivity is insensitive to the Taylor number provided Ta > 400 to ensure fully developed turbulence. In general, the product distribution does not depend explicitly on either the Schmidt number Sc or Reynolds number Re. Moreover, similitude with constant energy dissipation rate (power per unit mass) is only applicable for constant viscosity fluids in the micromixing regime. Attempts were made to increase the local mixing rates by introducing cavitation near the rotor surface. The effects of the cavitation, however, were not evident in terms of altering the selectivity.

Figure 10. Radius of vapor/liquid interface vs rotor frequency.

Acknowledgment The authors acknowledge the support of this work from Mr. Kelly Hudson the President and CEO of Hydro Dynamics Inc. Figure 11. Energy dissipation rate due to cavitation vs rotor frequency.

Appendix A.1. Vapor/Liquid Interface. Estimates of the decrease in the static pressure along the rotor radius were made by integrating a modified Bernoulli expression across the fluid stream lines. Since the total pressure drop from the stator to the vapor/liquid interface ∆Pt ) Ps - Pv as shown in Figure 5, one obtains

∫RR

∆Pt ) ∆Ps-r + F

o

v

ω2R dR

where

Lcav (dP/dR) = 1/2(Fv′2)

and v′2 is the mean square turbulent fluctuating velocity. Thus, the cavitation layer becomes

(A1)

where

(A5)

Lcav = v′2/(2ω2Ro)

(A6)

Here, v′2 is estimated from the expression 2

∆Ps-r = Fω dRo/4

(A2)

and ω ) 2πf and it is assumed that the mean velocity Um ≈ Ur/2 within the turbulent gap. The position of the vapor/liquid interface Rv is therefore defined by the expression

Rv ) [Ro2 + dRo/2 - 2∆Pt/Fω2]1/2

(A3)

where ∆Pt ≈ Ps (stator). Figure 10 demonstrates calculated values of Rv versus rotor frequency for two inlet pressures. A.2. Cavitation Layer. The thickness of the cavitation layer Lcav in Figure 5 is estimated from the static pressure gradient within the cylindrical hole

Lcav (dP/dR) ) Fω2RLcav

(A4)

v′2 = (lec)2/3

(A7)

and the Prandtl mixing length le = 0.1d. Equations A6 and A7 provide estimates of the thickness of the cavitation layer, and the results suggest a value of Lcav = 0.3-0.4 mm for decreasing rotation frequency from f ) 90 to 40 Hz (see Figure 11). Nomenclature Co ) limiting reactant concentration (kmol/m3) c ) universal constant (∼100) Cm ) torque coefficient Da ) Damkohler number df ) feed tube diameter (m) E ) activation energy (J) f ) rotor frequency (s-1)

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h ) length of rotor (m) k ) rate constant (m3 mol-1s-1) Lcav ) thickness of cavitation layer (m) l ) reactant feed length scale (m) le ) integral length scale (m) lk ) Kolmogoroff length (m) M ) torque on rotor [(kg m2)/s2] n ) number of feed tubes P ) pressure [psia; kg/(m s2)] Ps ) stator pressure [psia; kg/(m s2)] Pr ) rotor pressure [psia; kg/(m s2)] Pv ) vapor pressure [psia; kg/(m s2)] Q ) reactor volume flow rate (m3/s) Qf ) volume flow rate of limiting reactant feed (m3 s-1) Qc ) volume flow rate of vortex cell ()Umd2) (m3 s-1) Re ) gap Reynolds number () Urd/ν) Ro ) radius of rotor (m) Rv ) radius of vapor/liquid interface (m) Rec ) critical reactor Reynolds number Ref ) feed tube Reynolds number Sc ) Schmidt number T ) fluid temperature (K) T′ ) turbulent fluctuating temperature (K) Ta ) Taylor number [)Urd/ν(d/Ro)1/2] tk ) Kolmogoroff time (s) td ) eddy dissipation time (s) tm ) inertial time scale (s) tf ) reactant feed time (s) Ur ) rotor surface velocity (m s-1) Um ) average fluid velocity in gap (m s-1) v′ ) root-mean-square turbulent velocity fluctuation (m s-1) uf ) fluid velocity in feed tube (m s-1) V ) reactor hold up volume (m3) X ) selectivity or product distribution Greek Symbols τ ) reactor residence time (s) τo ) shear stress on rotor [kg/(m s2)] t ) turbulent energy dissipation from Taylor-Couette flow (m2 s-3) c ) turbulent energy dissipation from cavitation (m2 s-3) κ ) turbulent kinetic energy (m2 s-2) ω ) angular velocity of rotor (rad s-1) ν ) kinematic viscosity (m2 s-1) F ) fluid density (kg m-3) Subscripts i ) feed location in fluid gap t ) average in fluid gap

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Received for review March 14, 2005 Revised manuscript received June 27, 2005 Accepted July 12, 2005 IE050342J