Modeling Ozone Contacting Process in a Rotating Packed Bed

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Ind. Eng. Chem. Res. 2004, 43, 228-236

Modeling Ozone Contacting Process in a Rotating Packed Bed Y. H. Chen,† C. Y. Chang,*,† W. L. Su,† C. C. Chen,† C. Y. Chiu,‡ Y. H. Yu,† P. C. Chiang,† and Sally I. M. Chiang§ Graduate Institute of Environmental Engineering, National Taiwan University, Taipei 106, Taiwan, Department of Environmental Engineering, Lan-Yang Institute of Technology, I-Lan 261, Taiwan, and Tairex Environmental Technology Company, Ltd., Taichung 420, Taiwan

The process of the ozone dissolution with decomposition in a rotating packed bed (RPB) is studied for model establishment. The RPB, which provides a high gravitational force by adjusting the rotational speed, is taken as a novel ozone contactor because of its high mass-transfer coefficient. It has a high potential to improve the ozonation performance or reduce the ozone contactor volume by applying RPBs. The objective of this study is to investigate and model the dynamic ozone dissolution process with decomposition in a RPB with oxygen mass transfer. In addition, the empirical correlations between the system parameters (including liquid holdup, liquid film thickness, specific area of the gas-liquid interface a, and mass-transfer coefficient k0L) and operating conditions are presented. Furthermore, the variations of gas and liquid ozone concentration profiles are predicted to illustrate the dynamic behavior of mass transfer in a RPB. The validity of the model is demonstrated by comparing the predicted results with experimental data, indicating good agreement. As a result, the present model is useful and referable for the proper description of ozone contacting in a RPB system. Introduction Ozone is widely used as an oxidant applied in the water treatment and disinfect ion. The mixture of gases composed of oxygen and ozone is transferred to water by injecting it through the gas-liquid contactor. The efficiency of the ozonation processes is usually dependent on the dissolved ozone concentration. In addition, the rate-limiting step in many ozonation processes is attributed to gas-liquid mass transfer.1 This means that the performance of the ozonation treatment is generally restricted to the gas-liquid mass-transfer rate of ozone. Consequently, the promotion of a gas-liquid contactor with high mass-transfer efficiency is desired. The rotating packed beds (RPBs) were used as gasliquid contactors for the applications of adsorption, distillation, stripping, etc.2,3 According to the previous studies, RPBs have higher gas-liquid mass-transfer coefficients, which is an important factor for evaluating the gas-liquid mass-transfer rate. RPBs are designed to generate high acceleration due to centrifugal force. The novel technology is also named “Higee”. The volumetric gas-liquid mass-transfer coefficients achieved in the RPB are 1-2 orders of magnitude higher than those in a conventional packed bed.4 Recently, the RPB has been introduced by Lin and Liu5 as an ozonation contactor. Compared to the conventional ozone contactors such as bubble column reactors (BCRs) and mechanically stirred reactors (MSRs), there is a high potential to enhance the ozonation performance or reduce the contactor volume by using RPBs. The concentration of ozone is usually relatively low with respect to that of oxygen in the carrier gas. One of the advantages of ozone dissolution is its contribution to dissolved oxygen, which can be used in the biological * To whom correspondence should be addressed. Tel./fax: +886-2-2363-8994. E-mail: [email protected]. † National Taiwan University. ‡ Lan-Yang Institute of Technology. § Tairex Environmental Technology Co., Ltd.

process after the decomposition of ozone.6 The quantification of ozone and oxygen mass transfer associated with the operation condition is critical to the rational design of the ozonation treatment. Furthermore, there exists a temporary and unsteady period before the ozone contacting system reaches the steady state. However, the previous gas-liquid transfer model in RPBs for absorption or air stripping was commonly developed for the steady state.2,7 Moreover, the ozone self-decomposition reaction, which should be taken into consideration in the ozone dissolution, was neglected in modeling the RPBs. Therefore, the information about the dynamic process of ozone dissolution with decomposition reaction in RPBs needs to be elucidated. The objective of this study is to model and investigate the dynamic ozone and oxygen mass-transfer process in RPBs. Three major factors are included in the present model: (1) gas and liquid convections; (2) gas-liquid mass transfer; (3) ozone decomposition reaction kinetics. First, the liquid holdup volume (VL), relative liquid holdup fraction (L), mean liquid thickness (hL), and specific area of the gas-liquid interface (a) in RPBs are measured and calculated under various experimental conditions. Oxygen aeration and ozone dissolution experiments are sequentially carried out to determine the mass-transfer coefficients. Moreover, the corresponding correlations are derived to represent the effect of operating parameters in a RPB. The validity of the model is demonstrated by the agreement of the predicted results with experimental data. Consequently, the model proposed can provide useful information about the dynamic behavior of ozone dissolution with decomposition in a RPB. Theoretical Analysis Ozonation Kinetics and Film Model. Modeling the dynamic processes of ozone dissolution with decomposition in a RPB contactor requires quantification of the rates of the gas-liquid mass transfer and ozone decomposition reaction, simultaneously. Noting that the dif-

10.1021/ie030545c CCC: $27.50 © 2004 American Chemical Society Published on Web 12/10/2003

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fusivities of ozone and oxygen in the gas phase are about 104 times higher than those in the liquid phase, this study assumes that the resistance of the mass transfer is solely contributed by the liquid phase. As the ozone is dissolved in water, it may be consumed via the selfdecomposition reactions. For the spontaneous ozone decomposition to oxygen in water, the corresponding pseudo-first-order decomposition rate equation can be expressed as eq 1.8

-

dCALb 2 dCOLb ) ) kdCALb dt 3 dt

(1)

With the decomposition of ozone and formation of oxygen, the mass-transfer rates of ozone (A) and oxygen (O) may be enhanced and retarded, respectively. The ratios of the mass-transfer rates of ozone with the decomposition of ozone and of oxygen with the formation of oxygen to those without may be assigned by the enhancement factor of the ozone decomposition (ErA) and retarding factor of the oxygen formation (RfO), respectively. As indicated by Danckwerts,9 the predictions of the effects of chemical reactions based on the film, penetration, and surface renewal models are closely similar. However, the computation related to the film model is simpler because it involves the ordinary rather than partial differential equations. According to the film model for predicting ErA and RfO, ErA of ozone and RfO of oxygen are expressed as eqs 2 and 3, respectively.9,10

ErA ) RfO ) 1 -

Ha (CAGi/HA) cosh Ha - CALb sinh Ha CAGi/HA - CALb

[

(

)

(2)

CAGi 3 DA Ha cosh Ha - CALb 2 DO sinh Ha HA CAGi COGi - CALb / - COLb (3) HA HO

(

)] (

)

with Ha ) xkdDA/k0LA. Dynamic Ozone Dissolution Model. Assumptions of the model are as follows. 1. For the homogeneous regime of a RPB, the gas and liquid holdups (G and L), specific area of the gas-liquid interface (a), and mass-transfer coefficients of ozone (k0LA) and oxygen (k0LO) are constant. 2. The plug-flow condition is applicable to both gas and liquid phases. 3. The effect of the pressure variation on the gas concentration is negligible. 4. Henry’s law applies. 5. Chemical reactions in the gas phase are neglected. A small variation in L with the change of radius (r) may exist. Nevertheless, the assumption of a homogeneous regime has been satisfactorily used in the previous studies of Liu et al.3 and Kumar and Rao11 and is thus considered to be applicable for the RPB modeling. Although the gaseous flow configuration in the RPB may have the boundary effect, a one-dimensional (1D) mass-transfer model is found to be practicable in this work according to the predicted results of good agreement. However, the present 1D model still remains a simplification of reality regarding the smallness of RPBs. For a more vigorous description, a 2D or 3D modeling would be necessary. On the basis of the above assumptions, the overall mass balance of the gas phase may be described by eq

4. In eq 4, the left-hand-side term represents the

G

( (

) )

∂CG ∂(uGCG) uGCG CAGi ) + - ErAk0LAa - CALb ∂t ∂r r HA COGi - COLb (4) RfOk0LOa HO

variation of the local gas concentration, while the righthand-side terms stand for the gas convection and ozone and oxygen gas-liquid mass transfers, respectively. Applying the ideal gas equation and noting the constant gas pressure, one would have CG ) P/RT as a constant. This assumption could be deemed reasonable according to the results of Liu et al.,3 Kumar and Rao,11 and Sandilya et al.,12 which found that the pressure drop in RPBs is small and between 0.001 and 0.04 atm. Substituting CG ()P/RT) into eq 4 then gives eq 5 for the superficial gas velocity in the dimensionless form (UG, with UG ) uG/uG,2).

UG dUG )+ ErAStGAyA(θAGi - θALb) + dr* re* RfOStGOyO(θOGi - θOLb) (5) for rGB* e r* e 1, with the boundary condition (BC)

r* ) 1, UG ) 1

(6)

For 0 e r* < rGB*, UG ) 0. The dimensionless position of moving gas flow [rGB*, with rGB* ) (rGB - r1)/(r2 r1)] at time τ ()t/tL), which can be computed by eq 7, has a minimum value of zero.

drGB* ) -Rm1RuGLUG,r*)rGB* dτ

(7)

with the initial condition (IC)

τ ) 0, rGB* ) 1

(8)

The dimensionless superficial liquid velocity (UL, with UL ) uL/uL,1) can be calculated according to eq 9.

UL ) r1/r ) r1/[(r2 - r1)r* + r1]

(9)

with the BC

r* ) 0, UL )1

(10)

Further, the dimensionless governing equations of gas ozone (θAGi, with θAGi ) CAGi/CAGi0) and oxygen (θOGi, with θOGi ) COGi/COGi0) are expressed by eqs 11 and 12, respectively.

[

∂(UGθAGi) UGθAGi ∂θAGi ) Rm1RuGL + ∂τ ∂r* re*

]

ErAStGA(θAGi - θALb) (11)

[

∂(UGθOGi) UGθOGi ∂θOGi ) Rm1RuGL + ∂τ ∂r* re*

]

RfOStGO(θOGi - θOLb) (12)

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The dimensionless liquid-phase governing equations for ozone [θALb, with θALb ) CALb/(CAGi0/HA)] and oxygen [θOLb, with θOLb ) COLb/(COGi0/HO)] should consider the chemical reaction terms according to eq 1 and can be expressed as follows.

[

∂(ULθALb) ULθALb ∂θALb ) Rm1 + ∂τ ∂r* re*

]

ErAStLA(θAGi - θALb) - DaAθALb (13)

[

∂(ULθOLb) ULθOLb ∂θOLb ) Rm1 + ∂τ ∂r* re*

]

RfOStLO(θOGi - θOLb) + DaOθALb (14) The ICs of eqs 11-14 are

Table 1. Specification of RPB Contactors Used in This Study

τ ) 0, θAGi ) θALb ) 0, θOGi ) 0.21/yO, θOLb ) θOLb0 (15) The applicable BCs of eqs 11-14 are as follows.

At the inner radius, r* ) 0: θALb ) 0, θOLb ) θOLb0

(16)

At the outer radius, r* ) 1: θAGi ) θOGi ) 1

Figure 1. Schematic diagram of a RPB contactor. Components: 1, stationary liquid distributor; 2, seal; 3, packed-bed rotator; 4, housing case; 5, rotor shaft; 6, motor.

(17)

In the above equations, the definitions of the dimensionless variables and parameter groups are listed in the Nomenclature section. Note that the Stanton and Damko¨hler numbers stand for the significance of gasliquid mass transfer and chemical reactions, respectively. Because dissolved oxygen can be generated from the decomposition of ozone, the supersaturation of dissolved oxygen may occur near the outer layer of RPBs. Nevertheless, it only happens under the conditions of the high ozone feed concentration, mass-transfer coefficient, and ozone decomposition rate all together. For common cases, the occurrence of an inverse driving force of oxygen mass transfer is rare. Computation Algorithm for Solving Model Equations. Equations 2, 3, and 5-17 represent the governing equations of the ozone dissolution model for predicting the dynamic and radial variations of ozone and oxygen concentration profiles in a RPB gas-liquid contactor. The present work considers (1) the dynamic state, (2) the oxygen mass transfer, (3) the chemical reactions of self-decomposition of ozone, and (4) the effect of chemical reactions on mass transfer. The finite difference method based on the Taylor series is employed with the Turbo C program in this study. Equations 2 and 3 are first solved to yield the values of ErA and RfO at time τ. The obtained ErA and RfO are substituted into eq 5 along with eq 6 to compute UG. Equations 7-17 are then be solved using the forward-difference method to compute the values of the variables at the next time step of τ + ∆τ from the available values at τ. This is followed by the computation of ErA, RfO, and UG at τ + ∆τ. The computation is conducted up to the steady state. The grids along r* ) 0-1 and the size of the time step (∆τ) adopted in the program are 101 points and 10-5, respectively.

item

unit

inner radius of a packed bed, r1 outer radius of a packed bed, r2 axial height of a packed bed, ZB weight of packing in a packed bed volume of a packed bed, VB volume of the housing case specific area of packing per unit volume of a packed bed, ap total packing area, apVB voidage, 

m m m kg m3 m3 m2/m3

contactor A contactor B 0.0385 0.0825 0.02 0.131 3.35 × 10-4 3.37 × 10-3 840

m2 0.281 m3/m3 0.954

0.023 0.059 0.02 0.0686 1.85 × 10-4 1.04 × 10-3 793 0.147 0.956

Experimental Section Figure 1 shows a simple description of a RPB contactor, which consists of a rotator and stationary case. Two RPB contactors noted as contactors A and B with different dimensions are employed in this work. The specification of these two RPB contactors is given in Table 1. Note that the volume of a packed bed (VB) in contactor A is 1.81 times that in contactor B. The 304 stainless steel wire cut in the sharp of annular rings is stacked in the packed bed. The density (Fs) and diameter (ds) of the wire are 8478 kg/m3 and 2.2 × 10-4 m, respectively. The rotator is connected to a rotor shaft on two bearings, which are, in turn, mounted on a steel structure. The shaft is connected to a motor, which is controlled by a speed regulator. The rotational speed can be operated properly from 300 to 1500 rpm, which provides gravitational forces of 6-152g and 4-103g based on the arithmetic mean radii of RPBs for contactors A and B, respectively. Liquid enters the RPB through six holes in the liquid distributor. These six holes are arranged in a vertical group of three, and the groups are spaced 180° apart. The liquid is sprayed on the inside edge of RPB and thrown outward by the centrifugal force. The gas is introduced from the outside and flows countercurrently to the liquid in the RPB. The volume of liquid holdup (VL) in contactors A and B is determined by measuring the amount of retained liquid in the RPBs. The amount of retained liquid under different operating conditions is calculated by collecting the effluent liquid from the RPBs after the stop of fed liquid. Oxygen aeration experiments are employed to estimate the volumetric mass-transfer coefficients of oxygen (k0LOa) for various operating conditions. The variation of the effluent dissolved oxygen concentration (COLb,eff) is analyzed by the dissolved oxygen meter (model Oxi 340, Wissenschaftlich-Technische Werksta¨tten GmbH & Co. KG (WTW), Weinheim, Germany) with

Ind. Eng. Chem. Res., Vol. 43, No. 1, 2004 231

packed bed size (VB) and operation parameters. Further, the relative liquid holdup (L ) VL/VB) can be calculated with the range of 0.078-0.264 for the two RPB contactors. Using the multivariable regression, the correlation of L associated with the operation parameters can be derived from the data of Table 2 as eq 18.

L ) 21.3uL,10.646uG,2-0.015ω-0.148

Figure 2. Experimental apparatus sketch. s, - -, ‚‚‚: ozone gas stream, experimental solution, and isothermal water, respectively. Components: 1, oxygen cylinder; 2, drying tube; 3, ozone generator; 4, flowmeter; 5, three-way valves; 6, RPB gas-liquid contactor; 7, liquid-storage tank; 8, thermostat; 9, pumps; 10, dissolved oxygen sensor; 11, liquid ozone sensor; 12, gaseous ozone detector; 13, KI solution; 14, vent to hood; 15, water exists.

the sensor of model CellOx 325 (WTW, Weinheim, Germany). A liquid-storage tank is equipped with a thermostat to maintain constant temperatures for the solution at 18 or 25 °C in all experiments. Furthermore, the ozone dissolution is performed to provide data for the model verification. Ozone-containing gas is generated from pure oxygen by the ozone generator (Tairex Environmental Technology, Taiwan). Before the ozone dissolution experiments are started, the ozone-containing gas is bypassed to the photometric analyzer (model SOZ-6004, Seki, Tokyo, Japan) to ensure the stability and determine the ozone concentration. A portion of the gas stream at the preset flow rate is directed into the contactor when the set conditions are reached. The effluent dissolved ozone concentration (CALb,eff) is analyzed by the liquid ozone monitor (model 3600, Orbisphere Lab., Neuchaˆtel, Switzerland) with a sensor of membrane-containing cathode. All of the recorded data for examining the concentration variations are adjusted with the corresponding instrument response times of about 1-7 s and the retention times of the input gas and effluent liquid of the RPB contactor. All experimental solutions are prepared using the deionized water without other buffers. The conductivity of water used is less than 1 µs/cm. The values of ozone decomposition rate constant kd and Henry’s constant HA are determined as 1.45 × 10-4 s-1 and 4.18 at 25 °C from the experiments in batch and semibatch stirred reactors, respectively. The experimental apparatus employed in this work is shown in Figure 2. Results and Discussion The results concerning the dynamic ozone dissolution in the RPBs include two parts: (1) the estimation of the system parameters and (2) the oxygen and ozone dissolution experiments. The parameters of the RPB system including L, hL, and a are estimated. The oxygen and ozone dissolution experiments are carried out to determine the mass-transfer coefficients. The concentration variations of θALb,eff and θOLb,eff are monitored in order to provide the experimental data for model verification. Estimation of System Parameters in RPBs. The VL values are obtained from the two RPB contactors as listed in Table 2. Obviously, VL is dependent on the

with R2 ) 0.9631 (18)

As shown in Figure 3, the predicted values by eq 18 agree well with the experimental results. The L value increases with an increase of the inlet superficial liquid velocity (uL,1) and a decrease of the rotational speed (ω), respectively. However, the effect of the inlet superficial gas velocity (uG,2) on L is comparatively slight. This circumstance is similar to that in RPBs with packing material of nonporous glass beads.13 Moreover, the exponent of uL,1 for the G correlation as 0.646 indicates the flow regime categorized to the film flow regime according to the analysis of Burns et al.14 Furthermore, the mean liquid film thickness of packing (hL) for the steel wire can be estimated by eq 19.

L ) aw(hL + hL2/ds)

(19)

aw denotes the specific wetted area of packing per unit volume of a packed bed. The ratio of aw to ap stands for the proportion of the effective packing area, which is wetted by liquid film flow. According to the previous studies, aw is usually assumed to be equal to ap13 or accounted for by the empirical correlations.15 Because the validity of the two approaches has not been established, both approaches of aw estimation are adopted for the following discussion. As shown in Table 2, eqs 20 and 21 listed below, which were employed by Tung and Mah15 for estimating the values of aw/ap, give results close to the value of σc/σ of 0.97. The average

aw/ap ) 1 - exp[-1.45(σc/σ)0.75ReL0.1Fr-0.05We0.2] (20)

aw/ap ) 1.05ReL0.047We0.135(σc/σ)-0.206

(21)

value of aw/ap by eqs 20 and 21 is taken for computing the aw value. Accordingly, the hL values based on these two approaches can be estimated from eq 19 as listed in Table 2. It can be seen that the estimated hL value based on the partial wetting of packing is about 2.5 times that based on the complete wetting of packing. In addition, the corresponding empirical correlations for hL are expressed as follows.

For complete wetting of packing hL (µm) ) 4106uL,10.454uG,2-0.002ω-0.104

(22)

For partial wetting of packing hL (µm) ) 2055uL,10.183uG,20.003ω-0.109

(23)

Figure 4 shows that the values of hL calculated by the approaches of Basˇic´ and Dudukovic´13 and Tung and Mah15 lie within (20% of those predicted by the correlations of eqs 22 and 23. Apparently, the hL value is increased with an increase of liquid flow velocity (uL,1) and a decrease of the rotational speed (ω). The dependence of hL on the gas flow velocity (uG,2) is much weaker

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Table 2. Liquid Holdup (EL) in RPB Contactors aw/ap

run no.a

uL,1 × 103 (m/s)

uG,2 × 103 (m/s)

w (rpm)

VL (mL)

L (m3/m3)

eq 20

eq 21

hLb,d (µm)

hLc,d (µm)

ab,e (m-1)

ac,e (m-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1.77 1.77 1.77 4.44 4.44 4.44 0.865 0.865 1.73 1.73 1.73 3.46 3.46 3.46

3.12 3.12 7.07 3.12 3.12 7.07 4.36 4.36 4.36 4.36 4.36 4.36 4.36 4.36

900 1500 1500 900 1500 1500 900 1500 300 900 1500 300 900 1500

48.0 44.0 40.0 73.7 81.7 79.0 14.5 17.0 31.5 24.7 24.5 48.9 42.0 40.0

0.143 0.132 0.120 0.220 0.244 0.236 0.078 0.092 0.170 0.134 0.132 0.264 0.227 0.216

0.238 0.238 0.238 0.325 0.325 0.325 0.181 0.181 0.232 0.232 0.232 0.294 0.294 0.294

0.251 0.251 0.251 0.335 0.335 0.335 0.195 0.195 0.243 0.243 0.243 0.303 0.303 0.303

113 106 99 154 166 162 74 84 133 112 133 182 164 158

297 282 265 322 344 337 246 275 349 300 298 397 362 351

1702 1648 1592 2017 2106 2076 1326 1396 1755 1598 1755 2105 1973 1934

759 731 700 1088 1144 1125 483 522 786 702 699 1091 1016 992

a Run nos. 1-6: contactor A. Run nos. 7-14: contactor B. b Assume complete wetting of packing with a ) a . c Use the average value w p of aw/ap by eqs 20 and 21 for aw computation. d Apply eq 19 with aw to obtain hL. e Compute the value of a by eq 24.

Figure 3. Diagonal graph of experimental and predicted L by eq 18. O and 4: contactors A and B, respectively.

than those on uL,1 and ω. Therefore, the influence of gas flow on the pattern of the liquid flow is relatively negligible, which is consistent with the result of Guo et al.16 The thickness of hL in RPBs is mainly based on the balance of inertia and centrifugal forces for the liquid film flow. Moreover, the specific area of the gas-liquid interface (a) can be estimated as listed in Table 2 by eq 24 and then associated with the correlations of eqs 25 and 26.

a ) aw(1 + 2hL/ds)

(24)

For complete wetting of packing a (m-1) ) 9907uL,10.254uG,2-0.013ω-0.035

(25)

For partial wetting of packing a (m-1) ) 21360uL,10.491uG,2-0.018ω-0.053

(26)

As shown in Figure 5, the calculated values of a can be predicted very well by the empirical correlations of eqs 25 and 26. The value of a depends significantly on uL,1 and slightly on ω and uG,2 under the experimental condition. The calculated values of a based on the completely and partially wetted areas of packing are 1.67-2.65 and 0.61-1.37 times of the ap values in the RPBs, respectively. As a result, the calculated value of a from the complete wetting of packing is about 2 times that of the partial wetting of packing. The verification of the accuracy for these two calculations would need further future work. Note that the previous studies

Figure 4. Diagonal graph of calculated and predicted hL values. O and 4: contactors A and B, respectively. a and b: predicted by eqs 22 and 23, respectively.

usually neglected the effect of hL on a, yielding a ) aw (Liu et al.3 and Tung and Mah15) or a ) ap (Basˇic´ and Dudukovic´13). However, the value of a can be greater than aw or ap by counting the considerable thickness of the liquid film on the packing that may be referred to the study of Guo et al.16 It is worth addressing that the difference between a and aw (or ap) depends on the magnitude of hL and the geometric shape of packing. Oxygen and Ozone Dissolution in RPB. As for the oxygen aeration, eqs 5-10, 12, and 14 and their corresponding ICs and BCs are applied to describe the dimensionless concentration of oxygen (θOLb) in the RPB. A comparison of the effluent oxygen concentration (θOLb,eff) from the model prediction to that from the experiments then gives the proper value of the volu-

Ind. Eng. Chem. Res., Vol. 43, No. 1, 2004 233

Figure 6. Diagonal graph of experimental and predicted k0LOa by eq 27.

Figure 5. Diagonal graph of calculated and predicted a values. O and 4: contactors A and B, respectively. a and b: predicted by eqs 25, 26.

metric mass-transfer coefficient of oxygen (k0LOa). The employment of θOLb,eff for checking is because the predicted θOLb,eff value is sensitive to the Stanton numbers of oxygen (StGO and StLO), which are associated with k0LOa. Therefore, the value of k0LOa is determined by minimizing the difference of θOLb,eff between the model prediction and the experimental data. The k0LOa values along with the associated steady-state values of θOLb,eff in contactor A at various uL,1, uG,2, ω, and T are listed in Table 3. The correlation of eq 18 for the L estimation is applied in the simulation. Besides, the values of k0LOa can also be estimated by the packed-bed model proposed by Coulson and Richardson17 as well. Their model, which neglects the chemical reaction, is

applicable to oxygen aeration with the assumption of a constant gas oxygen concentration. The difference of estimated k0LOa values between these two models is below 0.5%, demonstrating that the present model can be properly simplified to describe the oxygen aeration processes and the numerical solution of this study should be reliable. Moreover, the height of the masstransfer unit (HTU) based on the liquid-phase resistance (HOL) can be calculated accordingly as listed in Table 3. The obtained values are within the order of 1-10 cm for HOL in RPBs reported by Keyvani and Gardner.18 The correlation for k0LOa can be derived from Table 3 as eq 27. In eq 27, the relationship between k0LOa and T

k0LOa (s-1) ) 3.83uL,10.747uG,20.282ω0.3111.016T-20 (27) (in °C) is referred to the expression of Adams et al.19 As shown in Figure 6, the predicted values of k0LOa agree with the experimental data satisfactorily. Note that 0LOa increases remarkably with increases of uL,1, uG,2, and ω. Referring to the above correlations for a as in eqs 25 and 26, the values of k0LO would increase likewise with increases of uL,1, uG,2, and ω. One may infer that the thickness of the liquid boundary layer (δL ) DO/k0LO) for the solute diffusion transport would be reduced with higher gas-liquid relative velocity and centrifugal acceleration. Furthermore, the values of k0LO can be calculated from the k0LOa values by applying

Table 3. Gas-Liquid Mass-Transfer Coefficients of Oxygen in a RPB Contactor run no.a

uL,1 × 103 (m/s)

uG,2 × 103 (m/s)

ω (rpm)

T (°C)

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

1.79 1.79 1.79 1.79 4.17 4.17 4.17 4.17 4.17 4.17 4.17 4.17 4.17 4.44 4.44 4.44 4.44

3.12 3.12 7.07 7.07 1.91 1.91 1.91 3.63 3.63 3.63 7.07 7.07 7.07 3.12 3.12 7.07 7.07

900 1500 900 1500 400 800 1200 400 800 1200 400 800 1200 900 1500 900 1500

18 18 18 18 25 25 25 25 25 25 25 25 25 18 18 18 18

a

θOLb,eff

k0LOa (s-1)

HOL (cm)

k0LO × 105 b (m/s)

k0LO × 105 c (m/s)

0.930 0.911 0.950 0.940 0.729 0.798 0.841 0.843 0.913 0.918 0.856 0.913 0.923 0.888 0.903 0.902 0.916

0.0642 0.0575 0.0725 0.0676 0.0642 0.0822 0.0964 0.0974 0.133 0.137 0.102 0.133 0.140 0.128 0.137 0.136 0.146

2.20 2.39 2.01 2.11 4.11 3.21 2.73 2.71 1.98 1.92 2.57 1.98 1.87 2.61 2.47 2.48 2.34

3.79 3.48 4.36 4.14 2.97 3.90 4.64 4.55 6.36 6.63 4.82 6.42 6.87 6.04 6.58 6.49 7.11

8.63 7.98 9.95 9.53 5.45 7.23 8.67 8.36 11.8 12.4 8.89 12.0 12.9 11.1 12.2 12.0 13.2

Run nos. 21-37: contactor A. b The values of a are predicted by eq 25. c The values of a are predicted by eq 26.

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Figure 7. θALb,eff or θOLb,eff versus τ for ozone dissolution in a RPB contactor. Symbols and lines: experiments and prediction, respectively. O and -‚-, 4 and - -: θALb,eff, θOLb,eff. DaA ) 3.26 × 10-4, DaO ) 7.94 × 10-5, k0LAa ) 0.0975 s-1, k0LOa ) 0.111 s-1, Rm1 ) 1.78, RuGL ) 0.368, StGA ) 0.192, StGO ) 0.028, StLA ) 1.01, StLO ) 1.15, tL ) 4.0 s, T ) 25 °C, uL,1 ) 3.46 × 10-3 m/s, uG,2 ) 4.36 × 10-3 m/s, ω ) 1500 rpm, yA ) 0.0204.

the correlations of eqs 25 and 26. This gives the k0LO values in the RPB in the range of 2.97 × 10-5-1.32 × 10-4 m/s, as listed in Table 3. In addition, the dimensionless groups of ReL and ReG are introduced in the correlation of k0LO as follows.

For complete wetting of packing k0LO (m/s) ) 1.30 × 10-6ReL0.557ReG0.278Gravg0.146 (28) For partial wetting of packing k0LO (m/s) ) 2.64 × 10-6ReL0.320ReG0.283Gravg0.155 (29) Note that the k0L correlation without the effect of gas flow rate proposed by Munjal et al.7 is k0L ) 2.6(Qw/∆X)Sc-1/2ReL,M-2/3Gravg1/6. Further, note that the ReL,M ) 4Qw/νL, implying that k0L is proportional to Qw1-2/3 ()Qw1/3). In eq 29, k0LO is proportional to ReL0.320 or QL0.320. Thus, the exponents of both liquid flow rate and Gravg for the k0L prediction are 0.333 and 0.167 by Munjal et al.7 comparable to 0.320 and 0.155 by eq 29. Apparently, the form of k0LO correlation in this study based on the assumption of partial wetting is closer to that of Munjal et al.7 In addition, the ozone dissolution results of θALb,eff and θOLb,eff versus τ with uL,1 ) 3.46 × 10-3 m/s, uG,2 ) 4.36 × 10-3 m/s, ω ) 1500 rpm, and yA ) 0.0204 are obtained in contactor B. The dynamic variations of ozone and oxygen concentration profiles can be simulated by solving eqs 5-17. The time variations of θALb,eff and θOLb,eff are predicted accordingly based on the present model and operating conditions with the proper values of k0LAa and k0LOa. As shown in Figure 7, the values of k0LAa and k0LOa are determined as 0.0975 and 0.111 s-1, respectively, by optimizing the prediction related to the experimental data. Note that the ratio of k0LA to k0LO is 0.88, which is closer to the value of (DA/DO)0.5 ) 0.89 (for which DA ) 2.0 × 10-9 m2/s and DO ) 2.5 × 10-9 m2/s at 25 °C) than that of DA/DO ) 0.80. The film theory also seems to be applicable with the relative error of about 9% [)(0.88 - 0.8)/0.88]. However, the comparison demonstrates that the penetration and surface renewal theories are more applicable than the film model for describing the gas-liquid mass-transfer behavior in the RPB.

Figure 8. Concentration profiles of θALb or θAGi at various τ for ozone dissolution in a RPB contactor. The conditions are the same as those in Figure 7. Lines: prediction. -‚-, - -, -‚‚-, - - -, and s: τ ) 0.5, 1.0, 1.2, 1.5, and steady state, respectively. (a) θALb; (b) θAGi.

Consequently, the k0LAa values can be calculated with the range of 0.0574-0.131 s-1 from the k0LOa values in Table 3 by applying the penetration and surface renewal theories. In comparison with the k0LAa values of 0.00588-0.0171 s-1 in the BCRs10 and of 0.020.04 s-1 in the stirred reactor,20 the values of k0LAa in the RPB are remarkably greater. Moreover, the time scale for reaching steady state is about 3tL according to the simulation results shown in Figure 7, which is similar to that in BCRs.21 However, the real time for the steady-state establishment in RPBs may be shorter than those of the other types of ozone contactors because of the small tL value in RPBs. In a comparison of the obtained results with the previous studies, one feature of the present modeling is that the transient period of the RPB can be predicted until reaching the steady state. The other one is that the present model considers the gas-liquid mass transfer of two diffusible species (O2 and O3) and the chemical reaction (decomposition of O3). Figure 8 depicts the concentration profiles of θALb and θAGi in the RPB as a function of the dimensionless radial coordinate r* and time τ. For both θALb and θAGi, the highest concentrations are at the outer radius of the RPB and decrease monotonically as r* decreases. It is seen that the shapes of the ozone concentration profiles in the transient state vary obviously with τ. Evidently, θALb and θAGi would approach the steady-state values faster near the liquid (r* ) 0) and gas (r* ) 1) inlets of the RPB, respectively. To characterize the effect of the ozone decomposition reaction on the values of ErA and RfO in RPBs, the Hatta number (Ha) associated with 0LA and kd should be addressed. When Ha < 0.3, the values

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of both ErA and RfO are close to unity. In this so-called slow reaction regime, the effect of the Ha value on the gas-liquid mass transfer is insensitive. On the other hand, the mass-transfer rates of ozone and oxygen would be remarkably enhanced and retarded, respectively, when Ha g 0.3. Regarding this fast reaction regime, it usually occurs in a high pH and ionic strength solution. Furthermore, the values of ErA and RfO would increase and decrease, respectively, with a greater Ha value. If the effect of Ha on ErA and RfO is neglected, the values of k0LAa and k0LOa may be overestimated and underestimated, respectively. The values of k0LA in the RPB can be estimated with the range of 2.64 × 10-51.17 × 10-4 m/s from the values of k0LO. Accordingly, ErA and RfO in the RPB would be significantly different from unity, while kd is greater than 0.03-0.6 s-1, yielding Ha g 0.3. Conclusions (1) The dynamic variations of the ozone and oxygen concentration profiles in a RPB can be well predicted by the present radial convection model from the beginning to steady state. The satisfactory empirical correlations for the system parameters in the RPBs are also developed. (2) The values of liquid holdup (L), mean liquid film thickness (hL), and specific area of the gas-liquid interface (a) in RPBs would increase strongly with an increase of the liquid flow rate (QL) and moderately with a decrease of the rotational speed (ω). However, the influences of the gas flow rate (QG) on L, hL, and a are found to be slight. (3) The volumetric mass-transfer coefficients (k0La) increase significantly with an increase of QG, QL, and ω. Note that 0La of ozone (k0LAa) in a RPB is remarkably higher than that in BCRs and MSRs. Furthermore, the penetration and surface renewal theories are demonstrated to be applicable for describing the mass-transfer behavior in a RPB. (4) The gas (θAGi) and liquid (θALb) concentrations of ozone decrease monotonically with a decrease of the radial coordinate from the outer to inner radii of the RPB. The RPB contactor is allowed to reach steady state for about three hydraulic retention times in this study. θAGi and θALb approach steady state faster near the gas and liquid inlets of a RPB, respectively. Nomenclature a ) specific area of the gas-liquid interface per unit volume of a packed bed, m2/m3 ap ) specific area of packing per unit volume of a packed bed, m2/m3 aw ) specific wetted area of packing per unit volume of a packed bed, m2/m3 CAGi, CAGi0 ) gas concentrations of ozone of holdup and inlet gases, M CALb, CALb,eff ) dissolved ozone concentrations in bulk and effluent liquids of a packed bed, M CALi ) dissolved ozone concentration of a liquid film at the gas-liquid interface, M CG ) total gas concentration in the gas phase, M COGi, COGi0 ) gas concentrations of oxygen of holdup and inlet gases, M COLb, COLb,eff ) dissolved oxygen concentrations in bulk and effluent liquids, M COLb0 ) COLb at the initial time, M

COLi ) dissolved oxygen concentration of a liquid film at the gas-liquid interface, M dp ) particle diameter of packing, m ds ) diameter of the packing material of stainless steel wire, m D ) molecular liquid diffusion coefficient of solute, m2/s DA, DO ) molecular liquid diffusion coefficients of ozone and oxygen, m2/s DaA ) Damko¨hler number of the self-decomposition reaction of ozone, Lkd(r2 - r1)/uL,1 DaO ) Damko¨hler number of oxygen, 3Lkd(r2 - r1)CAGi0HO/ 2uL,1COGi0HA ErA ) enhancement factor of ozone mass transfer defined by eq 2 Fr ) Froude number, L′2ap/FL2g g ) gravitational acceleration, m/s2 G′ ) gas mass flux, ∫rr12(QGFG/r) dr/[2πZB(r2 - r1)], kg/(m2‚s) Gravg ) Grashof number based on the average bed radius, ravgω2(ravg - r1)3/νL2 with ω in rad/s hL ) mean liquid film thickness of packing, m or µm Ha ) dimensionless Hatta number, xkdDA/k0LA HA, HO ) Henry’s law constants of ozone and oxygen, CAGi/ CALi, COGi/COLi, MM-1 HOL ) overall height of a mass transfer unit based on the liquid-phase resistance, cm kd ) self-decomposition rate constant of ozone, s-1 k0LA, k0LO ) physical liquid-phase mass-transfer coefficients of ozone and oxygen, m/s L′ ) liquid mass flux, ∫rr12(QLFL/r) dr/[2πZB(r2 - r1)], kg/(m2‚s) P ) gas pressure, atm QG ) gas flow rate, m3/s QL ) liquid flow rate, m3/s Qw ) liquid flow rate per unit width for a packed bed, m2/s r ) radial coordinate of a packed bed from the center, m ravg ) average (arithmetic) bed radius, m rGB ) position of moving gas flow at time t, m r1 ) inner radius of a packed bed, m r2 ) outer radius of a packed bed, m r* ) dimensionless form of r, (r - r1)/(r2 - r1) rGB* ) dimensionless form of rGB, (rGB - r1)/(r2 - r1) re* ) dimensionless effective radius, r/(r2 - r1) R ) gas constant, 0.082 atm‚L/K‚mol RfO ) retarding factor of oxygen mass transfer defined by eq 3 Rm1 ) ratio of the arithmetic mean radius to the inner radius of a packed bed, (r2 + r1)/2r1 RuGL ) gas-liquid velocity ratio, uG,2L/uL,1G ReG ) Reynolds number of the gas, G′/apµG ReL ) Reynolds number of the liquid, L′/apµL ReL,M ) Reynolds number of the liquid defined by Munjal et al.,7 4Qw/νL Sc ) Schmidt number, νL/D StGA, StGO ) gas Stanton numbers of ozone and oxygen, k0LAa(r2 - r1)/uG,2HA, k0LOa(r2 - r1)/uG,2HO StLA, StLO ) liquid Stanton numbers of ozone and oxygen, k0LAa(r2 - r1)/uL,1, k0LOa(r2 - r1)/uL,1 t ) time, s T ) temperature, °C or K tL ) liquid hydraulic retention time in a packed bed, LVB/ QL, s uG ) superficial gas velocity at r, m/s uG,2 ) inlet superficial gas velocity, m/s uL ) superficial liquid velocity at r, m/s uL,1 ) inlet superficial liquid velocity, m/s UG ) dimensionless superficial gas velocity, uG/uG,2 UL ) dimensionless superficial liquid velocity, uL/uL,1 VB ) volume of a packed bed, π(r22 - r12)ZB, m3 VL ) liquid holdup volume in a packed bed, mL or m3

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We ) Weber number, L′2/FLapσ yA, yO ) mole fractions of ozone and oxygen of an inlet gas ZB ) axial height of a packed bed, m νL ) kinetic viscosity, m2/s δL ) thickness of the liquid boundary layer, m ∆X ) surface renewal parameter (assumed to be dp/2 or dp by Munjal et al.7), m  ) voidage, G + L G ) relative gas holdup in a packed bed L ) relative liquid holdup in a packed bed θAGi ) dimensionless gas concentration of ozone of holdup gas, CAGi/CAGi0 θALb, θALb,eff ) dimensionless liquid concentrations of ozone in bulk and effluent liquids, CALb/(CAGi0/HA), CALb,eff/ (CAGi0/HA) θOGi ) dimensionless gas concentration of oxygen of holdup gas, COGi/COGi0 θOLb, θOLb,eff ) dimensionless liquid concentrations of oxygen in bulk and effluent liquids, COLb/(COGi0/HO), COLb,eff/(COGi0/HO) θOLb0 ) θOLb at the initial time, COLb0/(COGi0/HO) µG ) gas viscosity, kg/m‚s µL ) liquid viscosity, kg/m‚s FG ) density of gas, kg/m3 FL ) density of liquid, kg/m3 Fs ) density of packing materials of a steel wire, kg/m3 σ ) surface tension of the liquid, N/m σc ) critical surface tension of the packing material, N/m τ ) dimensionless time, t/tL ω ) rotational speed, rpm or rad/s (only for Gravg calculation)

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Received for review July 2, 2003 Revised manuscript received September 26, 2003 Accepted October 27, 2003 IE030545C