Modeling and Experimental Study of Oxygen Absorption in Sulfite

Article pubs.acs.org/EF

Modeling and Experimental Study of Oxygen Absorption in Sulfite Solution with Fine Particles Bo Zhao,* Meng Cao, Shujuan Wang, Yan Li, Yuqun Zhuo, and Changhe Chen Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Thermal Engineering, Tsinghua University, Beijing 100084, People’s Republic of China ABSTRACT: A steady bubble gas−liquid mass-transfer model for studying wet limestone scrubbing is established on the basis of the film theory and the principle of mass conservation. The characteristic, size, solid loading, and bubble size of the fine particles have a great influence on gas−liquid mass transfer. The present work is to explain how these factors influence the reaction between Na2SO3 and O2 and find the best reaction conditions through experiment. From the modeling study, the sulfite oxidation reaction rate is affected by the concentration of sulfite, bubble size, and mass diffusion coefficient. The reaction order with respect to the bubble radius is 0 during the kinetic-controlled process and −1 during the diffusion-controlled process. The oxidation rate was experimentally studied by contacting pure oxygen with sodium sulfite solution in the single-bubble system. Theoretical results show good agreement with the experimental data. It has been found that the fine particle and bubble size rather than the mass-transfer coefficient play a more significant role in controlling the overall rate of mass transfer in the system.

1. INTRODUCTION Bubbling systems are widely used in two-phase reactors in chemical, environmental, pharmaceutical, and petrochemical processes.1−3 The stirred tank is the most common reactor type for chemical reactions and energy conversion processes, such as the flue gas desulfurization process. This process requires a containing tank, where CaSO3 and CaSO4 are crystallized, and makeup CaSO3 is dissolved in the tank. Forced sulfite oxidation in the containing tank solves the main problem of the disposal of solid byproduct, i.e., sludge composed of calcium sulfite and sulfate. To reduce SO2 emission, the flue gas desulfurization equipment has been widely set up in many power plants. The stirred-tank system is a commonly used type of reactor for the flue gas desulfurization system. Generally, the presence of chemical reactions will enhance the mass transfer across the gas−liquid interface, which is the determining step of the whole mass-transfer process. Therefore, it is important to describe theoretically the gas−liquid mass-transfer characteristics. The gas−liquid mass-transfer models, such as two-film theory,4 penetration theory,5 surface renewal theory,6 and film-penetration theory,7 have been widely researched in recent decades. Nagy8 used a one-dimensional heterogeneous model to predict mass-transfer rates in three-phase systems. Tobajas et al.9 developed a mass-transfer expression based on Higbie’s penetration theory and Kolmogoroff’s theory of isotropic turbulence to predict the volumetric mass-transfer coefficient in an airlift marine sediment slurry reactor. Shimizu et al.10 proposed a mathematical model based on Higbie’s penetration theory using bubble breakup and coalescence to calculate the gas−liquid interfacial mass-transfer coefficient kL in bubble columns. Kittilsen et al.11 developed a model of gas−liquid mass transfer for a laboratory slurry-stirred reactor for olefin polymerization. Cockx et al.12 established a model to predict the mass-transfer coefficient kl in bubble columns based on a global correlation of the absorption coefficient. As indicated above, many research works on the gas−liquid mass-transfer model in gas−liquid systems have been reported. © 2012 American Chemical Society

Researchers who studied the reaction of sulfite oxidation pointed out the extreme sensitivity of its kinetics to experimental conditions, and its mechanism is very complex. The related experimental results show that the dissolved oxygen, sufite concentration, pH, temperature, and presence of catalyst affect the reation rate. Barron and O’Hern13 researched the oxidation of sulfite using the rapid-mixing method and revealed that the reaction order with respect to sulfite is 3/2 and the reaction order with respect to oxygen is 0; this reaction occurred in homogeneous conditions. Lancia et al. 14 studied the uncatalyzed heterogeneous oxidation of calcium bisulfate in a bubbling reactor and found that the reaction rate in the kineticcontrolled process is 0-order in oxygen and 3/2-order in bisulfate. Hjuler and Dam-Johansen15 studied the oxidation of calcium sulfite in aqueous slurries of residual product from spray absorption of sulfur dioxide for heterogeneous conditions typical of wet scrubbers and indicate a 0-order dependence from sulfite and a 1-order dependence from oxygen without additional catalyst; with manganous sulfate added as a catalyst, the order is about 0.5 and 1.0, respectively. Linek and Vacek16 had reviewed the results of sulfite oxidation reaction kinetics and mechanism and also the mass-transfer characteristic determination of absorption equipments during the last 40 years. An increase of the mass-transfer coefficient when adding fine particles, such as activated carbon, had been observed in a stirred cell with a flat gas−liquid interface.17−22 There were a lot of mechanisms proposed for the mass-transfer enhancement. In 1980, Alper et al. invented a mechanism called the “shuttle mechanism”. This mechanism was based on the assumption that the particles adsorb an additional amount of the absorbed gas at the gas−liquid interface or within the liquid Special Issue: 7th International Symposium on Coal Combustion Received: November 14, 2011 Revised: January 14, 2012 Published: January 16, 2012 3132

dx.doi.org/10.1021/ef201777n | Energy Fuels 2012, 26, 3132−3140

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film, after which this adsorbed gas desorbs from the particles in the liquid bulk. After that, various stationary and non-stationary models, using the shuttle mechanism, have been developed to describe the effects of particles on the mass-transfer enhancement.17,20,23−25 Although the “shuttle mechanism” is a very useful mechanism, it has some limits. For example, Holstvoogd et al.26 informed the validity of the shuttle mechanism showing that about a 100-fold higher activated carbon concentration, as compared to the bulk, is required to explain the effect. Kluytmans et al.21 also doubt this mechanism, because they found no dependency of the mass-transfer enhancement upon the carbon particle concentration in the stirred cell, which is not expected with the mechanism. On the basis of their experimental results, they proposed a new explanation, which is that the enhancement is caused by an increased level of turbulence at the gas−liquid interface caused by carbon particles present in a boundary layer. However, most of them are only concerned with the measurements of the gas−liquid volumetric mass-transfer coefficient and the interfacial area in the gas−liquid systems. Some of them also discussed mass-transfer modeling, but most masstransfer models were either confined to correlations within specific system operating conditions or take into account parameters with only limited influence. Although there are many models on the gas−liquid mass-transfer system, it seems that the effect of the bubble size has been ignored. In this study, a model has been developed considering not only the mass transfer and kinetics but also the influence of the bubble size on the reaction rate. The model has been verified by experiments via a new type image analysis method.27

The mass-transfer equation of a single bubble can be expressed in the differential equation according to Fick’s law. ṁ ″SO 2 − = Y SO 2 −(ṁ ″SO 2 − + ṁ ″SO 2 −) 3 3 3 4 dY SO 2 − 4 − ρD dr

Taking into account that the equation ṁ i = ṁ ″i4πr2 is fit to any kind of substances, when eq 2 is substituted into eq 3, the following expression can be obtained: ṁ O2 =

1 O2 → SO4 2 − 2

3

MW SO 2 − 3

MWO2

3

3

dr

⎧Y = Y SO 2 −,s r = rs ⎪ SO32 − 3 ⎨ ⎪ Y SO 2 − = Y SO 2 −, ∞ r → ∞ ⎩ 3 3

(4)

(5)

where D is the liquid-phase diffusion coefficient and r is the bubble radius. Equation 4 is a linear first-order differential equation with its solution as ⎛ ⎞ Y SO 2 −, ∞ − Y SO 2 −,s 3 3 ⎜ ⎟ ṁ O2 = 4πrsρD ln 1 + ⎜ 2MW SO 2 −/MWO2 + Y SO 2 −,s ⎟ ⎝ ⎠ 3 3 (6)

This logarithmic function can be expanded by Taylor expansion as ⎛ ⎞ Y SO 2 −, ∞ − Y SO 2 −,s 3 3 ⎟ ṁ O2 = 4πrsρD⎜ ⎜ 2MW 2− ⎟ 2 −/MWO + Y SO3 SO3 ,s ⎠ 2 ⎝

(7)

By changing the mass fraction to mole fraction, eq 6 can be converted into a linear equation. In turn, ṅO2 can be expressed as the ratio of “potential difference” over “resistance” nO ̇ 2=

(c SO 2 −, ∞ − c SO 2 −,s) Δc 3 3 = ⎛ 2ρ + c SO 2 −,s MWO2 ⎞ R diff 3 ⎜⎜ ⎟⎟ 4 r D π ρ s ⎝ ⎠

(8)

where (1)

R diff =

2ρ + c SO 2 −,s MWO2 3 4πrsρD

(9)

The term “Rdiff” is defined as the diffusion resistance. 2.2. Fine-Particle Influence. The mass-transfer enhancement factor is affected by the particle size and analyzed using a homogeneous steady-state mass-transfer model based on the SRPIA model,28 shown in Figure 1. The species on A in the coverage part of the interface is given by

ṁ O2 and ṁ SO 2 − 4

⎛ MW SO 2 − ⎞ 3 ⎟ = ⎜⎜1 + 2 ⎟ṁ O2 MW O2 ⎠ ⎝

d(Y SO 2 −MWO2/2MW SO 2 −)

with the following initial and boundary conditions:

and (5) the overall mass-transfer rate can be obtained by calculating the bubble volume decrease per unit time. 2.1. Mass Conservation. From eq 1, the relation between the mass flow rate of sulfite and sulfate radicals in terms of oxygen is obtained. ṁ SO 2 − = 2

4πr 2ρD (1 + Y SO 2 −MWO2/2MW SO 2 −) 3 3 ×

2. MATHEMATICAL MODEL AND SOLUTION The oxidation reaction of a single spherical pure oxygen bubble has been studied subject to the following assumptions. The solution phase acts as a continuous phase, and an oxygen bubble with adjustable size in the solution exists as a discrete phase. This model is focused on a single bubble with changing size controlled by the bubble generator. To model the masstransfer process of the single bubble, it is assumed that (1) the gas−liquid reaction process is pseudo-steady-state, (2) the shape of the bubble is a sphere of given radius, (3) the single bubble, full of pure oxygen in a quiescent, mass-transfer resistance, exists only in the liquid film of the liquid side (the effects of convection are ignored), (4) on the interface, the oxygen reacts kinetically with the sulfite radical to produce the sulfate radical according to the following reaction: SO32 − +

(3)

∂cA ∂ 2c = DA 2A ∂t ∂x (2)

(10)

where cA is the liquid-phase concentration of A. 3133


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In zone II: c ⎞ ∂cÃ (x , s) ⎛c = − s /DA ⎜ d − b ⎟e s / DA R2 − s / DA x ⎝ ∂x m s ⎠ (14)

The mass-transfer flux in zone I can be expressed as ji =

∫0

+∞

⎛ ∂c ⎞ ⎛ dc ̃ ⎞ −DA ⎜ A ⎟ se−st dt = −DA s⎜ A ⎟ ⎝ ∂x ⎠x = R ⎝ dx ⎠ x = R 1

1

(15)

When eq 13 is substituted into eq 15, the value of j1 can be abtained. In zone II: j2 =

∫0

+∞

⎛ d∼ ∂c c ⎞ −DA ( A )x = R2 se−st dt = −D As⎜⎜ A ⎟⎟ ∂x ⎝ dx ⎠ x = R

2

Figure 1. SPRIA model.

(16)

The boundary conditions are

In a similar manner, when eq 14 is substituted into eq 16, the value of j2 can be abtained. Therefore, the average mass-transfer flux on a single particle can be calculated.

t = 0, 0 ≤ x ≤ R1, cA = c b zone I: t > 0, x = 0, cA = c* t > 0, x = R1, cA = cd/m

j = j1 − j2 2md p

=

t = 0, R2 ≤ x , cA = c b

DA s (c* − c b)

zone II: t > 0, x = R2 , cA = cd/m t > 0, x → ∞ , cA = c b

δL

⎡ zone I zone II ⎤ ⎛ ∂cd ∂cA ⎞ ∂cA ⎞ 3 ⎢⎛ ⎥ = − ⎜ −DA ⎜ −DA ⎟ ⎟ ⎥ ∂t dP ⎢⎝ ∂x ⎠ ∂x ⎠ ⎝ x = R2 ⎦ x = R1 ⎣

If m = 1, then the mass-transfer flux in the absence of particles is given by 2d p

j0 =

DA s (c* − c b) dp δL

1

(e R1/ δL − e−R1/ δL) + 6e R1/ δL

The enhancement factor of a single particle is defined as

(11)

R /δ R /δ ⎤ −R / δ ⎢ (e 1 L − e 1 L) + 6e 1 L⎥ δL ⎣ δL ⎦

2md p ⎡ d p

Ep =

zone I

⎛ ⎛ ∂c ̃ ⎞ ∂c ⎞ L⎜ −DA A ⎟ = −DA ⎜ A ⎟ ⎝ ⎠ ⎝ ∂x ⎠x = R ∂x x = R

δL

(18)

Equation 11 can be solved using Laplace transform

1

(e R1/ δL − e−R1/ δL) + 6e R1/ δL (17)

where m is the distribution coefficient for partitioning of A between solid and liquid phases and cd is the solid-phase concentration of A. The balance for the accumulation of A on a single particle taken to be a sphere is given by

zone I

δL md p

j = j0

md p δL

(12)

(e R1/ δL − e−R1/ δL) + 6e R1/ δL (19)

In zone I:

Therefore, the total enhancement factor can be expressed as

c ⎞ ∂cÃ (x , s) ⎛ cd = ⎜ − b ⎟(( s /DA e s / DA x ⎝ ∂x m s ⎠ +

E = 1 − α + αE p

where α is the interface fractional coverage of particles. 2.3. Interface Kinetic Analysis. The interface kinetic was studied by many researchers. Lancia et al.14 used the numerical simulation method to obtain a kinetic equation of order 3/2 with respect to sulfite concentration and used the experimental29 method to obtain the same results. Therefore, the interface kinetics may be written as eqs 21 and 22.

s /DA e− s / DA x)/(e s / DA R1

− e− s / DA R1)) −

s /DA (((c* − c b)

(e s / DA R1− s / DA x + e− s / DA R1+ s / DA x)) /(s(e s / DA R1 − e− s / DA R1)))

(20)

v = ṁ ″O2,s /MWO2 = kc 3/22 − SO3 ,s

(13) 3134

(21)


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nO ̇ 2 = 4πrs2kc 3/22 − = SO3

(c SO 2 −,s − 0) 3

Then, the reaction rate could be obtained from eq 22.

1

,s

4πrs2kc 1/2 2 − SO3

,s

v (mol s−1 m−2) =

(22)

If assuming that R kin =

SO3

,s

(23)

The term “Rkin” is defined as the kinetic resistance. Then, eq 22 could be rewritten as eq 24. c SO 2 −, ∞ − c SO 2 −,s c SO 2 −,s − 0 3 3 3 nO = ̇ 2= R diff R kin

(24)

(29)

regime

Rkin/Rdiff

reaction rate law

diffussion controlled

≪1

• nO2 = c SO 2 −, ∞/R diff 3

intermediate

≈1

kinetic controlled

≫1

•

nO2 = c SO 2 −, ∞/(R diff + R kin) 3 •

nO2 = c SO 2 −, ∞/R kin 3

3. EXPERIMENTAL APPARATUS AND DATA PROCEDURE

3

(2ρ + c SO 2 −,s MWO2)c 1/2 2 − 3 SO3 ,s

,∞

SO3

the kinetic-controlled process, while the order with respect to the bubble radius is −1 during the diffusion-controlled process.

R kin ρD 1 = 1/2 R diff 2ρ + c SO 2 −,s MWO2 rskc 3 SO 2 −,s 1

= kc 3/22 −

Table 1. Summary of Sulfite Oxidation Regimes

2.4. Limiting Cases. Dependent upon the sulfite concentration and bubble size, primarily, one of the resistances may be much larger than the others, thus causing ṅO2 to depend essentially only upon that resistance. The definitions of Rkin and Rdiff (eqs 9 and 23) are used to take their ratio, and then eq 25 is obtained.

=

S

Table 1 summarizes our discussion of the limiting regimes of sulfite oxidation. According to a previous analysis, the conclusion of singleoxygen bubble oxidation in the sulfite liquid condition could be drawn. The order with respect to the bubble radius is 0 during

1 4πrs2kc 1/2 2 −

nO ̇ 2

⎛ ρD ⎞⎛ 1 ⎞ ⎜ ⎟⎜ ⎟ ⎝ k ⎠⎝ rs ⎠

3.1. Experimental Apparatus. Figure 2 shows the test rig for the oxygen bubble generation system and the bubble image recording system. A rectangular vessel made of transparent glass was used to hold the sulfite solution. The vessel was sealed to prevent contact between the air and the solution. Nitrogen was added to further protect the sulfite solution from oxidization by oxygen in the air. The vessel volume was about 8 L (200 × 200 × 200 mm). A pure oxygen bubble was generated at an orifice in the bottom plate. A gas-handling system provided precise control of the oxygen flow. At the beginning of the experiment, a pure oxygen bubble was generated in the sodium sulfite solution. Experiments showed that bubbles with diameters less than 4 mm stayed attached to the orifice because of the surface tension. The oxygen was then gradually absorbed by the sulfite reaction, which reduced the bubble volume. The CCD camera (DCR-TRV20E, Sony) recorded the magnified bubble image projected on a frosted glass plate with the images sent to the computer for later analysis. Sample bubble images captured by the CCD camera and the boundary identification images at different times are shown in Figures 3 and 4 during one reaction process. 3.2. Data Procedure. v is defined as the sulfite oxidation area reaction rate, and eqs 30 and 31 give the numeration method. Many researchers gave the volume reaction rate because they could not obtain the exact gas−liquid contact area. This method was experimentally studied by contacting a pure oxygen bubble with a sulfite solution and obtaining the exact contact area by the optical and mathematical method

(25)

2.4.1. Diffusion-Controlled Process. In the case of Rkin/Rdiff ≪ 1, the reaction rate is said to be diffusion-controlled. How individual parameters affect this quantity could be seen from eq 25. This ratio can be made small in several ways. First, k can be very large; this implies a fast reaction regime. It is also known 1/2 2− that a large bubble size, rs, or high sulfite concentration, cSO , 3 ,s has the same effect. The reaction rate could be written as eq 26. c SO 2 −, ∞ − 0 3 nO ̇ 2= R diff (26) The reaction is controlled by the diffusion; therefore, the sulfite concentration at the interface could be assumed to be zero.30 Then, the reaction rate could be obtained from eq 27. c SO 2 −, ∞ nO ̇ 2 3 v (mol s−1 m−2) = =D S 2rs (27) 2.4.2. Kinetic-Controlled Process. In the case of Rkin/Rdiff ≫ 1, the reaction rate is said to be kinetic-controlled. In this case, Rdiff is small and cSO32−,s is close to cSO32−,∞. The chemical kinetic parameters control the reaction rate, and the mass-transfer parameters are unimportant. Kinetically controlled reactions occur when bubble sizes are small, and some reasons causes k to be small, such as the low sulfite concentration and no catalyst condition. The reaction rate could be written as eq 28. c SO 2 −, ∞ − 0 3 nO ̇ 2= R kin (28) The reaction is controlled by kinetics; therefore, we could assume that the sulfite concentration at the interface is cSO32−,∞.30

v=

N=

dN S dt

(30)

pO V 2

8.31T

(p + p l + ps − pv )V = 0 8.31T =

(p0 + ρgh + 2σ/R − pv )V 8.31T

(31)

where N is the moles of oxygen in the bubble, pO2 is the pressure in the bubble, V is the bubble volume, T is the thermodynamic temperature, p0 is the atmospheric pressure, pl is the hydraulic pressure, ps is surface tension pressure, pv is vapor partial pressure, ρ is the sulfite concen3135


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Figure 2. Test rig of the oxygen bubble generating system and the bubble image recording system: 1, light source; 2, lens, pin holes, and filters; 3, oxygen bubble; 4, convex lenses and frosted glass; 5, charge-coupled device (CCD) video camera; 6, syringe and pump; 7, valve; 8, plastic tube; 9, heater; 10, thermometer; 11, analog-to-digital (A/D) and personal computer (PC); and 12, monitor. these parameters. Figure 5 showed a series of photos about one bubble absorption process. According to previous analysis, the bubble volume and surface changed by the reaction time were obtained. Figure 6 showed that the bubble volume and surface were nonlinearly changed with the reaction time. To investigate the relation between the reaction rate and the bubble size, requal was named as the character radius, which was shown in eq 32. The experimental oxygen bubble was not a standard sphere; therefore, the ratio between the 3 times volume and the bubble surface were used to express the character ratio.

Figure 3. Bubble image captured at 0 s and edge determined from the bubble image.

requal = 3VO2/AO2

(32)

From the above assumption, the profile of the bubble radius with the reaction time was obtained as Figure 7, which showed that the bubble radius was not linearly decreasing. Analizing the series of photos that were captured by the imagine capture system, as Figure 3 showed, the relation between the reaction rate with the bubble size was obtained. Figure 8 showed that the reaction rate was increasing with the bubble radius decreasing, which was an important result. Many kinetic studies13−15 were concerned about the relation between the reaction rate with the sulfite concentration, dissolved oxygen, pH, temperature, and even traces of catalysts (Co2+, Cu2+, and Mn2+) and inhibitors (alcohols, phenols, and hydroquinone). However, their experimental results were affected by the bubble size, because their system could not control the bubble in a close size. Zhao et al.27 had reported some conclusions that the sulfite oxidation reaction rate was affected by the mass-transfer characteristics of the gas−liquid absorbers with the cobalt ion catalyst as well as the reaction kinetics. In the condition of a high sulfite concentration and catalyst concentration, the sulfite oxidation reaction rate was controlled by the mass transfer, while in a low sulfite concentration and without catalyst condition, the sulfite oxidation reaction rate was controlled by the reaction kinetics. Therefore, the typical mass-transfer-controlled

Figure 4. Bubble image captured at 1080 s and edge determined from the bubble image. tration, σ is the sulfite surface tension coefficient, R is the bubble radius of curvature, and h is the bubble depth. The image processing software recognized the height and the bottom of the bubble and tried to fit the edge curve to the Young− Laplace equation. Thus, the bubble volume, the interfacial area, and the radius of curvature at the top of the bubble were derived from the image-processing results. Figures 3 and 4 showed two photos that have been recognized by the software. As the pure oxygen was filled in the bubble, the sulfite oxidation reaction rate were then determined from

Figure 5. Bubble’s life in the absorption process. 3136


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Figure 8. Reaction time with the bubble ratio.

In model calculations, the effect of the particle size on the distribution coefficient m is ignored, if assuming m = 54, and Table 2. Experimental Conditions size (μm)

loading (kg/m3)

temperature (°C)

1 5 10 20 100

0.1 0.2 0.4 0.6 0.8

40 50 60 70 80

then the comparison between experimental and calculation reFigure 6. Bubble volume and surface with the reaction time.

sults is shown in Figure 9.

Figure 7. Character radius against the reaction time. reaction process and typical kinetic-controlled reaction process were chosen to verify the single-bubble model theory.

Figure 9. Comparison between experimental and theoretical results.

4. RESULTS AND DISCUSSION 4.1. Effect of the Particle Size. Previous studies have shown that the mass transfer could be enhanced in the presence of fine active carbon particles, and our experiments were conducted in a stirred tank while adding active carbon particles into sulfite solution. The experimental conditions are listed in Table 2.

In Figure 9, the red curve represents calculation values using the model, while the black curve represent the experimental results; theoretical results agree well with the experimental data. 4.2. Effect of the Solid Loading. The effect of solid loading was studied at 50 °C. The concentration of sulfite is 0.5 mol/L. The stirring speed is 200 revolutions/min. Figure 10 shows the effect of the solid loading on the mass-transfer coefficient with the particle size of 10 μm. According to the 3137


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Therefore, the average enhancement factor can be calculated by eq 36. E=

1 δ − 2R

δ− R

∫R

E(χ)dχ

(36)

When the solid loading is too high, the particles will contact each other for adjacent cells and form a solid sheet that is impermeable to gas, as shown in Figure 12. Finally, the value of

Figure 12. Schematic of the plate model.

Figure 10. Effect of the solid loading.

RA(χ) will become small, which leads to the decrease of the mass-transfer enhancement factor. 4.3. Effect of the Bubble Size. To choose the typical mass-transfer-controlled reaction process, the reaction temperature of 308 K, the sulfite concentration of 0.3 mol/L, and the manganese ion concentration of 2.62 × 10−5 mol/L were set. The plots in Figure 13 were the experimental results, which was

“shuttle mechanism”, fine particles become into the gas−liquid boundary because of the stirrer and natural convection. Then, fine particles can take more gases, which prepare to transfer into the liquid than normal and release gas when particles become into the liquid bulk because of natural convection. That means more particles can take more gases. However, when there are too many particles, the mass-transfer coefficient decreases. To explain the phenomenon, a cell model was proposed by Karve and Juvekar.29 The schematic of the model is shown in Figure 11. A single particle is located at a random

Figure 11. Schematic of the cell model proposed by Karve and Juvekar.

distance χ from the particle center to the interface. The concentration profile of species A can be described by eq 33 ∇2 A − ρpk1A = 0

Figure 13. Reaction rate with the character ratio.

one reaction process in different bubble sizes. The curve was the linear equation relating requal and RO2 in the logarithmic coordinate. Equation 37 showed that the order with respect to the bubble radius is −1 during the diffusion-controlled process.

(33)

where ∇2 is the Laplacian in the (r,z) coordinates, assuming angular symmetry. In the model presented by Karve and Juvekar, the mass-transfer enhancement factor corresponding to a particular location χ is calculated by eq 34 R (χ) E(χ) = A kLA*

log R O2 = − 1.04 log requal − 6.24

Experimental results could improve the conclusion, which was noticed in the model analysis that the order with respect to the bubble radius is −1 during the diffusion-controlled process. This process occurred in the condition of a high sulfite concentration, high temperature, or using a catalyst. Many researchers did experiments on the sulfite-forced oxidation reaction, while their experimental condition was mostly controlled by the mass-transfer process. They mainly used

(34)

where RA is the rate of absorption of gas A per unit interfacial area. The distance χ is variable, because there is an equal probability of finding a particle at any diatance from the interface in the range R

Modeling and Experimental Study of Oxygen Absorption in Sulfite

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