Modeling Wet Air Oxidation of Sodium Acetate in a Bubble Column

Abstract

Scale-up bubble columns for wet air oxidation (WAO) represent a novel solution to the growing problem of sodium acetate-containing waste discharge. This study used an axial dispersion model to simulate a reactor at a high temperature of 320 °C. By minimizing the structure size and gas fed into the system, the estimated optimal reactor dimensions were obtained. At the optimized reactor diameter, total height, initial oxygen partial pressure, and superficial gas velocity of 1 m, 7 m, 40 bar, and 0.07 m/s, respectively, a degradation efficiency of over 90% was obtained, at which the residual concentration of sodium acetate degraded from 700 mol/m³ to 70 mol/m³. The axial distributions of the dissolved oxygen, sodium acetate, and oxygen concentrations in the gas were nearly uniform. The bubble column reactor exhibited an intermediate state between well-mixed flow and plug flow for both liquid and gas.

1. Introduction

A large amount of waste residue is produced in the industrial manufacture of dithianon, which is composed of sodium acetate (CH₃COONa), dimethyl sulfoxide, naphthalene, anthraquinone, and pigment. CH₃COONa is the primary component of the waste residue, with the other impurities uncharged. Currently, waste residue is treated via conventional methods such as burning and landfilling, both of which cause air, soil, or water pollution directly or indirectly [1]. In recent years, different kinds of powerful processes have emerged for mineralizing or degrading organic pollutants. The advanced catalytic oxidation processes through ozonation, hydrogen peroxide or catalyst is one of the typical techniques [2,3]. Other processes such as thermo oxidative degradation also garner popular attention [4,5].

The wet air oxidation (WAO) process, first patented by Zimmerman over 50 years ago, removes organic compounds in the liquid phase by completely oxidizing them to carbon dioxide and water using an oxidant such as oxygen or air [6]. WAO is carried out at elevated temperatures (150–320 °C) and pressures (2–20 MPa) [7]. The process is extremely clean because it does not involve the use of harmful chemical reagents, and the final products obtained are carbon dioxide and water if complete oxidation is achieved.

Bubble columns are one of the most efficient gas–liquid contactors, with optimal heat transfer [8], mass transfer [9] and chemical reaction [10]. In a bubble column, the gas phase is highly dispersed in the liquid phase and there is a large liquid loading capacity and interphase contact surface. The use of bubble columns for the WAO of sodium acetate represents a promising technological approach [11], but commercial-scale plants have not been developed or are still in the design stage. In addition, the WAO of sodium acetate in a bubble column has an uncertain scale-up disadvantage, and it is difficult to understand the reaction kinetics and flow patterns [12].

There is also a relative lack of literature concerning the mathematical modeling of sodium acetate WAO. Most reports are based on computational fluid dynamics models formulated in the steady-state and focus on the performance of the hydrodynamic variables inside the reactor and not on pollutant conversion [9,13,14]. As a result of the difficulties in obtaining proper information, all previous reactor models were validated using air–water systems under normal temperature and pressure conditions [15,16], although there are modeling reports on WAO in industrial and pilot plant units [17,18]. The effectiveness and economic viability with which a WAO process can be applied to an industrial problem are highly influenced by the detailed reactor design used [6]. For these reasons, the WAO process poses significant challenges in terms of chemical reactor engineering and design.

It is known that a kinetic model can be implemented independently within a reactor model if the reaction rate coefficients are obtained under a kinetic regime. In a typical WAO reaction, sodium acetate in an aqueous solution is oxidized in a bubble reactor. Pressurized air enters the reactor through the bottom of the reactor using a bubble generation device and the air bubbles flow upward. The oxygen in the air dissolves in the water phase and reacts with sodium acetate. This oxygen transfer process is illustrated in Figure 1.

2. Mathematical Modeling

The mathematical model developed in this study included reaction kinetics, a hydrodynamic model, and mass transfer behavior. The following assumptions were used in constructing the model: (1) The bubble size varies with the gas flow rate. (2) There are no radial velocity or density gradients in the liquid or gas phases. (3) The density of the gas phase remains constant. (4) The bubble diameter remains constant in the bubble column. We note that increasing temperature generally yields a dominant trend of associated reduction in liquid viscosity and surface tension, which in turn leads to average bubble sizes and a narrower bubble-size distribution [19], thus allowing the bubble diameter to be treated as a constant value in the reactor. (5) Gas holdup and the mass transfer coefficient are independent of the local height. (6) There are isothermal conditions in the reactor. Note that the temperature of the reactor is kept constant.

2.1. Reaction Kinetics

We referred to a bench-scale WAO experiment conducted in the literature [20], from which we adopted the reaction rate equation for the oxidation of sodium acetate. The rate of disappearance of sodium acetate is denoted with

$$r_1=k_1{e}^{-E/RT}{C}_{Na}{\left(C_{O_2}\right)}^{0.37}$$

(1)

where 270 < T < 320 °C and 36 mol/m³ < $C_{O_2}$ < 136 mol/m³. The kinetic parameter $k_1$ is equal to 5.6 × 10¹⁰ s⁻¹(mol/m³)^−0.37; activation energy E is equal to 167.7 kJ/mol. The activation energy is a function of temperature. There are isothermal conditions in the reactor. Note that the activation energy is kept constant.

The rate of disappearance of dissolved oxygen can be denoted with

$$r_2=2r_1$$

(2)

2.2. Hydrodynamic Model

The gas holdup and bubble size can used for calculating the interfacial area of gas and liquid. The terminal velocity of bubbles will be used for calculating the slip velocity of bubbles and the liquid side mass transfer coefficient.

The gas holdup of a bubble column reactor is expressed as [21]

$${\epsilon }_G=0.672{\left(\frac{U_G{\mu }_L}{{\sigma }_L}\right)}^{0.578}{\left(\frac{{\mu }_L^{4}g}{{\rho }_L{\sigma }_L^3}\right)}^{-0.131}{\left(\frac{{\rho }_G}{{\rho }_L}\right)}^{0.062}{\left(\frac{{\mu }_G}{{\mu }_L}\right)}^{0.107}$$

(3)

A model proposed by Jamialahmadi was used to predict the terminal velocity of bubbles over a wide range of gas and liquid properties [22]. This correlation for the bubble terminal velocity is suitable for high temperature regimes [23]:

$$U_{\infty }=\frac{U^{stokes}{U}^{Mendelson}}{\sqrt{{(U^{stokes})}^2+{(U^{Mendelson})}^2}}$$

(4)

$$U^{stokes}=\frac{1}{18}\frac{{\rho }_L-{\rho }_G}{{\mu }_L}g{d}_B^2$$

(5)

$$U^{Mendelson}=\sqrt{\frac{2{\sigma }_L}{d_B({\rho }_L+{\rho }_G)+\frac{g{d}_B}{2}}}$$

(6)

There are two well-known correlations of bubble size by Akita et al. and Wilkinson et al., respectively. Because the correlation proposed by Wilkinson et al. was obtained for a large-diameter pipe with large sparger openings and better predicts the Sauter mean diameter by considering the gas density [24], it was adopted to assess the bubble size in this study [25]:

$$\frac{g{\rho }_L{d}_B^2}{{\sigma }_L}=8.8{\left(\frac{{\mu }_L{u}_{sg}}{{\sigma }_L}\right)}^{-0.04}{\left(\frac{{\rho }_L{\sigma }_L^3}{g{\mu }_L^4}\right)}^{-0.12}{\left(\frac{{\rho }_L}{{\rho }_G}\right)}^{0.22}$$

(7)

It should be noted that smaller bubble size may be obtained by reducing the pore size in the sparger. The accompanying problem is that the smaller pore size of perforate plate increased the pressure drop of gas. Using the parameters in the equations above, the interfacial area can be expressed as

$$a=\frac{6{\epsilon }_G}{d_B}$$

(8)

2.3. Mass Transfer

Based on these simplifications and assumptions, it can be assumed that the consumption of sodium acetate in the liquid occurs only in the riser region. The axial dispersion models for the liquid and gas phases in a riser are expressed as follows [26].

For dissolved oxygen in liquid:

$$r_2=-\frac{U_L}{1-{\epsilon }_G}\frac{\partial C_{L,O_2}}{\partial z}+D_L\frac{{\partial }^2{C}_{L,O_2}}{\partial z^2}+\frac{k_{L}a}{1-{\epsilon }_G}(C_{O_2}^{\ast \ast }-C_{L,O_2})$$

(9)

For the oxygen component in gas:

$$0=-\frac{U_G}{{\epsilon }_G}\frac{\partial P_{O_2}}{\partial z}+D_G\frac{{\partial }^2{P}_{O_2}}{\partial z^2}-\frac{k_{L}a}{{\epsilon }_G}(C_{O_2}^{\ast \ast }-C_{L,O_2})$$

(10)

For sodium acetate in liquid:

$$r_1=\frac{U_L}{1-{\epsilon }_G}\frac{\partial C_{L,S}}{\partial z}-D_L\frac{{\partial }^2{C}_{L,S}}{\partial z^2}$$

(11)

where $U_L$ and $U_G$ are the superficial liquid and superficial gas velocities, respectively; ${\epsilon }_G$ is the gas holdup of the bubble column reactor; $z$ is the local height of the reactor; $D_L$ and $D_G$ are the axial dispersion coefficients in the liquid and gas phases, respectively; $C_L$ is the actual dissolved oxygen concentration at a given height; $C_G$ is the oxygen concentration in the gas phase at a given height; and $C_{O_2}^{\ast \ast }$ is the saturated dissolved oxygen concentration corresponding to the local oxygen partial pressure at a given height.

The axial dispersion model commonly yields results similar to those of the tank in series model. The tank in series model was also widely used for scaleup reactors [27]. More advanced calculation with tank in series model will be achieved in the future.

The axial dispersion coefficient of the liquid phase was calculated using a modified version of the formula by Deckwer et al. [28]:

$$D_L=0.678D_T^{1.4}{U}_G^{0.3}$$

(12)

The axial dispersion coefficient of the gas phase was calculated using a modified version of the formula by Pavlica and Olson [29]:

$$D_G=5D_T(U_G/{\epsilon }_G)$$

(13)

The volumetric mass transfer coefficient is a function of the interfacial area and transfer coefficient and is expressed as

$$k_{L}a=k_L\frac{6{\epsilon }_G}{d_B(1-{\epsilon }_G)}$$

(14)

The transfer coefficient $k_L$ can thus be estimated using the classical penetration theory of Higbie:

$$k_L=\frac{2}{\sqrt{\pi }}\sqrt{\frac{D_L}{\tau }}$$

(15)

where $\tau =d_B/U_{\infty }$. The mass transfer coefficient can be expressed using

$$k_L=\frac{2}{\sqrt{\pi }}\sqrt{\frac{D_{L,O_2}{U}_{\infty }}{d_B}}$$

(16)

where $D_{L,O_2}$ is the molecular diffusion coefficient and $U_{\infty }$ represents the bubble terminal velocity.

The local saturated dissolved oxygen concentration $C_{O_2}^{\ast \ast }$ varies with the partial pressure of oxygen $P_{O_2}$.

The saturated dissolved oxygen concentration and diffusion coefficient can be obtained using an empirical formula by Tromans [30]. It is based on a thermodynamic analysis for water. The Henry coefficient is a temperature-dependent function related to the chemical potential, entropy, and partial molar heat capacity. This empirical formula by Tromans has been widely used in chemical and environmental engineering [31,32].

$$C^{\ast \ast }=He{P}_{O_2}$$

(17)

where $He$ is the Henry coefficient, which can be calculated from the following temperature-dependent correlation:

$$He=\exp \left(\left(0.046T^2+203.35T\ln (T/298)-(299.378+0.092T)(T-298)-20591\right)/\left(RT\right)\right)$$

(18)

2.4. Numerical Solutions of the Mass Transfer Model

The system of partial differential equations was converted into a set of ordinary differential equations using the finite difference method for space coordinates [33]. The reasonable boundary condition was used to constrain the mass transfer model [34]. All simulations were performed using the MATLAB commercial software package with the bvp4c function solver. The solver code is provided in the supporting information (see Supplementary Information).

The reactor model had a reaction temperature of 320 °C, an initial sodium acetate concentration of 700 mol/m³, and a liquid flow rate of 1 m³/h. Other detailed conditions are listed in

Table 1. Variables used in modeling.

Variable	Nomenclature	Value	Unit
Reaction temperature	T	320	°C
Reaction pressure	P	20, 40, 60	bar
Diffusion coefficient of oxygen in water	D	5 × 10−8	m2/s
Viscosity of liquid		8 × 10−5	Pa·s
Viscosity of gas		2 × 10−5	Pa·s
Density of liquid		874	kg/m3
Density of gas		11,8, 23.6, 35.4	kg/m3
Surface tension of liquid		0.012	N/m
Henry coefficient		5.3	mol/(m3bar)

.

3. Results and Discussion

3.1. Effect of Reactor Total Height and Initial Oxygen Partial Pressure

A bubble column reactor with a diameter of 1 m was initially used for the WAO of sodium acetate. The superficial gas velocity was 0.08 m/s. The column diameter of 1 m and the superficial gas velocity of 0.08 m/s were the assumed values. The assumed values were kept constant and put into the equations, after which we could obtain the effect of reactor total height and initial oxygen partial pressure on the oxygenation performance.

The corresponding superficial liquid velocity was 3.54 × 10⁻⁴ m/s. The total reactor height was varied from 0.1 to 20 m, and the oxygen partial pressure at the gas feed was varied from 20 to 40 and 60 bar. The effect of reactor total height on the residual concentration of sodium acetate at the liquid outlet is shown in Figure 2, from which it is seen that the residual concentration decreased with increasing reactor total height; increasing the total reactor height caused the concentration reduction curve to flatten. At the oxygen partial pressure for the gas feed at 60 bar, a reactor with a total height of 5 m had a degradation efficiency of approximately 90%; at the oxygen partial pressure for the gas feed at 20 bar, a total reactor height of more than 15 m was needed to achieve this degradation efficiency, at which the residual concentration was 70 mol/m³. At the oxygen partial pressure for the gas feed at 40 bar, the total reactor height required to reach 90% efficiency was 7 m. Because the reaction rate is related to the concentration of dissolved oxygen, the oxygen partial pressure has an important influence on the total degradation efficiency.

The effect of the reactor total height on the oxygen partial pressure in the vent gas is shown in Figure 3. It is seen that, at lower total reactor heights, the oxygen partial pressure in the vent gas decreased with increasing height and increased slightly at larger heights. This indicates that the dissolved oxygen consumption per unit time was faster and slower, respectively, when the total reactor height was small and large, respectively. The results in Figure 2 indicate that, as the total reactor height increases, the reaction rate slows down and the residual concentration of sodium acetate decreases insignificantly. The Hatta number, the rate of reaction in an interfacial film to the rate of diffusion of species through the film, can explain this trade-off between residual concentration of sodium acetate and the oxygen partial pressure in the vent gas.

A higher initial oxygen partial pressure results in a lower oxygen utilization rate and increased operating costs but limits the initial cost of investment in the reactor equipment. The total height and initial oxygen partial pressure of the proposed optimized reactor were therefore set to 7 m and 40 bar, respectively.

3.2. Effect of Reactor Diameter

The effects of reactor diameter were then investigated. According to the results reported in the preceding section, for a reactor total height and initial oxygen partial pressure of 7 m and 40 bar, respectively. The superficial gas velocity of 0.08 m/s is kept as an assumed value and the feeding gas volume rate was maintained at 0.0628 m³/s in the preceding section. Keeping the total reactor volume and feeding gas volume rate constant, the reactor diameter was varied from 0.9 to 1, 1.1, 1.2, 1.3, and 1.4 m for which the aspect ratio of the bubble column is less than 10. The aspect ratio of the bubble column is defined as the ratio of the total reactor height to the reactor diameter. The superficial gas velocity and total reactor height were varied. The superficial gas velocity and reactor total height at each reactor diameter are listed in

Table 2. Relationship between reactor diameter and superficial gas velocity and total reactor height.

Reactor Diameter m	Total Reactor Height m	Superficial Gas Velocity m/s
0.9	8.65	0.0988
1	7.00	0.0800
1.1	5.79	0.0661
1.2	4.87	0.0556
1.3	4.15	0.0473
1.4	3.57	0.0408

, from which it is seen that reducing the reactor diameter increased the total height.

Conversely, increasing the reactor diameter reduced the superficial gas velocity and the aspect ratio of the bubble column. The residual concentration increased with reactor diameter, as shown in Figure 4 and, at reactor diameters above 1.2 m, the residual concentration exceeded 70 mol/m³, which represents a degradation efficiency of 90%. The reactor diameter should be less than 1.2 m. The smaller reactor diameter might explain the influence of the axial dispersion coefficient. The liquid axial dispersion and gas axial dispersion coefficients as a function of diameter are shown in Figure 5, from which it is seen that increasing the diameter increases the axial dispersion coefficient in both the gas and liquid owing to the reduction in superficial gas velocity. As the total reactor volume was kept constant, it is better when the aspect ratio of bubble column is large.

The effect of the reactor diameter on the oxygen partial pressure in the vent gas is shown in Figure 6. The oxygen partial pressure in the vent gas decreased with increasing reactor diameter. Increasing the reactor diameter led to a longer gas residence time in the reactor and more dissolved oxygen flux, which in turn reduced the oxygen partial pressure in the vent gas.

Too small reactor diameter and too large aspect ratio were not always reasonable due to the flow regime transition from bubbly flow to slug flow. There will be an instability of hydrodynamics of bubble column at this regime transition. When the gas holdup in a bubble column reaches a critical value, a flow regime transition occurs. The gas holdup at the regime transition can be expressed as [35]

$${\epsilon }_{trans}=0.59\times {3.9}^{1.5}\sqrt{\frac{{\rho }_G^{0.96}{\sigma }_L^{0.12}}{{\rho }_L}}$$

(19)

The effect of reactor diameter on the gas holdup is shown in Figure 7. The gas holdup at the regime transition for this reaction condition is approximately 0.283; the gas holdup at a reactor diameter of 0.9 m is 0.274, which is close to the gas holdup at the regime transition. To avoid regime transition, it is not necessary to reduce the reactor diameter to values of less than 0.9 m; thus, the proposed reactor diameter was set to 1 m.

3.3. Effect of Superficial Gas Velocity

To evaluate the effects of superficial gas velocity, the reactor diameter, total height, and initial oxygen partial pressure were maintained at 1, 7, and 40 bar, respectively, and the superficial gas velocity was varied from 0.05 to 0.1 m/s. The total feeding gas volume rate varied with superficial gas velocity. The effect of superficial gas velocity on the residual concentration of sodium acetate at the liquid outlet is shown in Figure 8, from which it is seen that the residual concentration decreased with increasing superficial gas velocity. Increasing the superficial gas velocity caused the concentration reduction to flatten. At superficial gas velocities of more than 0.07 m/s, the residual concentration was less than 70 mol/m³, which represents a degradation efficiency of 90%. To achieve a degradation efficiency of over 90%, we proposed a superficial gas velocity of 0.07 m/s. The point of the superficial gas velocity of 0.07 m/s seems to be the approximate critical value which distinguish large slope curves and flat curves. The larger superficial gas velocity will lead to the lower residual concentration and smaller superficial gas velocity will lead to the larger residual concentration which is beneficial for the practical operational flexibility in industry.

Because the reaction rate is related to the concentration of dissolved oxygen, the superficial gas velocity has an important influence on the total degradation efficiency. The effect of superficial gas velocity on the oxygen partial pressure in the vent gas is shown in Figure 9. It is seen that the oxygen partial pressure in the vent gas increased with superficial gas velocity. A large superficial gas velocity indicates a lower oxygen utilization rate, and thus, increased operating costs. The oxygen partial pressure in the vent gas was roughly proportional to the superficial gas velocity when the superficial gas velocity was less than 0.08 m/s. When the superficial gas velocity was 0.05 m/s, the oxygen partial pressure in the vent gas was near zero which exhibits the near complete oxygen utilization.

3.4. Optimized Reactor Performance

Based on the results of the previous sections, we derived an optimized reactor diameter, total height, initial oxygen partial pressure, and superficial gas velocity of 1 m, 7 m, 40 bar, and 0.07 m/s, respectively. The results shown in Figure 10, Figure 11 and Figure 12 demonstrate the predictive capability of the model. Those data explained that the detailed concentrations varied along with local height in this optimized reactor.

It is seen that the amount of residual sodium acetate decreases from 71.7 to 69.2 mol/m³ along with local height from the bottom to the top. The injected sodium acetate-containing waste water was diluted from 700 mol/m³ to about 70 mol/m³ at the bottom of the column. The decrease rate with increasing local height becomes gradually flat at the top of the column. The dissolved oxygen concentration in the liquid decreases in a comparable manner.

Furthermore, the difference between the sodium acetate content at the bottom and top of the reactor is small and the oxygen concentration in the gas phase remains nearly constant from the top to the bottom of the column. The axial distributions of the dissolved oxygen, sodium acetate, and oxygen concentrations in the gas are nearly uniform. The bubble column reactor exhibited an intermediate state between well-mixed flow and plug flow for both liquid and gas.

4. Conclusions

In this study, the effects of reactor diameter, total height, initial oxygen partial pressure, and superficial gas velocity on the WAO of sodium acetate in a bubble column were investigated. To determine optimum parameter settings, the values of reactor total height were varied from 0.1 to 20 m, the reactor diameter was varied from 0.9 to 1, 1.1, 1.2, 1.3, and 1.4 m, the oxygen partial pressure at the gas feed was varied from 20 to 40 and 60 bar, and the superficial gas velocity was varied from 0.05 to 0.1 m/s. The reactor parameters were varied to investigate their effects on the residual concentration of sodium acetate and the oxygen partial pressure. The reactor fluid dynamics and mass transfer characteristics were investigated extensively and approximated using models.

Step 1, the column diameter of 1 m and the superficial gas velocity of 0.08 m/s were the assumed values when the effects of the reactor total height and the initial oxygen partial pressure were investigated. The suggested reactor total height and the initial oxygen partial pressure were provided. Step 2, the suggested reactor total height, the initial oxygen partial pressure, the total reactor volume, and feeding gas volume rate were kept constant when the effects of column diameter was optimized. Step 3, the reactor diameter, total height, and initial oxygen partial pressure were maintained at 1 m, 7 m, and 40 bar when the superficial gas velocity was investigated.

The WAO reactor parameters were determined one by one. There may be some uncertainty during the assumption process which can be avoided with repeated verification. It should be noted that the aim of this work is to provide a reasonable design rather than to provide a best performance of WAO reactor. For example, the initial oxygen partial pressure was set as 40 bar instead of 60 bar after balancing the performance pros and cons of compressors cost. It is a common sense that the larger aspect ratio of the bubble column can improve the reaction performance because the gas will exhibit more residence time and oxygen mass transfer flux. Meanwhile, the aspect ratio of the bubble column of 7 rather than other larger values was adopted in this work considering the flow regime transition from bubbly flow to slug flow.

By optimizing the structural size and feeding gas used in a reactor system, the estimated optimal reactor dimensions can be obtained. The optimized reactor diameter, total height, initial oxygen partial pressure, and superficial gas velocity were found to be 1 m, 7 m, 40 bar, and 0.07 m/s, respectively. The proposed optimized reactor had a degradation efficiency of more than 90%, with a degradation in residual concentration of sodium acetate from 700 mol/m³ to 70 mol/m³.