Mass Transfer between Different Phases in a Mechanically-stirred Vessel and its Comparison with that in a Gas-stirred One

Abstract

In order to understand effect of operating conditions on mass transfer between different phases in a mechanically stirred vessel, cold model study was carried out with liquid paraffin as a dispersion phase and ion-exchanged water as a continuous phase. Inner diameter of vessel, D, was varied in conjunction with both depth, H₀, as D=H₀=400 and 300 mm. Rotation speed, N, was changed between 50–240 rpm, volume ratio, V_oil/V_w, of dispersed to continuous phase was 5.9×10^–2 and 1.2×10^–1.

Liquid/liquid mass transfer rate showed characteristic trend depending on liquid/liquid mixing pattern. It was kept nearly constant at lower level in the region I, monotonically increased in the region II except near the region III and its increasing rate decreased in the region II near the region III. Liquid/liquid mixing pattern was grouped into three regimes. I: the region where liquid/liquid interface did not arrive at the impeller, II: the region where liquid/liquid interface attained at impeller position, III: the region where gas/liquid interface touched impeller.

Under the same supply rate of mixing energy, liquid/liquid mass transfer rate of mechanical stirring corresponded to that of gas stirring at a point in the region II. In the region I and the first half of II, liquid/liquid mass transfer rate of gas stirring is larger than that of mechanical stirring, whereas that of gas stirring is smaller than that of mechanical stirring in the region III and the latter half of II.

Gas/liquid mass transfer rate increased remarkably with an increase in N in the region III.

1. Introduction

Slag/metal reaction caused by solid/liquid or liquid/liquid system is one of the most basic and important practices to remove the impurities in steel melt such as gas injection¹⁾ or mechanical stirring.^2,3,4) In order to research the slag/metal reaction operation, cold model experiment is effective for allowing setup of the wide range of experimental conditions and understanding of the influence of the operation factors easily. Thus, many studies on mechanical stirring practice have been conducted. Kuroyanagi et al.⁵⁾ and Ito et al.⁶⁾ showed effect of baffles on solid/liquid stirring, and Sukawa et al.⁷⁾ and Nomura et al.⁸⁾ clarified a behavior of particle dispersion in water. Observing solid/liquid flow patterns by cold model experiment, Nakai et al.⁹⁾ divided the behavior of particle dispersion into three stages, (1): the stage where a vortex does not arrive at impeller position and has no dispersion in liquid, (2): the stage where a vortex bottom exists between top and bottom of the impeller and particles begin to disperse into liquid, and (3): the stage where a vortex bottom attains at the position deeper than the impeller and there has complete dispersion in liquid. They confirmed that the stage (3) is desirable to accelerate solid/liquid reaction by hot model experiment. However, there are few studies on the relation between liquid/liquid mass transfer rate and its mixing pattern.

In this study, to find out the effects of operating factors and mixing pattern¹⁰⁾ on liquid/liquid and gas/liquid mass transfer rate, cold model experiments were carried out with liquid paraffin as dispersed phase and ion-exchanged water as continuous phase in a mechanical stirred vessel. Additionally, liquid/liquid mass transfer of mechanical stirring was compared with that of gas stirring under the equal supply rate of mixing energy.

2. Experimental

Schematic view of an experimental apparatus is shown in Fig. 1. Liquid paraffin as dispersion phase and ion-exchanged water as continuous phase were charged at total bath depth, H₀ mm in a vessel, inner diameter, D mm and vessel height, L mm.

Fig. 1.

Schematic view of experimental apparatus of mechanical stirring.

Four blades of impeller whose diameter was expressed as d_i mm, thickness as b_i mm and width as w_i mm was used as shown schematically in Fig. 2. The impeller was set in the central axis of the vessel.

Fig. 2.

Schematic view of impeller used for experiment.

In order to obtain benzoic acid transfer rate between liquid paraffin and water, temporal change in benzoic acid concentration in water partly transferred from liquid paraffin to water was calculated from the result of electrical conductivity measurement by a conductance meter (Automatic system research, mk-250EC). Liquid temperature was 290±3 K.

On the other hand, a dissolved oxygen analyzer (Wissenschaftlich- Technische- Werkstätten, Multi 3410 SET 5) was used to find oxygen transfer rate between air and water in addition to the liquid-liquid mass transfer experiment. After injecting nitrogen into ion-exchanged water to remove oxygen and injecting air into liquid paraffin to saturate oxygen, oxygen concentration in ion-exchanged water was tracked over time during mechanical stirring. Liquid temperature was 298±3 K.

Experimental conditions including underlined baselines are shown in Table 1. D=H₀=400 mm was a standard for vessel and impeller size, (d_i,b_i,W_i) was fixed to (116 mm, 67 mm, 31 mm). Rotation speed, N, was varied from 50 to 240 rpm and distance, H, between free surface of liquid paraffin and impeller upper position from 22 to 313 mm. Volume ratio, V_oil/V_w, of dispersed to continuous phase was 5.9×10^–2 and 1.2×10^–1 (baseline). Density and viscosity of liquid paraffin were 828 kg/m³, 8.20×10^–3 Pa·s, respectively. Depths of liquid paraffin and water defined as H_oil and H_w, respectively, were shown in Table 2.

Table 1. Experimental condition for mechanical stirring.

Variables
Vessel inner diameter, D (mm)	300, 400
Bath depth, H0 (mm)	300, 400
Impeller diameter, di (mm)	116
Impeller thickness, bi (mm)	67
Impeller width, wi (mm)	31
Rotating speed, N (rpm)	50–240
Impeller depth, H (mm)	22–313
Ratio of dispersion phase to water volume, Voil/Vw (–)	5.9 × 10–2, 1.2 × 10–1
Dispersion phase	liquid paraffin
Continuous phase	Ion-exchanged water

Table 2. Static bath depths of oil and under a given V_oil/V_w.

Voil/Vw (–)	D = H0 (mm)	Hoil (mm)	Hw (mm)
5.9 × 10–2	400	22	378
1.2 × 10–1	400	42	358
1.2 × 10–1	300	31	269

3. Results and Discussion

3.1. Liquid/Liquid Mixing Pattern

In the previous paper,¹⁰⁾ we described that three types of liquid/liquid mixing pattern were observed as shown schematically in Fig. 3 and varied depending on operating factors such as N, Hand V_oil/V_w. I (the left side of Fig. 3) is the region where each liquid phase exists separately and has no dispersion, II (the middle of Fig. 3) is the region where vortex of dispersed phase (liquid/liquid interface) arrives at impeller position and its dispersion starts into continuous phase, and III (the right side of Fig. 3) is the region where gas/liquid interface in addition to liquid/liquid one arrives at impeller position and dispersion occurs hard. Moreover, it was also found that the mixing pattern transited from I to II, and from II to III according to an increase in N and a decrease in H.

Fig. 3.

Schematic view of mixing pattern.

3.2. Calculation of Liquid/Liquid Mass Transfer Capacity Coefficient

According to double film theory, mass transfer rate of benzoic acid from liquid paraffin to water is given by   

$${\text{V}}_{\text{w}}{\text{dC}}_{\text{w}}{\text{/dt =K}}_{\text{w}}\text{a }\left({\text{C}}_{\text{w}}{ }^{\text{*}}-{\text{C}}_{\text{w}}\right)$$(1)

where V_w: water volume (m³), C_w: concentration of benzoic acid in water (kg/m³), K_wa: overall capacity coefficient (m³/s), K_w: overall mass transfer coefficient (m/s), a: interfacial area between liquid paraffin and water (m²), C_w^*: concentration of benzoic acid in water in equilibrium with that in liquid paraffin (kg/m³). When both of concentrations of benzoic acid in liquid paraffin and water are dilute enough, final partition ratio, h^*(=C_w^*/C_oil), of benzoic acid between water and liquid paraffin keeps constant, where C_oil: benzoic acid concentration in liquid paraffin (kg/m³).

Mass balance of benzoic acid is expressed by,   

$${\text{C}}_{\text{oil}}{\text{V}}_{\text{oil}}{\text{+ C}}_{\text{w}}{\text{V}}_{\text{w}}\text{=}\alpha$$(2)

where V_oil: volume of liquid paraffin (m³), α: total amount of benzoic acid dissolved in liquid/liquid system (kg). Thus, using Eqs. (1) and (2), Eqs. (3), (4), (5) are obtained:   

$${\text{dCw / dt = m}}_{\text{1}}\left({\text{m}}_{\text{2}}-{\text{C}}_{\text{w}}\right)$$(3)

  

$${\text{m}}_{\text{1}}\text{= }\left({\text{K}}_{\text{w}}{\text{a / V}}_{\text{w}}\right)\left({\text{h*V}}_{\text{w}}{\text{/ V}}_{\text{oil}}\text{+1}\right)$$(4)

  

$${\text{m}}_{\text{2}}\text{= }\left(\text{h*}\alpha {\text{ / V}}_{\text{oil}}\right)\text{ / }\left({\text{h*V}}_{\text{w}}{\text{/ V}}_{\text{oil}}\text{+1}\right)$$(5)

Solving Eq. (4) under an initial condition of C_w= 0,   

$$\left({\text{m}}_{\text{2}}-{\text{C}}_{\text{w}}\right){\text{/ m}}_{\text{2}}\text{= exp}\left(-{\text{m}}_{\text{1}}\text{t}\right)$$(6)

K_wa is obtained from a temporal change in C_w and Eq. (6). h^* value was 0.445 from V_oil=2.22×10^–3 m³, V_w=1.90× 10^–2 m³ and α=3.48×10^–3 kg.

On the other hand, K_w is given by   

$${\text{1/K}}_{\text{w}}{\text{= 1/k}}_{\text{w}}{\text{+ h}}^{\text{*}}{\text{/k}}_{\text{oil}}$$(7)

where k_w: water film coefficient of mass transfer (m/s), k_oil: oil film coefficient of mass transfer (m/s). Using the Wilke-Chang equation¹¹⁾ which permits the diffusivities of benzoic acid in water and oil to be estimated, ratio of benzoic acid diffusivity in water to that in liquid paraffin, D_w/D_oil, became about 3.2. Assuming that mass transfer coefficient is proportional to square root of diffusivity, k_w/(k_oil/h^*) was calculated to be (3.2^1/2)(0.455)=0.81. It follows that the rate-determining step in the mass transfer rate in this system was mixture of benzoic acid diffusion in water and oil phases.

3.3. Effect of Mixing Pattern on Liquid/Liquid Mass Transfer Rate

The relation between the overall capacity coefficient, K_wa, of benzoic acid transferred from liquid paraffin into water and H for N=146 and 189 rpm are shown in Fig. 4. V_oil/V_w was fixed to 1.2×10^–1. K_w and a in the symbol of overall capacity coefficient denote overall mass transfer coefficient and liquid/liquid interfacial area, respectively. Mass transfer rate increased with an increase in N. Depending on mixing pattern described in Fig. 3, distinct change in k_ia was found as follows. K_wa was almost kept constant for different H in the region I where mass transfer rate was less affected by liquid/liquid interface because it moved like rigid body. On the other hand, K_wa began to increase with the decrease in H in the region II and the increasing rate of k_ia became lower toward the region III and furthermore in the region III. The reason why K_wa increased in the region II can be attributed to the fact that liquid/liquid interfacial area increases with the increase in dispersion of liquid paraffin into water. Moreover, the decrease in the increasing rate of K_wa in the region II near III as well as III is because most liquid paraffin has already dispersed into water and the increase in liquid/liquid interfacial area decreases.

Fig. 4.

Relation between K_wa and impeller position.

The relation between K_wa and H for V_oil/V_w=1.2×10^–1 and 5.9×10^–2 is shown in Fig. 5 where N was fixed to 146 rpm. Mass transfer rate increased with an increase in V_oil/V_w for the equal N and H, which seems to be caused by the fact liquid/liquid interfacial area increases with the increase in V_oil/V_w as dispersion phase volume becomes large. The dependence of K_wa on the mixing pattern was similar between Figs. 4 and 5.

Fig. 5.

Relation between K_wa and impeller position.

The relation between K_wa and N for H=83 and 233 mm is shown in Fig. 6 where V_oil/V_w of liquid/liquid was fixed to 1.2×10^–1. Mass transfer rate increased with an increase in N because the it caused the transition of mixing pattern to I→II→III.

Fig. 6.

Relation between K_wa and rotating speed.

The calculated value of supply rate, ε_i, of mixing energy which will be described in section 3.4 became 1.18×10^–2 W/kg for N=146 rpm and D=H₀=400 mm. On the other hand, N of the same ε_i value was 117 rpm for D=H₀=300 mm. Figure 7 shows that the relation between K_wa/V_w and H for the above two vessels. V_oil/V_w was kept to 1.2×10^–1, although liquid volume between two vessels was different from one another. K_wa values of two vessels were almost equal in the regions II near III and III, and differed in the region I and II except near III. Assuming that liquid paraffin is fully dispersed into water at the same particle radius, r and number of particles, n, a and V_w become to 4πr²n and (4πr³n/3)/(1.2×10^–1), respectively. Therefore, the following equation is given,   

$${\text{K}}_{\text{w}}{\text{a/V}}_{\text{w}}{\text{= k}}_{\text{i}}\left(\text{4}\pi {\text{r}}^{\text{2}}\text{n}\right)\left(\text{1}{\text{.2×10}}^{-\text{1}}\right)\text{/}\left(\text{4}\pi {\text{r}}^{\text{3}}\text{n/3}\right)\text{ =0}{\text{.36k}}_{\text{i}}\text{/r}$$(8)

When supply rate of mixing energy is equal, mass transfer coefficient, K_w and fully dispersed liquid radius, r are also the same. Thus, the values of K_wa/V_w expressed in Eq. (8) was equal in the region II near III and III of Fig. 7. On the other hand, K_wa/V_w values in the region II except near III was not equal because it was affected by the difference in liquid paraffin dispersion into water due to different transition points from I to II between two vessels.

Fig. 7.

Relation between K_wa/V_w and rotating speed.

A multi-regression analysis was applied to the relation between K_wa and operating variables for the region II. The equation became as follows:   

$${\text{K}}_{\text{w}}\text{a=6}{\text{.31×10}}^{-\text{21}}{\text{N}}^{\text{4}\text{.61}}{\left(\text{H}-{\text{H}}_{\text{oil}}\right)}^{-\text{0}\text{.88}}{\left({\text{V}}_{\text{oil}}{\text{/V}}_{\text{w}}\right)}^{\text{0}\text{.44}}{\text{D}}^{\text{2}\text{.74}}$$(9)

where R²=0.62. As can be seen in Figs. 4 ,5 ,6 ,7, the transition of mixing pattern from II to III was accelerated along with the increase in rotation speed, the decrease in depth from free surface to impeller position and the increase in ratio of dispersion phase volume to water. Equation (9) has also the same tendency as the transition of II–III. To apply these results to the extended system such as slag/metal system quantitatively, dimensional analysis is preferable. However, it was difficult as this study was carried out by only liquid paraffin and water system. It is an issue in the future to rearrange it with the added data of several liquid/liquid systems in view of dimensional analysis.

3.4. Comparison between Gas and Mechanical Stirrings for Liquid/Liquid Mass Transfer Rate

Supply rate of mixing energy of gas stirring expressed as ε_g (W/kg) is given by Eq. (10),   

$${\mathit{\epsilon }}_{\text{g}}\text{= 8}\text{.58}\times {\text{10}}^{\text{5}}{\text{Q’T /W}}_{\text{w}}\hspace{0.17em}\text{log}\left(\text{1+H/10}\text{.3}\right)$$(10)

where Q’: gas flow rate (m³/s), T: bath temperature (K), W_w: mass of water (kg), H: depth from free surface to the position of gas injection (m).

On the other hand, supply rate of mixing energy of mechanical stirring expressed as ε_i (W/kg) is given by Eq. (4),¹²⁾   

$${\mathit{\epsilon }}_{\text{i}}{\text{= N}}_{\text{P}}\cdot {\rho }_{\text{m}}\cdot {\text{N}}^{\text{3}}\cdot {\text{d}}_{\text{i}}{ }^{\text{5}}{\text{/W}}_{\text{w}}$$(11)

where N_P: power number (–), ρ_m: mean density (kg/m³), N: rotation speed (rps), d_i: impeller diameter (m). N_P is calculated from Eq. (12) given by the Nagata’s formula,¹³⁾ where H₀: bath depth (m), D: vessel diameter (m), b_i: blade thickness (m), n_P: number of blade and μ_m: mean viscosity (Pa·s).

  

$$\begin{array}{l}{\text{N}}_{\text{P}}{\text{= A/R}}_{\text{e}}\text{+}\hspace{0.17em}\text{B}\{\left({\text{10}}^{\text{3}}\text{+1}{\text{.2 R}}_{\text{e}}{ }^{\text{0}\text{.66}}\right)\text{/}\\ {\left({\text{10}}^{\text{3}}\text{+ 3}{\text{.2 R}}_{\text{e}}{ }^{\text{0}\text{.66}}\right)\}}^{\text{P}}{\left({\text{H}}_{\text{0}}\text{/D}\right)}^{\text{(0}\text{.35+b’/D)}}\end{array}$$(12)

The variables of A, B, P, b’ and R_e in Eq. (12) are calculated from the following Eqs. (13), (14), (15) ,(16) ,(17).   

$$\text{A = 14 +}\left(\text{b’/} \text{D}\right)\text{{670}{\left({\text{d}}_{\text{i}}\text{/} \text{D – 0}\text{.6}\right)}^{\text{2}}\text{+185}$$(13)

  

$$\text{B = 10^}\left\{\text{1}\text{.3 – 4}{\left(\text{b’/} \text{D – 0}\text{.5}\right)}^{\text{2}}\text{–1}{\text{.14 d}}_{\text{i}}\text{/} \text{D}\right\}$$(14)

  

$$\text{P = 1}\text{.1 + 4}\left(\text{b’/} \text{D}\right)\text{– 2}\text{.5}{\left({\text{d}}_{\text{i}}\text{/} \text{D – 0}\text{.5}\right)}^{\text{2}}\text{–7}\left({\text{d}}_{\text{i}}\text{/} \text{D}\right)$$(15)

  

$${\text{b’= b}}_{\text{i}}\times {\text{n}}_{\text{P}}\text{/}\hspace{0.17em}\text{2}$$(16)

  

$${\text{R}}_{\text{e}}\text{=}{\rho }_{\text{m}}{\text{N d}}_{\text{i}}{ }^{\text{2}}\text{/}\hspace{0.17em}{\mu }_{\text{m}}$$(17)

Here, ρ_m and μ_m in Eqs. (11) and (17) are calculated from volume fraction of water and liquid paraffin. As shown in Eqs. (4), (5), (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16), (17), the supply rate of mixing energy of mechanical stirring is not affected by H, depth from free surface to the impeller position, although that of gas stirring is dependent on H in Eq. (3).

Both of ε_g for Q=10 and Q=20 L/min were calculated from Eq. (10) for H₀=0.4 m, H=0.28 m, V_oil/V_w=1.2×10^–1 and T=298 K. Rotation speed for mechanical stirring was obtained by substituting ε_i values corresponding to the calculated ε_g ones into Eqs. (11), (12), (13), (14), (15), (16), (17). As shown in Table 3, the rotation speeds, N=146 and 189 rpm correspond to gas flow rates, Q=10 and 20 L/min at H=0.28 m, respectively.

Table 3. Calculation of supplied rate of stirring energy, gas flow rate and rotating speed.

Q (L/min)	εg (W/kg)	N (rpm)	εi (W/kg)
10	1.18 × 10–2	146	1.18 × 10–2
20	2.36 × 10–2	189	2.35 × 10–2

Comparison between K_wa of mechanical stirring and gas stirring for the same supply rate of mixing energy is shown in Fig. 8. Gas stirring was carried out through two-holes nozzle immersed into water and air was blown horizontally. K_wa values of mechanical stirring in the region I and up to a point of the region II was smaller than those of gas stirring, whereas they were larger than above the point of the region II and the region III. From the standpoint of slag/metal reaction practice carried out under high temperature, the region III may have potential to damage an impeller due to its partial exposure to atmosphere. Therefore, the mechanical stirring in the region II closed to the region III is more preferable in order to support enhancement in slag/metal mass transfer rate and stable slag/metal operation at the same time.

Fig. 8.

Comparison of K_wa between mechanical stirring and gas injection for the equal energy supplied rate into bath.

3.5. Oxygen Transfer Rate from Gas to Continuous Liquid Phase in Mechanical Stirring

In a mechanical stirring practice, oxygen transfers from air to water by means of two routes: one is that oxygen in air moves into water directly and the other that oxygen moves through liquid paraffin. Thus, oxygen balance in water is expressed by Eq. (18),   

$$\begin{array}{l}{\text{V}}_{\text{w}}{\text{dC’}}_{\text{w}}{\text{/dt = K}}_{\text{w,G-L}}{\text{a}}_{\text{G-L}}\left({\text{C’}}^{\text{*}}{ }_{\text{w,G-L}}\text{–} {\text{C’}}_{\text{w}}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}{\text{+ K}}_{\text{w,L-L}}{\text{a}}_{\text{L-L}}\left({\text{C’}}^{\text{*}}{ }_{\text{w,L-L}}\text{–} {\text{C’}}_{\text{w}}\right)\end{array}$$(18)

where K_w,G-L: overall mass transfer coefficient of oxygen between air and water (m/s), a_G-L: interfacial area between air and water (m²), C’^*_w,G-L: oxygen concentration in water in equilibrium with oxygen in air (kg/m³), C’_w: oxygen concentration in water (kg/m³), K_w,L-L: overall mass transfer coefficient of oxygen between liquid paraffin and water (m/s), a_L-L: interfacial area between liquid paraffin and water (m²), and C’^*_w,L-L: oxygen concentration in water in equilibrium with oxygen in liquid paraffin (kg/m³).

Oxygen balance in liquid paraffin is given by oxygen input from air/liquid interface and oxygen output to water shown in Eq. (19).   

$$\begin{array}{l}{\text{V}}_{\text{oil}}{\text{dC’}}_{\text{oil}}{\text{/dt = K}}_{\text{oil,G-L}}{\text{a}}_{\text{oil,G-L}}\left({\text{C’}}^{\text{*}}{ }_{\text{oil,G-L}}\text{–} {\text{C’}}_{\text{oil}}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}{\text{ – K}}_{\text{w,L-L}}{\text{a}}_{\text{L-L}}\left({\text{C’}}^{\text{*}}{ }_{\text{w,L-L}}\text{–} {\text{C’}}_{\text{w}}\right)\end{array}$$(19)

where C’_oil :oxygen concentration in liquid paraffin (kg/m³), K_oil,G-L: overall mass transfer coefficient of oxygen between air and liquid paraffin (m/s), a_oil,G-L: interfacial area between air and liquid paraffin (m²), C’^*_oil,G-L: oxygen concentration in liquid paraffin in equilibrium with oxygen in air (kg/m³). Assuming that oxygen transfer in liquid paraffin becomes a steady state quickly, Eq. (19) can be expressed as Eq. (20).   

$${\text{K}}_{\text{oil,G-L}}{\text{a}}_{\text{oil,G-L}}\left({\text{C’}}^{\text{*}}{ }_{\text{oil,G-L}}\text{–} {\text{C’}}_{\text{oil}}\right){\text{=K}}_{\text{w,L-L}}{\text{a}}_{\text{L-L}}\left({\text{C’}}^{\text{*}}{ }_{\text{w,L-L}}\text{–} {\text{C’}}_{\text{w}}\right)$$(20)

Overall gas-liquid mass transfer coefficient, K_G-L (m/s) is given by Eq. (21).   

$${\text{1/K}}_{\text{G-L}}\text{= 1/}\left({\text{H’k}}_{\text{G,G-L}}\right){\text{+1/k}}_{\text{L,G-L}}$$(21)

where k_G,G-L is gas film coefficient of oxygen transfer (kg/(Pa·m³·s)), k_L,G-L is liquid film coefficient of oxygen transfer (m/s), and H’ is Henry’s constant (Pa·m³/kg) defined by Eq. (22).   

$${\text{P}}_{\text{oxy}}{\text{= H’C}}_{\text{oxy}}$$(22)

In Eq. (22), p_oxy is oxygen partial pressure (Pa) and C_oxy is oxygen concentration in liquid (kg/m³). When liquid took sufficient time to contact with air, C’^*_oil,G-L in liquid paraffin became 11.6×10^–3 kg/m³, whereas C’^*_w,G-L in water was 9.3×10^–3 kg/m³. Oxygen solubilities in liquid paraffin and water were approximately same and small enough. Substituting P_oxy=1.013×10⁵×0.21=2.31×10⁴ Pa and the above C’^*_oil,G-L and C’^*_w,G-L values into Eq. (22), H’ values for liquid paraffin and for water were calculated to be 1.99×10⁶ and 2.48×10⁶ Pa·m³/kg, respectively. Large Henry’s constant means that K_G-L≒ k_L,G-L in Eq. (21) and rate determining steps for oxygen transfer between gas and liquid is liquid film diffusion.

For liquid-liquid mass transfer rate, K_w,L-L is given by Eq. (23),   

$${\text{1/K}}_{\text{w,L-L}}{\text{= 1/k}}_{\text{w,L-L}}{\text{+ h}}^{\text{**}}{\text{/k}}_{\text{oil,L-L}}$$(23)

where k_w,L-L: water film coefficient of oxygen transfer (m/s), k_oil,L-L: oil film coefficient of oxygen transfer (m/s), h^**: oxygen partition ratio (=C’^*_w,L-L/C’_oil). The h^** value is obtained from the experiment with a closed vessel filled with water and liquid paraffin. Using the Wilke-Chang equation,¹¹⁾ ratio of oxygen diffusivity in water to that in liquid paraffin, D_w,oxygen/D_oil,oxygen became about 3.2 as well as benzoic acid diffusivity in section 3.2. If mass transfer coefficient is proportional to square root of diffusivity, k_w,L-L/(k_oil,L-L/h^**) was calculated to be (3.2^1/2)(1.80)=3.2. It resulted in the rate-determining step of mixture of oxygen diffusion in water and oil phases as well as benzoic acid diffusion.

Thus, Eq. (20) can rewritten for C’^*_w,L-L, using constant parameters under a given operating factor such as liquid film coefficient of oxygen transfer through liquid paraffin, k_L,G-L (m/s), a_oil,G-L, K_w,L-La_L-L, C’^*_oil,G-L (=11.6×10^–3 kg/m³), h^**(=1.80), and time-dependent C’_w.   

$$\begin{array}{l}{\text{C’}}^{\text{*}}{ }_{\text{w,L-L}}\text{=}\left({\text{k}}_{\text{oil,G-L}}{\text{a}}_{\text{oil,G-L}}{\text{C’}}^{\text{*}}{ }_{\text{oil,G-L}}{\text{+ K}}_{\text{w,L-L}}{\text{a}}_{\text{L-L}}{\text{C’}}_{\text{w}}\right)\text{/}\\ \hspace{1em}\hspace{1em}\hspace{1em}\mathrm{\hspace{0.17em} }\hspace{0.17em}\left({\text{k}}_{\text{oil,G-L}}{\text{a}}_{\text{oil,G-L}}{\text{/h}}^{\text{**}}{\text{+K}}_{\text{w,L-L}}{\text{a}}_{\text{L-L}}\right)\end{array}$$(24)

When resistance to oxygen transfer between gas and liquid paraffin is ignored compared with that between liquid paraffin and water, that is, k_oil,G-La_oil,G-L»K_w,L-La_oil,L-L, C’^*_w,L-L, Eq. (24) can be expressed as Eq. (25).   

$${\text{C’}}^{\text{*}}{ }_{\text{w,L-L}}{\text{= h}}^{\text{**}}{\text{C’}}^{\text{*}}{ }_{\text{oil,G-L}}$$(25)

Eq. (25) means liquid paraffin is saturated with oxygen because oxygen supply interface between air and liquid paraffin is sufficient during mechanical stirring practice. However, as oxygen transfer resistance usually exists between gas and liquid paraffin, Eq. (24) must be used instead of Eq. (25).

According to the visual observation, water contacted with air directly in the region III as well as with liquid paraffin, whereas it did not meet air in the regions I and II. Thus, the first term of the right side in Eq. (18) can be ignored for the region I and II as follows:   

$${\text{V}}_{\text{w}}{\text{dC}}_{\text{w}}{\text{/dt = K}}_{\text{w,L-L}}{\text{a}}_{\text{L-L}}\left({\text{C’}}^{\text{*}}{ }_{\text{w,L-L}}\text{–} {\text{C’}}_{\text{w}}\right)$$(18’)

By substituting Eq. (24) into Eq. (18’), it can be integrated with C’_w=C’₀,_w at t=0.   

$$\begin{array}{c}\left({\text{h}}^{\text{**}}{\text{C’}}^{\text{*}}{ }_{\text{oil,G-L}}\text{–} {\text{C’}}_{\text{w}}\right)\text{/}\left({\text{ h}}^{\text{**}}{\text{C’}}^{\text{*}}{ }_{\text{oil,G-L}}\text{–} {\text{C’}}_{\text{0,w}}\right)\\ \text{=exp[–} \text{(}\alpha {\text{K}}_{\text{w,L-L}}{\text{a}}_{\text{L-L}}{\text{/V}}_{\text{w}}\text{)t]}\end{array}$$(26)

  

$$\alpha \hspace{0.17em}\text{= }\left({\text{k}}_{\text{oil,G-L}}{\text{a}}_{\text{oil,G-L}}{\text{/h}}^{\text{**}}\right)\text{/}\left({\text{k}}_{\text{oil,G-L}}{\text{a}}_{\text{oil,G-L}}{\text{/h}}^{\text{**}}{\text{+K}}_{\text{w,L-L}}{\text{a}}_{\text{L-L}}\right)$$(27)

When k_oil,G-La_oil,G-L/h^** approaches infinite, resistance of oxygen mass transfer between gas and liquid does not exist and α becomes unit.

For the region III, Eq. (18) can be integrated with C’_w=C’₀,_w at t=0.   

$$\left({\text{A – C’}}_{\text{w}}\right)\text{/}\left({\text{A – C’}}_{\text{0,w}}\right)\text{=exp}\left(\text{–Bt}\right)$$(28)

  

$$\begin{array}{c}{\text{A = (K}}_{\text{w,G-L}}{\text{a}}_{\text{G-L}}{\text{C’}}^{\text{*}}{ }_{\text{w,G-L}}\text{+}\alpha {\text{K}}_{\text{w,L-L}}{\text{a}}_{\text{L-L}}{\text{h}}^{\text{**}}{\text{C’}}^{\text{*}}{ }_{\text{oil,G-L}}\text{)/}\\ {\text{(K}}_{\text{w,G-L}}{\text{a}}_{\text{G-L}}\text{+}\alpha {\text{K}}_{\text{w,L-L}}{\text{a}}_{\text{L-L}}\text{)}\end{array}$$(29)

  

$${\text{B = (K}}_{\text{w,G-L}}{\text{a}}_{\text{G-L}}\text{+}\alpha {\text{K}}_{\text{w,L-L}}{\text{a}}_{\text{L-L}}{\text{)/V}}_{\text{w}}$$(30)

where C’^*_w,G-L and C’^*_oil,L-L are 9.3×10^–3 and 11.6×10^–3 kg/m³, respectively, as mentioned above.

Typical examples of temporal change in oxygen concentration in water are shown in Fig. 9. Mixing patterns of N=200 and 300 rpm represented those of the region II and III, respectively. A in Eq. (28) for the region III was obtained from K_w,G-La_G-L calculated by trial-and-error method and αK_w,L-La_L-L calculated by extrapolation of the region II. The slopes, (K_w,G-La_G-L+αK_w,L-La_L-L)/V_w, of temporal change in oxygen concentration were linear, which means the overall volumetric coefficient was kept constant. Oxygen absorption rate in the region III was faster than that in the region I and II due to gas/liquid mass transfer in addition to liquid/liquid one.

Fig. 9.

Typical examples of temporal change in oxygen concentration in water.

The relation between K_w,G-La_G-L+αK_w,L-La_L-L and N for H=83 mm and V_oil/V_w=1.2×10^–1 is shown in Fig. 10 where the other operating factors were a standard, and the vertical axis of the region I and II indicates αK_w,L-La_L-L. αK_w,L-La_L-L was kept nearly constant from the region I to the first half of the region II and it began to increase from the second half of the region II. αK_w,L-La_L-L values in the region III were given by extrapolating αK_w,L-La_L-L in the second half of the region as indicated in a dotted line in Fig. 10. It was found that K_w,G-La_G-L+αK_w,L-La_L-L of the region III increased in a discontinuous manner.

Fig. 10.

Relation between oxygen absorption rate and rotating speed.

By subtracting the extrapolated αK_w,L-La_L-L from the K_w,G-La_G-L+αK_w,L-La_L-L in Fig. 10, increasing rate of K_w,G-La_G-L for rotation speed is shown in Fig. 11. As presented in the visual observation, the direct oxygen transfer between immersed air bubbles and water remarkably increased K_w,G-La_G-L in the region III.

Fig. 11.

Increase in direct oxygen absorption rate between air and water in the region III.

αK_w,L-La_L-L became 0.7×10^–5 m³/s at N=200 rpm as shown in Fig. 10, whereas K_wa was estimated to be 2.9×10^–5 m³/s under the same experimental condition as known from Fig. 6. The value of K_wa was 4.1 times larger than that of αK_w,L-La_L-L. Using Eqs. (7) and (23), k_w/(k_oil/h^*) = 0.81 and k_w,L-L/(k_oil,L-L/h^**) = 3.2, and assuming that mass transfer coefficient is proportional to square root of diffusivity, Eq. (31) was obtained as follows.   

$$\begin{array}{c}{\text{K}}_{\text{w}}{\text{a/K}}_{\text{w,L-L}}{\text{a}}_{\text{L-L}}\text{= }\left({\text{k}}_{\text{w}}\text{/1}\text{.81}\right)\text{/}\left({\text{k}}_{\text{w,L-L}}\text{/4}\text{.2}\right)\\ \text{=}\hspace{0.17em}\text{2}\text{.3}\left({\text{k}}_{\text{w}}{\text{/k}}_{\text{w,L-L}}\right)\hspace{0.17em}\text{=}\hspace{0.17em}\text{2}\text{.3}{\left({\text{D}}_{\text{w}}{\text{/D}}_{\text{w,oxygen}}\right)}^{\text{1/2}}\end{array}$$(31)

Substituting benzoic acid diffusivity in water, D_w=0.9×10^–9 m²/s¹³⁾ at 298 K and oxygen diffusivity in water, D_w,oxygen= 2.2×10^–9 m²/s¹²⁾ at 290 K into Eq. (30), K_wa/K_w,L-La_L-L became 2.3(0.9/2.2)^1/2=1.5. Thus,   

$$\begin{array}{c}\alpha \text{=}\left({\text{k}}_{\text{oil,G-L}}{\text{a}}_{\text{oil,G-L}}{\text{/h}}^{\text{**}}\right)\text{/}\left({\text{k}}_{\text{oil,G-L}}{\text{a}}_{\text{oil,G-L}}{\text{/h}}^{\text{**}}{\text{+K}}_{\text{w,L-L}}{\text{a}}_{\text{L-L}}\right)\\ \text{= 1}\text{.5/4}\text{.1 =}\mathrm{\hspace{0.17em} }\text{0}\text{.37}\end{array}$$(32)

Equation (32) gives k_oil,G-La_oil,G-L/K_w,L-La_L-L=1.0, which means oxygen transfer rate between air and liquid paraffin is almost the same value as that between water and liquid paraffin at N=200 rpm in the region II.

4. Conclusions

Cold model study on liquid/liquid mass transfer was carried out with liquid paraffin as a dispersed phase and with ion-exchanged water as a continuous phase in a mechanical stirred vessel.

(1) Liquid/liquid mass transfer rate had almost the same trend depending on mixing pattern. The mass transfer rate was kept nearly constant on a low level in the region I, monotonically increased in the region II except near the region III and the increasing rate was decreased in the region II near the region III and the region III. Here, I is the region where liquid/liquid interface does not arrive on an impeller, II the region where liquid/liquid interface attains at an impeller and a part of liquid paraffin disperses in water and III the region where gas/liquid interface as well as liquid/liquid touches with an impeller and much liquid paraffin disperses into water phase.

(2) Liquid/liquid mass transfer rate increased with a decrease in depth from free surface to impeller position and increases in rotation speed and ratio of dispersion phase volume to continuous one, because these factors accelerated the transitions of mixing pattern from I to II.

(3) A multi-regression equation on liquid/liquid mass transfer rate in the region II was as follows:   

$${\text{K}}_{\text{w}}{\text{a=10}}^{-\text{20}\text{.2}}\times {\text{N}}^{\text{4}\text{.61}}{\left(\text{H}-{\text{H}}_{\text{oil}}\right)}^{-\text{0}\text{.88}}{\left({\text{V}}_{\text{oil}}{\text{/V}}_{\text{w}}\right)}^{\text{0}\text{.44}}{\text{D}}^{\text{2}\text{.74}}$$

where K_wa: overall capacity coefficient (m³/s), N: rotation speed (rpm), H: depth from free surface to impeller position (mm), V_oil/V_w: ratio of dispersion phase volume to continuous one (–), D: vessel diameter (mm).

(4) Mass transfer rate of gas stirring was faster than mechanical stirring in the region I and the first half of the region II, whereas that of gas stirring was slower than mechanical stirring in the second half of the region II and the region III.

(5) Gas/liquid mass transfer rate increased remarkably with an increase in rotation speed in the region III.

Acknowledgements

This work was carried out under the project of NEDO (New Energy and Industrial Technology Development Organization), entitled “Research and development project on enhancement of usage of hard-to-use ferrous scrap”.

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