1 Introduction

Solvent extraction is one of the most common separation processes used in the nuclear industry [1-2]. Tri-butyl phosphate (TBP) diluted with dodecane is widely used as a solvent in the nuclear fuel cycle [3-5]. TBP is a preferred solvent due to its excellent chemical resistance, very low vapor pressure, and favorable physical properties. For example, the PUREX process uses 30%(v/v) TBP diluted with dodecane as a solvent due to its favorable chemistry for selective extraction of uranium (U) and plutonium (Pu), as well as splitting of Pu from U followed by stripping of U [6]. It is also used for extraction of other metal ions relevant to the nuclear fuel cycle [7]. Solvent extraction involves the transfer of a species from a dispersed liquid phase to a continuous liquid phase or vice versa through a liquid-liquid interface. The mass transfer rate depends on the mass transfer coefficient, interfacial area, concentration gradient and important interfacial phenomena, such as flow conditions in the vicinity of interface and Marangoni convection. The diameter of the drop is one of the most important parameters as it controls the interfacial area and the diffusion–convection process, which are central to mass transfer [8]. The drop formation mechanism depends on the type of the equipment used. For example, in a mixer–settler, drops form due to turbulent eddies generated by a rotating impeller [9, 10], whereas in a centrifugal extractor or in a Taylor-Couette extractor, the drops form due to shearing between a rotating inner cylinder and a stationary outer cylinder [11-14]. In air pulsed columns, which are widely used for solvent extraction in the nuclear fuel cycle, drop formation occurs when the dispersed phase is pushed through perforations in sieve plates under the effect of pneumatic pulsing [15-18]. However, a prerequisite for understanding the drop formation phenomenon for such complex cases is a fundamental understanding of drop formation phenomenon for simple flow conditions. The present study focuses on drop formation at the tips of nozzles submerged in a quiescent immiscible phase. It is a step in the direction of understanding drop formation phenomena in complex flow conditions, predictive model design, and simulation of solvent extraction equipment in the nuclear fuel cycle. Many more steps are required to reach the final goal. These steps include studying drop formation for different flow conditions such as in pulsating flow, counter-current flow etc. We intend to study these cases in the future.

Drop diameter primarily depends on the geometry of the perforation (nozzle or orifice), the velocity of dispersed phase, interfacial tension, viscosity ratio, and the density difference between the two liquid phases. Harkins & Brown calculated the drop volume at very low flow rates by equating the buoyancy and interfacial tension forces and incorporating a correction factor to account for the drop volume that remains attached to the nozzle at the time of break-off [19]. The Harkins-Brown correction factor is empirical in nature [19, 20]. Hayworth and Treybal [21] as well as Null and Johnson [22] extended the work of Harkins and Brown. They proposed models that take into account the velocity effect and incorporate drag and inertial forces into the force balance equation. Scheele and Meister, Meister and Scheele, and Hamad and co-workers proposed a two-stage drop formation mechanism in which the drop formation process is divided into two stages, that is, drop growth during the first stage and necking, followed by drop detachment in the second stage [23-27]. In addition to the above studies, numerical investigations have also been reported to study the hydrodynamics of single drop formation in the laminar flow regime [28, 29]. Drop formation phenomenon depends significantly on the physical properties of the two liquids used. Though there are several studies on drop formation at nozzles, data on drop formation for TBP diluted in dodecane–nitric acid system is scarce. This study aims to fill this gap.

In this work, experiments were carried out to determine the drop diameter, necking/jetting length, and drop detachment time as a function of the dispersed phase velocities at different nozzle diameters and for different nitric acid concentrations. Variations in the nitric acid concentration cause variations in the physical and interfacial properties. High speed imaging was used to capture the phenomenon of drop formation. The data obtained by carrying out image analysis were regressed to determine correlations of drop diameter and minimum jetting velocity.

2 Experimental setup and procedure

A schematic diagram of the experimental setup is shown in Figure 1. The experimental set-up was made of borosilicate glass and stainless steel to ensure that interfacial contamination is avoided. De-ionized water and commercial grade TBP, dodecane, and nitric acid were used. While the dispersed phase was always 30%(v/v) TBP in dodecane, nitric acid of three different concentrations (0.01N, 1N and 3N) were used as the aqueous phase. The organic (dispersed) and aqueous (continuous) phases were mutually equilibrated prior to experiments. Density, viscosity, and interfacial tension were experimentally determined using mutually saturated phases. The use of mutually saturated phases helped eliminate the effect of mass transfer on drop formation. The glass column was filled with the continuous phase. 30% (v/v) TBP in dodecane was pumped through the nozzle at the desired flow rate using a precision syringe pump, and drops were generated at the tip of the nozzle. The nozzle was observed on a computer screen in real time using a high speed camera. The camera was set to capture video at a frame rate of 100 to 200 fps. The images acquired by the high speed imaging system were stored on a PC as video clips. These clips were analyzed using ImageJ, which is a powerful image processing software tool. The time of formation of 10 drops was determined from video playback. From the knowledge of the volumetric flow rate of the dispersed phase as measured by a precision syringe pump, and the time of formation of 10 drops, the average volume of one drop and its equivalent diameter were calculated. The nozzle tip was in the field of view in each video. The image scale in terms of pixel per mm was obtained from the nozzle outer diameter. This scale was then used to measure the drop detachment height, i.e. the necking length and the jetting length. The drop formation time could be easily calculated from the recording speed (frames per second) and number of frames captured between the detachment of two successive drops.

Figure 1Full-screen View

(Color online) Schematic diagram of the experimental setup

The drop formation phenomenon at the nozzles tips submerged in a quiescent continuous phase depends on the nozzle diameter, nozzle velocity, and physical properties of the liquids. Thus, experiments were devised so that effects of these three variables on the drop formation phenomenon could be studied. To understand the effect of nozzle size, three different stainless steel nozzles were used in these experiments. The inner diameters (ID) of the nozzles were 1.91 mm, 3.04 mm, and 4.88 mm. For each nozzle, the experiments were conducted with nitric acid of three different concentrations (0.01N, 1N and 3N) as the aqueous phase so that the effects of physical properties could be studied. For each combination of nozzle diameter and nitric acid concentration, the flow rate of the dispersed phase was varied in the range of 4-40 ml/min in order to understand the effect of dispersed phase velocity. Thus, 88 experiments were conducted in total.

The physical and interfacial properties of mutually saturated test systems used in the experiments are presented in Table 1.

3 Results and discussion

The high speed imaging system allowed visualization of drop formation phenomena, i.e., drop growth at the nozzle tip, onset of necking at intermediate values of velocity of the dispersed phase, and jet formation at higher velocities. The images acquired for the 1.91 mm inner diameter nozzle for Phase System – I (see Table 1) are shown in Figure 2. At lower velocities (⪯ 116 mm/s, or flow rate of ⪯ 20 mlpm), the drop detachment point is very close to the nozzle tip, and the drop size is governed by the balance between buoyancy and interfacial tension forces. The onset of necking is observed at about 140 mm/s. The necking phenomenon observed in experiments agrees with that reported in the literature [28, 30]. Velocity effects in the form of inertial and drag forces contribute in the force balance equation as explained by Scheele and Meister [23, 24] and Hamad et al. [27].

Figure 2Full-screen View

(Color online) Images of drop formation at the tip of the 1.91 mm inner diameter nozzle for different dispersed phase flow rates (velocities). The images are for Two-Phase System-I {30% (v/v) TBP and dodecane - 0.01 N nitric acid}.

Drop detachment height is measured from the tip of the nozzle. It is clearly observed from the captured images in Figure 2 and later on in Figure 3 that the increase in drop detachment height with increased velocity is gradual in the dripping region and steep in the jetting region. This is consistent with the observations of Scheele and Meister [23,24], Meister and Scheele [25,26], and Homma et al. [31]. A similar pattern was observed for the 3.04 mm inner diameter nozzle, whereas the jetting phenomena could not be captured due to the limited velocity range in the 4.88 mm inner diameter nozzle.

Figure 3Full-screen View

(Color online) Effect of the dispersed phase velocity on drop diameter for the three nozzles used in experiments with Two-Phase System -I.

3.1 Effect of dispersed phase velocity and nozzle diameter

Figure 3 shows the dependence of drop diameter on the dispersed phase velocity through the nozzle. The drop diameter first increases up to a maximum value with increase in the dispersed phase velocity, then reduces with continued increase in the dispersed phase velocity. This trend is in agreement with the two-stage drop formation model proposed in the literature [23, 24, 32-34]. At a lower dispersed phase velocity, the drop size is governed by the balance between the buoyancy {V_AN∆ρg} and interfacial tension {πd_Nγ} forces. As the velocity increases, the kinetic {4/3ρ_dU_NQ} and drag {C_DµQ(d_N/d)ⁿ/d} forces also contribute to the force balance equation. Among these forces, the interfacial force and drag force are the restraining forces, whereas the buoyancy force and the kinetic force are the lifting forces. The onset of necking is attributed to a larger lifting force. The dispersed phase continues to flow into the drop through the neck until the break off time. Eventually the diameter of the detached drop depends on the complex balance between the forces and the necking phenomenon. This complex force balance causes the drop diameter to first increase and then decrease with an increase in the dispersed phase velocity. Figure 3 shows that the drop diameter increases with an increase in the nozzle diameter. An increase in the interfacial force due to an increased nozzle diameter requires a higher lifting force, which is realized only with a larger drop. This causes drop diameter to increase with an increase in the nozzle diameter.

Figure 4 shows the effects of the dispersed phase velocity and nozzle diameter on the drop detachment height. It is observed that the drop detachment height first moderately increases linearly with increased dispersed phase velocity as can be seen from the slope of the initial part of the curves in Figure 4. Beyond a certain dispersed phase velocity, the drop detachment height increases steeply with increased dispersed phase velocity. This transition marks the transition from the dripping to the jetting regime. The velocity at which this transition occurs is called the minimum jetting velocity. For the same velocity, the drop detachment height increases with an increase in nozzle diameter. The above observations are in agreement with the simulation results reported by Soleymani et al. [28]. For a specific system, increased nozzle diameter results in an increased restraining force due to interfacial tension, which leads to an increased neck length before detachment.

Figure 4Full-screen View

(Color online) Effect of the dispersed phase velocity on drop detachment height for the three nozzles used in experiments with Two-Phase System-I.

The drop detachment time, shown in Figure 5, is the time between the detachment of two successive drops. Actually, it is the total of separation, spreading, growth, and necking/jetting period, as suggested by Soleymani et al. [28]. Drop detachment time depends strongly on the nozzle geometry, as well as the wettability of the materials. In the present case, there is no drop spreading as the stainless steel nozzle is not wetted by the organic phase. In the range of velocity values studied, the observed trend of drop detachment time is the same for all nozzles. As the velocity increases, the time interval between detachment of two successive drops decreases. This is due to the increased kinetic force with increased dispersed phase velocity. An increased nozzle diameter results in reduced drop detachment time, which is in agreement with the observation of Soleymani et al. [28].

Figure 5Full-screen View

(Color online) The effect of dispersed phase velocity on drop detachment time for the three nozzles used in experiments with Two-Phase System-I.

The experimental data was highly reproducible, as shown in Figure 6. This figure depicts drop diameter variation with flow rate for two different experimental datasets. The error bars represent ±5% of the values of the first experimental data set.

Figure 6Full-screen View

(Color online) Reproducibility of the data showing drop diameter variation with dispersed phase flow rate for the 1.91 mm ID nozzle used in experiments with Two-Phase System -II.

3.2 Effect of nitric acid concentration

The extraction and stripping chemistry of the TBP-dodecane solvent depends on the nitric acid concentration, which typically ranges from 0.01 N to 3 N [14-16, 35]. The extraction of heavy metals is carried out at about 3N nitric acid due its preferentially higher distribution coefficient, whereas the stripping is carried out at 0.01 N nitric acid as the TBP–metal complex is unstable at lower concentrations. Physical properties of the aqueous phase change with nitric acid concentration. Thus, drop formation dynamics is also affected by nitric acid concentration. Therefore, it is necessary to study the effect of nitric acid concentration on drop diameter, drop detachment height, and drop detachment time. Figure 7 shows how the dispersed phase velocity affects the drop diameter for three different aqueous phases with different nitric acid concentrations. The trend of increase and subsequent decrease of drop diameter with increase velocity is observed for all nitric acid concentrations and is in conformity with the two-stage drop formation model. For nitric acid concentrations ranging from 0.01N to 3N, the interfacial tension increases marginally from 9.38 mN/m to 10.99 mN/m, whereas the density difference between the phases increases from 194 kg/m³ to 262 kg/m³. Thus, an increased density difference is significant compared to the marginal increase in interfacial tension. Thus, with an increase in nitric acid concentration, the increase in the buoyancy force is more pronounced than the increase in the interfacial tension force. This causes the drop diameter to reduce with increasing nitric acid concentration.

Figure 7Full-screen View

(Color online) Effect of dispersed phase velocity on drop diameter for three nitric acid concentrations in the quiescent continuous phase (nozzle diameter = 1.91 mm).

Figure 8 shows the effect of dispersed phase velocity on the drop detachment height for three different nitric acid concentrations. As nitric acid concentration increases, the transition from dripping to jetting occurs at a higher velocity. Higher jetting velocity at higher nitric acid concentration is attributed to higher values of interfacial tension. However, the effect is not significant for the range of interfacial tension values in this study. Figure 9 shows that for a given dispersed phase velocity, the drop detachment time is lower at higher nitric acid concentration. A lower drop detachment time at higher nitric acid concentration is attributed to higher values of density difference, leading to an increased buoyancy force and resulting in a faster detachment of drops.

Figure 8Full-screen View

(Color online) The effect of velocity on drop detachment height for three nitric acid concentrations in the quiescent continuous phase (nozzle diameter = 1.91 mm).

Figure 9Full-screen View

(Color online) The effect of velocity on drop detachment time for three nitric acid concentrations in the quiescent continuous phase (nozzle diameter = 1.91 mm).

3.3 Correlation for drop diameter

Since drop diameter is a very important hydrodynamic variable which directly affects specific interfacial area for mass transfer, its accurate prediction is necessary for given combinations of independent variables, such as dispersed phase velocity and nozzle diameter. The experimental data obtained in this study were regressed to obtain correlations ofdrop diameter. The non-linear regression was done by using Lab Fit curve fitting software [36]. When all the experimental data are used for regression, the drop diameter could be correlated by Eq (1).

$\frac{d}{d_{\text{N}}}=\frac{2.1962}{Eo}+0.76739\exp (-0.3976Fr)-\frac{0.82523}{E{o}^2}$ (1)

The average absolute relative error in the fit of Eq (1) is 5.23%. The parity plot is shown in Figure 10. Eo is the Eötvös number, which represents the ratio of buoyancy force to interfacial tension force. Fr is the Froude number, which represents the ratio of inertial force to the gravitational force. Eo and Fr are expressed by Eq (2) and (3), respectively. These dimensionless numbers account for all the restraining and lifting forces that affect the drop formation phenomenon.

Figure 10Full-screen View

(Color online) Parity plot of the experimental values of d/d_N and d/d_N values predicted by the correlation of Eq (1) considering all experimental data.

$Eo=\frac{\left({\rho }_{\text{c}}-{\rho }_{\text{d}}\right)g{d}_{\text{N}}^{\text{2}}}{\gamma }$ (2) $Fr=\frac{U_{\text{N}}^2}{g{d}_{\text{N}}}$ (3)

If only experimental data in the dripping regime are used, regression leads to the correlation given by Eq (4). The average absolute relative error in the fit of Eq (4) is 4.24% which is slightly better than the fit of Eq (1). The parity plot of the correlation given by Eq (4) is shown in Figure 11.

Figure 11Full-screen View

(Color online) Parity plot of the experimental values of d/d_N and d/d_N values predicted by the correlation of Eq (4) by considering experimental data of the dripping regime only.

$\frac{d}{d_{\text{N}}}=2.0347e^{\left(0.0058342Eo-0.10727F{r}^2\right)}-0.57065\ln (Eo)$ (4)

3.4 Correlation for minimum jetting velocity

The jetting velocity (U_j) is the minimum velocity of the dispersed phase through the nozzle at which the jet forms. It is important to know the minimum jetting velocity for a given phase system (physical properties) and nozzle diameter as it separates the dripping and jetting regimes. In this work, we have tested two of the previously reported correlations for minimum jetting velocity and adapted them for use with our experimental data. The previously reported correlations of Ryan [37] and de Chazal and Ryan [33] were tested. These correlations are given by Eq (5) and Eq (6), respectively.

$U_{\text{j}}=1.16{\left(\frac{\gamma }{\Delta \rho g{d}_{\text{N}}^2}\right)}^{0.95}{(d_{\text{N}}g)}^{0.5}\left(\frac{\Delta \rho }{{\rho }_{\text{d}}}\right)$ (5) $U_{\text{j}}=\sqrt{\frac{2\sigma }{{\rho }_{\text{d}}{d}_{\text{N}}}}\left(1.07-0.75\sqrt{\frac{\Delta \rho g{d}_{\text{N}}^2}{4\gamma }}\right)$ (6)

The above correlations are tested with our limited experimental data (6 data points) for minimum jetting velocity. The correlation of Ryan is found to give an average absolute relative error of about 64%, whereas the correlation of de Chazal and Ryan is found to give an average absolute relative error of about 40%. Since the correlation of de Chazal and Ryan is found to give a better match, it is adapted to fit our experimental data. Due to the limited experimental data, we attempted a two parameter model with $\sqrt{2\sigma /{\rho }_{\text{d}}{d}_{\text{N}}}$ and $\sqrt{\Delta \rho g{d}_{\text{N}}^2/4\gamma }$ as the independent variables. The correlation given by Eq (7) is obtained with a very low average absolute relative error of only 2.97%. Figure 12 shows the parity plot of the jetting velocity obtained experimentally with that predicted by Eqs (5) - (7).

Figure 12Full-screen View

(Color online) Parity plot of the experimental and predicted values of minimum jetting velocity.

$U_{\text{j}}=13.8138\left(\frac{2\gamma }{{\rho }_{\text{d}}{d}_{\text{N}}}\right)-0.0364321\sqrt{\frac{\Delta \rho g{d}_{\text{N}}^2}{4\gamma }}.$ (7)

4 Conclusion

Hydrodynamics of formation of a single drop of TBP-dodecane solvent in a quiescent dilute nitric acid phase is studied with varying dispersed phase velocity and using nozzles of different diameters. Nitric acid concentration is also varied from 0.01N to 3N. The drop formation process was captured by a high speed video technique, and various stages of drop formation, i.e., drop growth, necking, jetting, and drop detachment, are distinctly observed in the images. The drop diameter is found to first increase and subsequently decrease with dispersed phase velocity. The effect of velocity on drop detachment height clearly indicates a transition from the dripping to the jetting regime. The drop detachment time is found to reduce with an increase in dispersed phase velocity. For a given dispersed phase velocity, an increased nozzle diameter leads to an increased drop diameter and drop detachment height and a reduced drop detachment time. The drop diameter and drop detachment time are found to increase with increase in nitric acid concentration. The drop detachment height does not vary significantly with change in nitric acid concentration. The minimum jetting velocity is found to be lower for a smaller diameter nozzle. Similarly, the minimum jetting velocity is found to be lower for a lower nitric acid concentration. Nonlinear regression is used to obtain the correlations for drop diameter and minimum jetting velocity. The resulting correlations are found to be very accurate with average absolute relative errors less than or equal to 6%. The proposed correlations will be useful for estimating drop size, as well as to quantify the onset of jetting in a two-phase system involving TBP-dodecane and nitric acid, which is widely used for liquid-liquid extraction-based separation processes in the nuclear fuel cycle.