**3. Dimensional Analysis**

The purpose of this study was to establish a relationship between the mixing time and other valid variables. Their dimensions were considered through dimensional analysis. In this method, the first step is to select the appropriate initial parameters, including the input parameters and output parameters of the mixing time. The second step is to group these initial parameters into a dimensionless group and organize the new relationships between the various parameters. In particular, it is important to select the initial parameters precisely, since there is a need for a unique relationship between the chosen parameters [28,29].

Several factors may affect the value of mixing time. The variables that may affect the mixing time are density ρ, kinematic viscosity υ, surface tension σ, stirring energy ε [30], etc. In this study, 20 initial parameters were selected to identify the dimensionless groups of the parameters which can quantify the mixing time, as shown in Table 2. Based on this, 17 corresponding dimensionless groups were obtained from the selected initial parameters, according to the Vaschy–Buckingham theorem.


**Table 2.** Selected parameters for the dimensional analysis of the mixing time.


**Table 2.** *Cont*.

In Table 2, M, L, t are three basic dimensions, where M stands for the mass dimension, kg, L for the length dimension, m, and t for the time dimension, s.

The mixing time can be represented by the following functional relationships:

$$
\pi = F(\rho, \upsilon, \sigma, \tau, \varrho, \varepsilon, H\_f, D\_f, H\_{\text{oil}}, H\_{\text{us}}, H\_{\text{us}}, H\_{\text{ls}}, D\_{\text{ts}}, H\_{\text{is}}, R\_{\text{is}}, H\_{\text{ts}}, R\_{\text{ns}}, R\_{\text{ns}}, h\_{\text{ss}}, V\_{\text{es}}, q\_{\text{s}}) \tag{1}
$$

Meanwhile, since many factors are constant in this study, such as the height of the furnace, the height of the high vacuum oil, the height of the side nozzle, the diameter of the side nozzle and so on, the functional relationships can be further simplified to

$$
\pi = \mathcal{F}(\rho, \upsilon, \sigma, \tau, \lg, \varepsilon, D\_{\mathfrak{s}\iota} H\_{\mathfrak{s}\iota} h\_{\mathfrak{s}\iota} V\_{\mathfrak{s}\iota} q\_{\mathfrak{s}\iota}) \tag{2}
$$

The equation can be expressed in the following dimensionless equation:

$$
\pi = q(\pi\_1, \pi\_2, \pi\_3, \pi\_4, \pi\_5, \pi\_6, \pi\_7, \pi\_8) \tag{3}
$$

where π1~π<sup>8</sup> is a dimensionless group, and ϕ is a functional symbol. After substituting in each variable, it gives

$$
\pi = \rho^a \upsilon^b \sigma^c \tau^d g^c \varepsilon^f D\_s^g H\_i^h h\_s^i V\_s^j q\_s^k \tag{4}
$$

$$M^0L^0t^0 = \left(ML^{-3}\right)^a \left(L^2t^{-1}\right)^b \left(Mt^{-2}\right)^c \left(t\right)^d \left(Lt^{-2}\right)^c \left(ML^2t^{-3}\right)^f \left(L\right)^g \left(L\right)^h \left(L\right)^i \left(Lt^{-1}\right)^j \left(L^3t^{-1}\right)^k \tag{5}$$

By replacing the other variables with ρ, *Ds* and *Vs*, we get

$$
\pi = \rho^{(-c-f)} \upsilon^b \sigma^\epsilon \tau^d g^\epsilon \varepsilon^f D\_s^{(-b-c-d+c-2f-h-i-2k)} H\_l^h h\_s^i V\_s^{(-b-2c+d-2c-3f-k)} q\_s^k \tag{6}
$$

$$
\pi = \left(\frac{\upsilon}{D\_S V\_S}\right)^b \left(\frac{\sigma}{\rho D\_s V\_s^2}\right)^c \left(\frac{\tau V\_s}{D\_s}\right)^d \left(\frac{\mathcal{g} D\_s}{V\_s^2}\right)^e \left(\frac{\varepsilon}{\rho D\_s^2 V\_s^3}\right)^f \left(\frac{H\_i}{D\_s}\right)^h \left(\frac{h\_s}{D\_s}\right)^i \left(\frac{q\_s}{D\_s^2 V\_s}\right)^k \tag{7}
$$

Since the side-blown nozzle is located in the high vacuum oil, the kinematic viscosity here is selected as the high vacuum oil kinematic viscosity, *voil*, m2·s−1. The tracer feeding position and the insertion position of the side nozzle are closely related to the total height of the water phase and oil phase, the *DS* in the tracer feeding position number and the insertion depth number of side nozzle are replaced by the total height of the water phase and oil phase *Hoil*<sup>+</sup>*w*, which can be converted into the following equation:

$$
\pi = \left(\frac{\nu\_{\rm oil}}{D\_{\rm S}V\_{\rm S}}\right)^{\rm b} \left(\frac{\sigma}{\rho D\_{\rm s}V\_{\rm s}^{2}}\right)^{c} \left(\frac{\tau V\_{\rm s}}{D\_{\rm s}}\right)^{d} \left(\frac{gD\_{\rm s}}{V\_{\rm s}^{2}}\right)^{e} \left(\frac{\varepsilon}{\rho D\_{\rm s}^{2}V\_{\rm s}^{3}}\right)^{f} \left(\frac{H\_{\rm i}}{H\_{\rm oil+w}}\right)^{h} \left(\frac{h\_{\rm s}}{H\_{\rm oil+w}}\right)^{i} \left(\frac{q\_{\rm s}}{D\_{\rm s}^{2}V\_{\rm s}}\right)^{k} \tag{8}
$$

In this work, the angle between the upper side nozzle and the lower side nozzle is also a relatively important factor, including the relative angle and horizontal angle, so the angle factor should be taken

into account when studying the dimensionless groups. The momentum dimensionless group is studied based on different angles. δ is introduced here, that is,

$$\delta = \left(\frac{\pi}{4}\right)^{\frac{1}{2}} \times \left(2 \times \left(V\_s \times \cos\alpha\right)^2\right)^{\frac{1}{4}} \times \left(\frac{D\_f}{\left(Q\_s\right)^{\frac{1}{2}}}\right) \tag{9}$$

where α is combined the the relative angle between the upper side nozzle and the lower side nozzle and the horizontal angle of the upper side nozzle and the lower side nozzle. This parameter synthetically considers the influence of flow velocity with various angles, *Vs* is flow velocity of the side nozzle, <sup>m</sup>·s<sup>−</sup>1. *Df* is the diameter of the furnace, m, and *Qs* is the volumetric flow rate, m3·s<sup>−</sup>1.

Through the above dimensionless group derivation, a series of dimensionless groups can be obtained as follows in Table 3.


**Table 3.** Expression of dimensionless groups based on Equations (8) and (9).

The original dimensionless groups mentioned above can be substituted into Equation (10) to obtain

$$
\pi = \wp(\frac{\upsilon\_{\rm oil}}{D\_{\rm S}V\_{\rm S}}, \frac{\sigma}{\rho D\_{\rm s}V\_{\rm s}^{2}}, \frac{\tau V\_{\rm s}}{D\_{\rm s}}, \frac{\mathcal{g}D\_{\rm s}}{V\_{\rm s}^{2}}, \frac{\varepsilon}{\rho D\_{\rm s}^{2}V\_{\rm s}^{3}}, \delta\_{\prime} \frac{H\_{\rm i}}{H\_{\rm oil+w}}, \frac{h\_{\rm s}}{H\_{\rm oil+w}}, \frac{q\_{\rm s}}{D\_{\rm s}^{2}V\_{\rm s}}) \tag{10}
$$

Extracting the τ from Equation (10) and rearranging, we get

$$\tau = \frac{D\_s}{V\_s} (\frac{\upsilon\_{o\bar{u}l}}{D\_S V\_S}, \frac{\sigma}{\rho D\_s V\_s^2}, \frac{gD\_s}{V\_s^2}, \frac{\varepsilon}{\rho D\_s^2 V\_s^3}, \delta, \frac{H\_i}{H\_{o\bar{u}l+w}}, \frac{h\_s}{H\_{o\bar{u}l+w}}, \frac{q\_s}{D\_s^2 V\_s}) \tag{11}$$

where <sup>υ</sup>*oil* is the kinematic viscosity of the high vacuum oil, m2·s<sup>−</sup>1, <sup>σ</sup> is the surface tension of the high vacuum oil, kg·s−2, *g* is the acceleration of gravity, m·s−2, <sup>ε</sup> is the stirring energy of the side nozzle, kg·m2·s−3, <sup>δ</sup> is the dimensionless groups of momentum, *qs* is the flow rate of the single side nozzle, <sup>m</sup>3·s<sup>−</sup>1, *hs* is the insertion depth of the side nozzle, m, *Hoil*<sup>+</sup>*<sup>w</sup>* is the total height of the high vacuum oil and water, m, *Ds* is the diameter of the single side nozzle, m, and *Vs* is the flow velocity of the side nozzle, m·s<sup>−</sup>1.

The dimensionless groups related to Equation (11) based on a physical chemistry handbook are shown in Table 4:


**Table 4.** The dimensionless groups associated with Equation (11) based on the physical chemistry handbook [31].

Combining Tables 3 and 4, the dimensionless groups in Equation (11) can be sorted into Table 5.

**Table 5.** The dimensionless groups in Equation (11).


Based on the fact that KF Z2 is constant in this condition, it will be removed here. At the same time, *Njm* is named in order to simplify the expression of <sup>j</sup> <sup>M</sup>×Nsh N 5 3 sc , that is Njm <sup>=</sup> <sup>j</sup> <sup>M</sup>×Nsh N 5 3 sc . According to Tables 3–5 and the actual working conditions of this study, the equation of dimensionless groups can be expressed as follows:

$$\pi = \frac{D\_s}{V\_s} \Big( \text{Re}^a, \text{Ca}^b, \text{La}\_1{}^c, \delta^d, \text{N}\_{jm}^c \left( \frac{H\_i}{H\_{oil+w}} \right)^f, \left( \frac{h\_s}{H\_{oil+w}} \right)^{\mathcal{S}} \right) \tag{12}$$

As can be seen from Table 4, Re Ca <sup>=</sup> <sup>σ</sup>*Ds* ρυ*oil*<sup>2</sup> is a constant. Since Ca is the dimensionless group representing the relative effect of viscous drag forces versus surface tension forces acting across an interface between a liquid and a gas, or between two immiscible liquids, so the Capillary number (Ca) is retained here, thus the following dimensionless groups equation can be further obtained:

$$\tau = \frac{D\_s}{V\_s} \Big( \mathbf{C} \mathbf{a}^a, \mathbf{L} \mathbf{a}\_1^b, \mathbf{\delta}^c, \mathbf{N}\_{jm'}^d \Big( \frac{H\_i}{H\_{oil+w}} \Big)^e \Big( \frac{h\_s}{H\_{oil+w}} \Big)^f \Big) \tag{13}$$

where *a*, *b*, *c*, *d*, *e*, and *f* are the empirical coefficient, to be derived from the experimental data.

Through the analysis of the data obtained in the hydraulic simulation experiment, the influence of the dimensionless quantities and the mixing time was obtained.
