Predicting transition from selective withdrawal to entrainment in two-fluid stratified systems

Selective withdrawal is a desired phenomenon in transferring oil from large caverns in the U.S. Strategic Petroleum Reserve (SPR), because entrainment of oil at the time during withdrawal poses a risk of contaminating the environment. Motivated to understand selective withdrawal in an SPR-like orientation, we performed experiments in order to investigate the critical submergence depth as a function of critical flow rate. For the experiments, a tube was positioned through a liquid-liquid interface that draws the lower liquid upward, avoiding entrainment of the upper fluid. Analysis of the normal stress balance across the interface produced a Weber number, utilizing dynamic pressure scaling, that predicted the transition to entrainment. Additionally, an inviscid flow analysis was performed assuming an ellipsoidal control volume surface that produced a linear relationship between the Weber number and the scaled critical submergence depth. This analytical model was validated using the experimental data, resulting in a robust model for predicting transition from selective withdrawal to entrainment.

Figure 1

(a) Primary orientation in an initial state. (b) Primary orientation in a steady, intermediate subcritical flow rate state. (c) Primary orientation just after the inception of entrainment. (d) Secondary orientation just after the inception of entrainment. All the tube inner diameters in the figure are 10.8 mm.

Figure 2

Schematic diagram of the experimental setup. Cameras (not shown) were located perpendicular to the two faces of the hexagonal test tank [1].

Figure 3

Plots showing the nondimensional submergence depth as a function of various nondimensional parameters. The secondary orientation data are presented by “filled” symbols, whereas the primary orientation data are presented by “hollow” symbols. The symbols are defined in Table 1. (a) Nondimensional submergence depth as a function of capillary number as in Ref. [4]. The dotted red line shows Cohen's correlation. (b) Nondimensional submergence depth as a function of capillary number as in Ref. [6]. (c) Nondimensional submergence depth as a function of Froude number as in Ref. [8]. (d) Nondimensional submergence depth as a function of Froude number as in Ref. [2]. The dotted red line shows Lubin's correlation; see Eq. (15).

Figure 4

Weber number as a function of Ohnesorge number. The data from Cohen's experiment are shown using the symbol $*$. The data from Lubin and Springer's experiment are shown using the symbol $+$. The data from this experiment had two orientations. The fluid combinations for the primary and the secondary orientation are expressed using the symbols mentioned in Table 1.

Figure 5

Schematic of symbols and control volume used in the analysis of inviscid model. (a) Primary orientation. (b) Secondary orientation. The critical flow rate is denoted by $Q_{cr}$, the tube diameter by $d$ and the submergence depth by $S$. $H_d$ and $b$ are the axes of the 2D ellipsoid. $U_s$ is the average velocity normal to the control volume surface and $a$ is the offset of the ellipsoidal control volume from the tube opening

Figure 6

An example multiphase Eulerian-Eulerian CFD simulation for system 4, withdrawal of bottom layer fluid at subcritical flow rate using Ansys fluent [16]. Label (1) denotes the initial position of the undisturbed interface; label (2) denotes an equilibrium position of the interface; label (3) denotes the control volume where the velocity vectors are perpendicular to the ellipsoidal control surface (dashed line); and label (4) denotes isovelocity contours.

Figure 7

Critical submergence depth as a function of critical Weber number. The “hollow” symbols represent the primary orientation data, whereas the “filled” ones represent the secondary orientation data. The solid line through the hollow symbols is the best linear fit for the primary orientation and the solid line through the filled symbols is the best linear fit for the secondary orientation. The dotted lines are 95% prediction intervals. The region above the 95% upper bound of entrainment depth is selective withdrawal zone, whereas the area below is entrainment zone. The symbols are defined in Table 1.

Figure 8

Ellipsoidal control volume parameters $(H_d,b)$ as a function of the inverse Bond number for (a) primary orientation, (b) secondary orientation, and (c) Lubin and Springer model. The fluid combination of system 3 (Table 1) was chosen. The flow rate was set to be 0.35 ${\text{m}}^3/\text{s}$. The surface tension coefficient was varied from 0.001 to 0.1 N/m and eight selected diameters were chosen from a very wide range of 0.001 to 0.30 m. The solid line is for plotting $H_d$ and the dashed line is for plotting $b$.