METROLOGICAL ANALYSIS OF DIFFERENT TECHNIQUES FOR MEASURING INTERFACE TENSION BETWEEN TWO FLUIDS BASED ON SPINNING DROP METHOD

. The spinning drop method foundations of measuring interface tension between two immiscible liquids are considered. Different techniques of the spinning drop method and their metrology evaluation are compared. The dimensionless parameters of spinning drop are calculated using the fourth-order Runge–Kutta procedure and they are approximated by the seventh-order polynomial dependence. The relative errors of the different techniques and the approximate dependence are obtained. modeling processes; processing interpretation measured data.


Introduction
Interface tension (IT) at the interface of two insoluble liquids is a significant parameter of the technological processes where surface characteristics at the interface are essential. This is especially important in the oil production methods with the help of reservoir pressure maintenance using surfactants (SAA) [3]. It should also be noted that IT can vary in the range of 0.01÷20 mN/m. Measurement of such IT values is usually carried out with the help of the devices that implement the spinning drop method (SD) [5]. The essence of the SD method consists in the following: a horizontally placed glass tube is filled with such a heavier fluid under study as aqueous surfactant solution; after that a drop of such a lighter fluid under investigation as oil is injected into this fluid; then the tube is revolved around its horizontal axis with a certain angular velocity  . Both the appropriate SD dimensions (for example, its largest diameter, length, and volume) and the density difference of the interfacial fluids are measured depending on the selected techniques for determining IT; the IT values  [4,[6][7][8] are calculated with the help of the corresponding dependencies [4,[6][7][8].
Among such dependencies, regardless of the date when their authors published them, the following are wide spread now B. Vonnegut's dependence [1]: where r dimensionless parameter which is determined on the basis of the appropriate J. Slattery's

Theoretical Part
Let us conduct theoretical calculation of the SD geometrical dimensions in order to evaluate method errors of the abovementioned techniques.
Let us consider the horizontal rotating tube, inside of which there is fluid 2 with higher density same time, we neglect the gravitational force, which allows us to suggest that the rotation axes of the tube 1 and drop 3 coincide.

Fig. 1. Rotating tube with investigated heavier and lighter fluids
Then the pressure 1 A P inside the drop in pt. А is as the following: ydistance from pt. А to the х axis. Correspondingly, the pressure outside the drop in pt. А is as follows: Hence, the pressure difference along the interface of two fluids in pt. А is as the following: In case there is gravitational force, the drop rotation axis shifts in relation to the tube rotation axis by the value which is equal to , where ggravitational acceleration, dynamic viscosity of the heavier fluid. However, the SD form doesn't change therewith. On the other hand the pressure difference along the interface in pt. A will be as follows: Rcurvature radii of the drop surface in pt. А in the plane of fig. 1 and in the plane that is perpendicular to the plane of Fig. 1 respectively [2].
Besides, the pressure difference 0 P  along the interface on the level of the horizontal rotation х axis in pt. О will be as the following [1]: where 0 Rcurvature radius of the SD interface surface in pt. О (Fig. 1).
Then, when we take into account dependencies (8) and (9), dependence (7) will be as the following: Equation (10) Having multiplied both the left and the right parts of (12) by a , we will obtain an equation in a dimensionless form that describes the SD surface:   When solving (13) and (14) for different specified values of 0 / Ra at the moment when the angle reaches  = 90°, we find the corresponding SD geometrical parameters. The initial boundary conditions are the following: (15) and the final boundary conditions are as follows:

RR
(16) When the final conditions of (16) are reached, there isn't any further increase in the parameters according to (16) and the SD surface becomes strictly cylindrical, i. e.

Results and Discussion
Some of the results of the SD dimensionless parameters ( The results of such error calculation are provided in table 3.  (18), have a small method error in the indicated range of values / 2R l . However, when implementing B. Vonnegut and J. Slattery's techniques there is a necessity to measure the largest SD radius 2R , which is significantly influenced by the optical zoom factor Ì of the tube with the fluids under study that can vary in the range from 1.332 to 1.34 [1]. Calculation of a certain Ì value depends on many factors and it can lead to significant additional errors of the obtained results.
Therefore, it is advisable to use the techniques that do not involve measurement of the largest SD diameter 2R (S. Torza and H. Princen's techniques and approximate dependence (18)). However, S. Torza and H. Princen's techniques are characterized by significant method errors.
Therefore, it is recommended to use approximate dependence (18) given that modern means for IT  measurement are equipped with computer aids. This allows to easily develop the appropriate software that would consider dependence (18).