THEORY AND MODELING OF THE INTERFACIAL TENSION FORCE ON THE OIL SPREADING UNDER ICE COVERS

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1 Ice in the Environment: Proceedings of the 6th IAHR International Symposium on Ice Dunedin, New Zealand, 2nd 6th December 2002 International Association of Hydraulic Engineering and Research THEORY AND MODELING OF THE INTERFACIAL TENSION FORCE ON THE OIL SPREADING UNDER ICE COVERS A. Konno and K. Izumiyama 2 ABSTRACT Interfacial tension force which acts on the edge of an oil slick was believed to be derived from oil-water, ice-water, and ice-oil interfacial tensions and the contact angle, but its precise formulation was not known. This study investigated interfacial tension force theoretically and presents an accurate and original model of this force. Oil-spill experiments were carried out at the ice model basin at National Maritime Research Institute to verify this model and to observe the behavior of oil under ice covers. Theoretical estimation of oil slick size was also carried out. These results conform closely to one another. INTRODUCTION Risk of oil pollution has risen in the Sea of Okhotsk from the start of the crude-oil/gas production at the northern Sakhalin offshore. Winter production has not begun, but is planned to start from Therefore, an oil spill in the frozen ocean is not a groundless concern. However, research and preparation to cope with this serious situation seems to be insufficient in Japan. To take countermeasures against a spill hazard, this research project was organized by five research institutes, including the authors; it seeks to elucidate behavior and methods of withdrawal of spilled oil from a frozen ocean. We are responsible for theoretical analyses and fundamental experiments. This study is a part of that research project. If oil spills under a flat ice cover, the oil forms a round slick; thus the slick size can be represented by its radius. The relationship between interfacial tension and final slick radius was discussed by Yapa and Chowdhury (989). They introduced net interfacial tension σ n, which they considered to be derived from oil/water, ice/water, and ice/oil interfacial tensions and the contact angle. They also carried out a laboratory experiment, finding that these estimations agreed with experimental results. However, the physical meaning of interfacial tension and how to estimate it was not given. Konno and Izumiyama (2002) applied the theory of ADSA (Axisymmetric Drop Shape Analysis) to analysis of Yapa and Chowdhury (989); they found that net interfacial ten-

2 sion is twice the oil/water interfacial tension, under the assumption that the contact angle of the oil/ice/water interface is nearly equal to 80. THEORY AND APPLICATION OF THE ADSA This study made use of the theory of ADSA (Axisymmetric Drop Shape Analysis) constructed by Rotemberg et al. (983). They introduced the reasonable assumptions that external forces, other than gravity, are absent, and that sessile/pendant drop is axisymmetric. Under these assumptions, the meridian curve of a sessile/pendant drop can be represented by arc length s measured from the apex, as x = x(s) and z = z(s); these obey the following differential equations: d x d s = cos φ d z d s = sin φ dφ d s = 2 + ( ρ)g R2 0 z sin φ σ x. Here, ρ is the density difference between the drop and the surrounding fluid, R 0 is the radius of curvature at the apex, σ is interfacial tension, g is gravity acceleration, and φ is the turning angle measured between the tangent to the interface at the point (x, z) and the datum plane; x, z and s are non-dimensionalized by R 0, as s = s/r 0, x = x/r 0 and z = z/r 0. Also, ( ρ)gr 2 0 /σ is the shape parameter and is often written as β. It should be noted that the above equations represent both pendant and sessile drops, depending on axis arrangement and the choice of plus and minus values of ρ. The above differential equations in () determine the drop shape with given R 0 and β = ( ρ)gr 2 0 /σ. To determine interfacial tension σ from the drop shape, the curvefitting problem must be solved with parameters R 0 and β to fit the calculated drop shape to the measured one. Solution methods for the differential equations and optimization method used in this research were explained by Konno and Izumiyama (2002); they are not explained here for the lack of space. Verification of the method To verify the method, measurement and related programs, the surface tension of the water (the interfacial tension of the water/air interface) was measured by the pendant-drop method. Figure shows the shape of the water drop and measured results of this drop. Measured surface tension was 99.7 mj/m 2, at 4 C while the real surface tension is mj/m 2 at 5 C (National Astronomical Observatory, 2000). Considering difficulties of surface tension measurement of the water because of its high sensitivity to surfactant additives, this result appears reasonable. Further verification continues. ()

3 R 0 =.38 mm, σ = 99.7 mj/m 2 Figure : An example of the ADSA measurement: water in air for verification height controller screws to control the ice plate level syringe to inject oil under the ice plate ice plate oil drop water basin (30cm x 30cm x 30cm) digital camera with a macro lens Figure 2: Experimental apparatus to measure oil drop shapes Measurement of the oil/water interfacial tension Figure 2 shows the experimental apparatus for measuring the oil drop meridian shape. Machine oil #0 was provided for measurement after it was blackened by oil-solvent dye. The drop size was varied so that drops of different shapes could be measured. Figure 3 shows examples of drop shapes and measured results. The average oil/water interfacial tension of this oil was J/m 2.

4 Drop #49 0 Measured ice bottom 8 z [mm] 6 Regressed curve 4 Measured surface points x [mm] R0 = 4.0 mm, σ = 25.6 mj/m2 Figure 3: An example of the ADSA measurement: oil in water Flat line in the right graph of Figure 3 represents measured bottom of the ice sheet. To measure the contact angle, differential equations in () are integrated until z = R0 z reaches the ice sheet; φ at that time is the contact angle. In this measurement, however, the curvature sometimes did not reach the line before φ reached 80. That occurred because of the difficulty of accurately determining the ice bottom from photographs such as Figure 3. From limited results of measurement, we concluded that the contact angle was near 80. Further explanation and measurement results are reported in Konno and Izumiyama (2002). MODELING OF INTERFACIAL TENSION FORCE FOR THE OIL SPILL PROBLEM If oil spills under an ice cover, the oil slick may behave under the effects of: buoyancy; interfacial tensions between oil/water and oil/ice interfaces; friction from water and ice; and adhesion to the ice bottom. However, these important parameters were not yet formulated clearly, except buoyancy. This section presents a model equation for interfacial tension force. The following matters are assumed: () the bottom of the oil slick is flat, except near the edge, and the edge of the oil slick is smooth; (2) thickness of the oil slick, h, is far smaller than radius R; and (3) the contact angle on the oil/water/ice interface is 80. We start from the classical Laplace equation that describes the pressure difference across a curved interface, = P, (2) + σ R R2 where σ is the interfacial tension, R and R2 represent the two principle radii of curvature of the interface, and P is the pressure difference across the interface. Using the radius of curvature on the edge r = r(s), which is defined as shown in Figure 4, Eq. (2) can be transformed as follows: σ. + (O(r/R2 ) = O(h/R) ) P = σ R R2 r

5 ice oil slick h water oil P φ r(s) s water Figure 4: Illustration of the edge shape of the oil slick and the relationship between the position, the radius of curvature and the perpendicular direction of the surface In addition, from the geometrical relationship, dφ/ds = /r. Therefore, P = σ dφ ds. To calculate the interfacial tension force on a unit length of the slick edge f t, the level component of this pressure difference should be integrated. f t = P sin φ ds = σ dφ sin φ ds arc arc ds π (3) = σ sin φ dφ = 2σ We thereby obtain the simple model: f t = 2σ. 0 It should be mentioned that the upper limit of the interval of integration in Eq. (3) is equal to the contact angle. Therefore, the net interfacial tension and the final slick radius are functions of the oil/ice interfacial tension and the contact angle. Verification of the model by theoretical and experimental results To verify this model by theory and experiments, we formulate the approximate relationship between volume and radius of the oil slick under flat ice covers. Suppose the oil slick is large and flat so that volume V can be approximated to be the product of thickness and bottom area, as V = πr 2 h. The relationship between oil thickness h and the interfacial tension force on a unit length f t comes from consideration of the balance of buoyancy and f t, under assumption of hydrostatic conditions. This reasoning yields 2 ( ρ)gh2 = f t.

6 From the above equations and Eq. (3), h is eliminated and the relationship between V and R is derived as follows. ( ) 2 V 2 ( ρ)g = 2σ πr 2 ( ) ( ρ)gv 2 /4 R = (4) 4π 2 σ On the other hand, if the radius of curvature at apex R 0 is given, the shape of the corresponding oil drop (slick) can be calculated by integrating equations in () from s = 0 until φ = 80. The radius of the slick is the maximum value of x = R 0 x. The volume of the slick can also be calculated by again integrating the shape of the meridian. By varying R 0 parametrically, the relationship between the oil-slick volume and radius is obtained. This relationship was adopted as the theoretical relationship. We carried out oil-spill experiments using the ice model basin at the National Maritime Research Institute. Experimental facilities, conditions and results are reported in Izumiyama and Konno (2002). In this paper we limit our discussion to a consideration of results of experiments under flat ice covers and comparison of the model and theoretical relationship. Figure 5 shows measured results, theoretical estimation and the model estimation of oil slick radii. The theoretical line ( Theory in Figure 5) and model line ( Model ) are very close to each other. These meet the experimental result ( Experiment in Figure 5) very well, especially in the small volume region. In the large volume region, there are small discrepancies; experimental result values are less than those achieved by theoretical estimation. This might be because the radii of the oil slicks were not yet converged; these were therefore smaller than final slick radii. Application of the model to an oil spill under uneven ice covers The discussion above addresses oil spills under flat ice covers. The above estimations are inappropriate for oil spills under uneven ice covers, but the model of the interfacial tension force can be applied. The model equation says that the interfacial tension force is twice that of the oil/ice interfacial tension per unit length. Figure 6 illustrates interfacial tension acting an oil slick on the interface. Strength of the interfacial tension force is a function only of length of the interface; the direction is always from the outside to the inside, normal to the interface. Therefore we should consider only the two-dimensional problem. The key question is how to determine the oil-slick boundary. Numerical modeling and simulation of the oil spill under uneven ice covers are in progress as a part of our continuing research project.

7 Figure 5: Comparison of theoretical estimation and experimental results for oil slick radii Figure 6: The interfacial tension forces of an oil spill are inwardly directed and normal to the boundary of the spill CONCLUSIONS Theoretical analyses were carried out to clarify the relationship between oil/water interfacial tension, the oil/water/ice contact angle, and the final slick radius. Results were verified by experiments. The following conclusions were obtained.. Oil/water interfacial tension was successfully measured by the ADSA method. 2. A simple but accurate model equation of interfacial tension acting on an oil slick

8 on the edge was constructed: f t = 2σ. 3. The final slick radius under a flat ice cover can be accurately estimated using the above model equation. 4. Model application to the oil spill problem under uneven ice covers was discussed. ACKNOWLEDGMENT This study was supported by the Program for Promoting Fundamental Transport Technology Research from the Corporation for Advanced Transport & Technology (CATT). REFERENCES Izumiyama, Koh and Konno, Akihisa. Laboratory test on spreading of oil under ice covers. In The 7th International Symposium on Okhotsk Sea & Sea Ice (2002) Konno, Akihisa and Izumiyama, Koh. On the relationship of the oil/water interfacial tension and the spread of oil slick under ice cover. In The 7th International Symposium on Okhotsk Sea & Sea Ice (2002) National Astronomical Observatory, ed. Rika Nenpyo (Chronological Scientific Tables 200), Maruzen Co., Ltd. (2000) 064p. Rotemberg, Y., Boruvka, L. and Neumann, A.W. Determination of surface tension and contact angle from the shapes of axisymmetric fluid interfaces. Journal of Colloid and Interface Science 93(): (983). Yapa, P.D. and Chowdhury, T. Oil spreading under ice covers. In Proceedings of 989 Oil Spill Conference (989) 6 66.