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Bubble Penetration Through a Single Layer Sphere Bed Robert E. Manning Steven H. Collicott Purdue University Supported in part by NASA research grant NNC05GA25G, Dr. Walter Duval, Program Manager School of Aeronautics and Astronautics

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Capillary Phenomena in a Sphere Packed Bed Sphere packed beds are used as a model for porous media. Porosity can vary from approximately 48 to 26%. Used for infiltration problems. –Mayer & Stowe (2006) –Hilden & Trumble (2003) –Slobozhanin, Alexander, Collicott, & Gonzalez (2005, unpublished) Here we examine a non-wetting liquid droplet passing through the pore (or a gas bubble surrounded by wetting liquid)

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Problem Formulation Single layer with four spheres of fixed radius r Droplet can impinge on any number of spheres Hexagonal (δ=90º) to square (δ=90º) packing angles Droplet is above the pore under a uniform gravitational field (g). Under what conditions will the droplet pass through the pore? What is the topology progression of the droplet? -g

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Topologies Droplet will impinge on spheres and form contact lines. Seven unique topologies exist. Clearly unique topologies with one contact line and four contact lines exist. Due to symmetry, the two and three contact line cases are noted below. Below, gray denotes wetted sphere.

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Solution Methodology Assume quasi-static. Both gravitational and capillary energies are considered. Use Surface Evolver to solve for minimum energy based on volume, packing angle, and liquid contact angle. Program seven different topologies and find the Bond number intervals where droplet is stable. Two and one contact line solutions cannot exist for negative Bond numbers (driving droplet into pore).

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Convergence Study 1 Examined how energy and center of mass are affected by refinement. For accuracy of 0.01 Bond, 8000 facets needed. For 0.001, over 12000 needed.

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Convergence Study 2 Examined criteria for convergence. If both energy and center of mass are within 1e-4, then we assume convergence.

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Convergence Study 3 & 4 Considered how many facets are needed to resolve a droplet impinging on another dry sphere. –As few as 1000 facets resolved this to within 0.001 Bond number. Also approximately 40-50 iterations were needed to detect instability due to contact line collapse (droplet pulling away from a sphere). For results presented today, 15000 facets were used to model the liquid-gas interface of the droplet.

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A Single Packing Angle A droplet with unity volume and liquid contact angle of 135º. The sphere layer is near square packed with δ=88º. The vertical distance to the droplet’s center of mass (z) was calculated for a range of Bond numbers. Repeated for all seven topologies. Interestingly this packing exhibits all seven possible droplet types. z

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88º Packing Angle

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Stability Regions Four Contact Lines 90º packing angle 70º packing angle

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Stability Regions Four Contact Lines Three A 70º Packing Angle

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Stability Regions Four Contact Lines Three A Two B 70º Packing Angle

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Stability Regions Contact Lines One Two A

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Stability Regions Four Contact Lines Three A Two B One Two A

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Near Square Packed Stability Regions Four Contact Lines Three A Two B One Two A

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Near Square Packed Stability Regions Four Contact Lines Three A Two B One Two A Three B Two C

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Conclusion Necessary criteria were computed for both droplet penetration of the pore and “dripping” from the sphere layer. A variety of droplet shapes exist for different regions. Future work will examine different contact angles and volumes.

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