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TECHNICAL PAPERS: Fluids/Heat/Transport

Effects of Inertia and Gravity on Liquid Plug Splitting at a Bifurcation

[+] Author and Article Information
Y. Zheng, H. Fujioka, J. B. Grotberg

Department of Biomedical Engineering, University of Michigan, Ann Arbor, MI 48109

J. C. Grotberg

Department of Biomedical Engineering, University of Michigan, Ann Arbor, MI 48109grotberg@umich.edu

J Biomech Eng 128(5), 707-716 (Apr 19, 2006) (10 pages) doi:10.1115/1.2246235 History: Received October 11, 2005; Revised April 19, 2006

Abstract

Liquid plugs may form in pulmonary airways during the process of liquid instillation or removal in many clinical treatments. During inspiration the plug may split at airway bifurcations and lead to a nonuniform final liquid distribution, which can adversely affect treatment outcomes. In this paper, a combination of bench top experimental and theoretical studies is presented to study the effects of inertia and gravity on plug splitting in an airway bifurcation model to simulate the liquid distributions in large airways. The splitting ratio, $Rs$, is defined as the ratio of the plug volume entering the upper (gravitationally opposed) daughter tube to the lower (gravitationally favored) one. $Rs$ is measured as a function of parent tube Reynolds number, $Rep$; gravitational orientations for roll angle, $ϕ$, and pitch angle, $γ$; parent plug length $LP$; and the presence of pre-existing plug blockages in downstream daughter tubes. Results show that increasing $Rep$ causes more homogeneous splitting. A critical Reynolds number $Rec$ is found to exist so that when $Rep⩽Rec$, $Rs=0$, i.e., no liquid enters the upper daughter tube. $Rec$ increases while $Rs$ decreases with increasing the gravitational effect, i.e., increasing $ϕ$ and $γ$. When a blockage exists in the lower daughter, $Rec$ is only found at $ϕ=60deg$ in the range of $Rep$ studied, and the resulting total mass ratio can be as high as 6, which also asymptotes to a finite value for different $ϕ$ as $Rep$ increases. Inertia is further demonstrated to cause more homogeneous plug splitting from a comparison study of $Rs$ versus $Cap$ (another characteristic speed) for three liquids: water, glycerin, and LB-400X. A theoretical model based on entrance flow for the plug in the daughters is developed and predicts $Rs$ versus $Rep$. The frictional pressure drop, as a part of the total pressure drop, is estimated by two fitting parameters and shows a linear relationship with $Rep$. The theory provides a good prediction on liquid plug splitting and well simulates the liquid distributions in the large airways of human lungs.

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Figures

Figure 1

Schematic of the experimental setup. The roll angle, ϕ, and pitch angle, γ describe the orientation of the bifurcation plates with respect to gravity. The branch angle of the daughter tube with respect to the parent tube is indicated by θ. 0, 1, 2, 3, 4, and 5 indicate 6 IR sensors for the parent and two daughter tubes.

Figure 2

(a) Schematic of the measurement of plug velocity/length by two pairs of IR sensors and the sample binary signal of meniscus recording. (b) The calibration chart of the front meniscus velocity measured from IR sensors versus that from the pump flow rate in a 4mm glass tube. (c) The sample signal of the experiment. 0 and 1 is the binary signal of the menisci of the parent plug; 2 and 3 is the signal for the plug menisci in the upper daughter; 4 and 5 is for the plug menisci in the lower daughter; and 6 is the pressure signal.

Figure 3

Rs versus Rep for γ=0deg and different ϕ using deionized distilled water. ϕ=15deg: ◆ (experiments),—(theory); ϕ=30deg: ∎ (experiments),---(theory); ϕ=60deg: ▴ (experiments), ⋯ (theory).

Figure 4

Rs versus Rep for ϕ=30deg and different γ using deionized distilled water. γ=−15deg: ◆ (experiments),—(theory); γ=0deg: ∎ (experiments), --- (theory); γ=15deg: ▴ (experiments), ⋯ (theory).

Figure 5

Experimental results of effect of parent plug volume on splitting ratio, Rs versus Rep for deionized distilled water at ϕ=15deg, γ=0deg. L0=1cm(L0∕a1=5): ∎ (expt),—(theory); L0=2cm(L0∕a1=10): ◆ (expt), --- (theory).

Figure 6

Experimental results of Rs versus Rep with the presence of the lower plug blockage for γ=0deg and different ϕ using deionized distilled water. ◆: ϕ=15deg; ∎: ϕ=30deg; ▴: ϕ=60deg.

Figure 7

Experimental results of the total liquid ratio, Rt versus Rep with the presence of the lower plug blockage for γ=0deg and different ϕ using deionized distilled water. ◆: ϕ=15deg; ∎: ϕ=30deg; ▴: ϕ=60deg.

Figure 8

Schematic of plug flow in the bifurcation plates. a1, a2, and a3 are the radius of the parent, upper, and lower daughter tube, respectively. P1 is the pressure of the pumped air. P2 and P3 are atmospheric pressure. L1, L2, and L3 are the plug lengths in the parent, upper, and lower daughter tube, respectively. π1, π0, π2, and π3 are the pressures in the liquid plug at the rear interface, bifurcation zone, front meniscus of the upper daughter, and front meniscus of the lower daughter, respectively. Up, U2, and U3 are the plug velocities of the rear meniscus in the parent tube and front menisci in the upper daughter and lower daughter tubes.

Figure 9

Rs versus Cap for γ=0deg and ϕ=30deg with three liquid materials. Deionized distilled water: ◆ (expt),—(theory); 60% glycerin: ∎ (expt); LB-400X (17): ▴ (expt), ⋯ (theory).

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