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TECHNICAL PAPERS: Evaporation, Boiling, and Condensation

Effects of Gravity, Shear and Surface Tension in Internal Condensing Flows: Results From Direct Computational Simulations

[+] Author and Article Information
Q. Liang, X. Wang, A. Narain

Department of Mechanical Engineering-Engineering Mechanics, Michigan Technological University, Houghton, MI 49931

J. Heat Transfer 126(5), 676-686 (Nov 16, 2004) (11 pages) doi:10.1115/1.1777586 History: Received October 04, 2003; Revised April 22, 2004; Online November 16, 2004
Copyright © 2004 by ASME
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Figures

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Flow geometry for simulations. The film thickness has been exaggerated for the purposes of clarity in discussing the algorithm and the nomenclature. In the figure, all the neighboring points are affecting a flow variable at a typical point P in the “elliptic” vapor flow.
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The liquid domain calculations underneath δshift(x,t) with prescribed values of (u1si,v1si1si) on δshift(x,t) satisfy the shear and pressure conditions on the actual δ(x,t). Discarding all other calculations, only calculations underneath δ(x,t) are retained. The vapor domain calculations above δ(x,t) with prescribed values of (u2i,v2i2i) on δ(x,t) satisfy ṁVK=ṁEnergy and the requirement of continuity of tangential velocities.
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For flow situation specified in Table 1 with α=90 deg and xe=30, the figure depicts two sets of steady solutions C1 (for Ze=0.51) and C3 (for Ze=0.44) that provide the initial conditions for solutions to be obtained for t>0 without any specification of exit conditions. The figure shows the resulting δ(x,t) predictions for t>0, the set of δ(x,t) curves C1 start at Ze=0.51 at t=0, and tend, as t→∞, to the solution for which Ze|Na=0.47. The other set of curves C3 start at Ze=0.44 at t=0 and also tend, as t→∞, to the same Ze|Na solution.
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Qualitative nature of the attracting steady/quasi-steady solution. The figure shows that different steady solutions associated with different exit conditions (Ze at t=0) are attracted, under unconstrained exit conditions and as t→∞, to a special steady solution with Ze=Ze|Na.
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(a) For the flow situations specified in Table 1, xe=35, and vertical configuration; the figure shows steady and time-averaged values of shear stress at the wall (τ̄w) and the tangential stress (τ̄nt0i) at the steady interface location; and (b) For the flow situations specified in Table 1, xe=35, and 0g configuration; the figure shows steady and time-averaged values of shear stress at the wall (τ̄w) and the tangential stress (τ̄nt0i) at the steady interface location
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For flow situation specified in Table 1 with α=90 deg and xe=50, the above figure depicts two different film thickness, wall heat flux, and natural exit condition Ze|Na predictions for two different wall temperature specifications Tw(x) (and hence ΔT ) which is, for case 1: ΔT1=7.5°C and case 2: ΔT2(x)=5°C⋅(1+x/xe)
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(a) For flow situations specified in Table 1, the figure shows smooth steady condensate film thickness for vertical, horizontal and zero-gravity cases which have, for xe=20,Ze|Na=0.691, 0.877, and 0.91 respectively. (b) For the flow situations specified in Table 1, the figure shows different exit pressure π̄2(xe) values associated with different Ze values. In particular, it also shows Ze|Na and their corresponding exit pressure values for the cases considered in Fig. 5(a). (c) For flow situations specified in Table 1, the figure shows phase-speeds ūsteady(x) for the cases considered in Fig. 5(a). (d) For flow situation specified in Table 1, the characteristics curves x=xc(t) denote curves along which infinitesimal initial disturbances naturally propagate (see 11) on the stable steady solutions of Fig. 5(a). On characteristics that originate on x=0 line, there are no disturbances as δ(0,t)≅0 implies δ′ ≅0.
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For flow situations specified in Table 1, the above δ(x,t) predictions (Δt=10) are for vertical and 0g cases with initial data δ(x,0)=δsteady(x)+δ(x,0), where a nonzero disturbance δ(x,0) has been superposed at t=0 on the steady solution δsteady. Here δ(x,0)=0.004⋅sin(2πx/5) for 3.5≤x≤13.5 and δ′=0 elsewhere.
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Closer to the interface, the figure shows the interface location and vapor/liquid streamlines at t=300 for a resonant bottom wall noise given as v1(x,0,t)=ε⋅sin(2πx/λ)⋅sin(2πt /T) where ε=0.32E−5, λ=10, and T(x)=λ/ūsteady(x). The underlying steady solution is for a zero gravity flow in Table 1 with xe=30,Ze|Na=0.91 and σ0=15E−03 N/m.
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(a) For flow situations specified in Table 1 and xe=35, the above δ(x,t) values at t=0 (with Ze=Ze|Na) and t=20 are for resonant and non-resonant bottom wall noise given as v1(x,0,t)=ε⋅sin(2πx/λ)⋅sin(2πt/T), ε=0.24E−05 and λ=10. For nonresonant cases T=24 and for resonant cases T=T(x)=λ/ūsteady. (b) For flow situations specified in Table 1 and xe=35, the steady and unsteady values of exit pressures are shown as a function of x. The predictions are for case 1: vertical (α=90 deg), case 2: horizontal in 1g (α=0 deg, Fry−1=0.233) and case 3: 0g (Fry−1=0). The waves are due to bottom wall noise specified in Fig. 8(a).
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For flow situations specified in Table 1, the δ(x,t) values at t=0 (with Ze=Ze|Na) and t=25 are for different bottom wall noises defined as: v1(x,0,t)|I=2ε⋅sin(2πx/λ)⋅sin(2πt /T), and v1(x,0,t)|II=ε⋅sin(2πx/λ)⋅sin(2πt/T)+ε⋅sin(2πx/(λ/2))⋅sin(2πt/T) where ε=0.24E−05, λ=10, T=T(x)=λ/ūsteady, and σ0=0.015 N/m.
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For flow situations specified in Table 1, the δsteady(x) and δ(x,t) predictions are for resonant and nonresonant bottom wall noise with different amplitudes ε (ε* or 5ε* ) and different surface tensions σ0 (σ* or 30σ* or σ* /30). The noise in the legend are specified by v1(x,0,t)=ε⋅sin(2πx/λ)⋅sin(2πt /T), ε* =0.24E−05, λ=10, and σ* =0.015 N/m. For nonresonant cases, T=24 and for resonant cases, T=T(x)=λ/ūsteady.
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(a) For the flow situations specified in Table 1, xe=35, and vertical configuration; the figure shows steady and time-averaged values of heat flux at the wall and at the interface; and (b) for the flow situations specified in Table 1, xe=35, and 0g configuration; the figure shows steady and time-averaged values of heat flux at the wall and at the interface

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