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TECHNICAL PAPERS

Direct Computational Simulations for Internal Condensing Flows and Results on Attainability/Stability of Steady Solutions, Their Intrinsic Waviness, and Their Noise Sensitivity

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

Department of Mechanical Engineering–Engineering Mechanics, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931

J. Appl. Mech 71(1), 69-88 (Mar 17, 2004) (20 pages) doi:10.1115/1.1641063 History: Received December 12, 2002; Revised June 09, 2003; Online March 17, 2004
Copyright © 2004 by ASME
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References

Figures

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Flow geometry for simulations
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The solenoid valve V1 (actively controlled by flow meter F1) and constant heat input Q̇in≈Ṁin⋅hfg(pB) to the boiler fixes inlet pressure p0 and inlet flow rate to the test section. The mass flow rate through pump P is adjusted to a value that matches the corresponding value at F1. A high flow rate of the coolant (water) flow around the test section fixes condensing surface temperature at a nearly uniform value of Tw while it still allows for different heat removed rates for different exit qualities Ze.
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Computational grids for flow simulation. For chosen xui lines, yvj lines in grid A are first generated by points Pi on δ(x,t). Above the “highest” yvj line thus obtained, the remaining yvj lines are independently generated with suitable unequal spacings. Grid B lines at x=xudi are different from xui lines and are used for tracking the interface δ(x,t).
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(a) The liquid domain calculations underneath δshift(x,t) with prescribed values of (u1si,v1si1si) on δshift(x,t) satisfy the shear and pressure condition on δ(x,t). (b) 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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The above predictions are for vertical channel flows of saturated R-113 vapor. The flow cases are specified in Table 1 with α=90 deg, xe=50 and two different exit conditions, viz. Ze1=0.5 and Ze2=0.38.
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For the flow situations specified in Table 1 with α=90 deg, xe=50, the figure shows the equivalence of specifying exit vapor quality Ze or exit pressure π̄e≡1/(1−δ)∫δ1π2dy to specify exit conditions. It is computationally more convenient to specify exit condition Ze.
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With all remaining flow parameters specified as in Table 1 with α=90 deg, the above figure shows that exit condition specified by the number Ze at a given xe must lie within two well-defined values, viz. Ze|min(xe)≤Ze≤Ze|max(xe). This restriction, presumably, arises from the fact (see Carey 37) that the assumed annular/stratified flows only occur within certain parameter ranges.
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(a) For flow situation specified in Table 1 with α=90 deg and xe=30, the figure depicts two sets of δ(x,t) predictions for t>0. One curve C1 starts at Ze=0.51 at t=0, and tends, as t→∞, to the solution for Ze|Na=0.47. The other curve C2 starts at Ze=0.44 at t=0 and tends, as t→∞, to the same steady Ze|Na solution. (b) For flow situations considered in Fig. 7(a), the above predictions for t>0 starts at t=0 from the same curves C1 and C2 in Fig. 7(a). However, at t>0, there is a condensing surface noise given by v1(x,0,t)=ε⋅ sin(2πx/λ)⋅ sin(2πt /T), with ε=0.3E-6,λ=10, and T=24. As t→∞, the mean part of wavy quasi-steady solutions coincides with the smooth solution, shown in Fig. 8(a) for Ze=Ze|Na=0.47.
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Qualitative nature of the stable, steady/quasi-steady solutions
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For flow situations specified in Table 1 with α=90 deg and xe=20, the above δ(x,t) predictions (Δt=2.5) are for 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 (shown as curve C1 above for t<0). The steady solution corresponds to Ze=Ze|Na=0.5. Here δ(x,0)=0 except in the interval x*<x<x*+10, where x*=3.5 and δ(x,0)=0.5⋅δsteady(x)⋅ sin(2πx/5). It is clear that even this large a disturbance damps at later times.
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(a) For flow situation specified in Table 1 with α=90 deg and xe=40, the characteristics curve C1 denote curves along which infinitesimal initial disturbances naturally propagate on the stable steady solution. Curve C1 denotes characteristics along which finite disturbance arising from forced bottom wall noise actually propagate. On characteristics originating at x=0,δ(0,t)≅0 implies δ≅0. (b) For flow situations defined in Table 1 with α=90 deg and xe=48, the above ū(x,t) predictions for t≥0 are for (i) steady flow with Ze|Na=0.524, (ii) resonant case in Fig. 13, (iii) nonresonant case in Fig. 13, (iv) large initial disturbance of Fig. 10 and (v) small initial disturbance δ(x,0) which is one-fifth of δ(x,0) in Fig. 10. (c) For flow situations defined in Table 1 with α=90 deg and xe=50, the above v̿(t) values are along actual characteristics curves like C1 in Fig. 11(a). The predictions are for (i) the steady and stable flow with Ze|Na=0.578,(ii) the resonant case of Fig. 13, (iii) the nonresonant case of Fig. 13, and (iv) the small initial disturbance case (scaled up and shown in the lower figure) in Fig. 11(c).
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For the above flow situation specified in Table 1 with α=90 deg and xe=50, the steady solutions are obtained for Ze=Ze|min=0.26 and Ze|Na=0.36
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(a) For the flow situations specified in Table 1 with α=90 deg, xe=50 and Ze=0.578; the above δ(x,t) predictions compare the nonresonant noise with a resonant noise of the same amplitude (ε=0.9E-4). The noise is given by: v1(x,0,t)=ε⋅ sin(2πx/λ)⋅ sin(2πt /T), where (i) λ=10 and T=24 for the nonresonant case, and (ii) λ=10 and T=T(x)=λ/ūsteady(x). (b) For the flow situations considered in Fig. 13(a), the above depicts the wall heat flux qw(x,t), in W, at t=25 for the resonant case, and its time-averaged values q̄w(x), in W, for all other cases.
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(a) The above δ(x,t) predictions for t>0 are for the steady solution curve C1=C2 at t=0 and initial noise specified in Fig. 10. The t=0 solutions are obtained on two grids I and II with (ni×nj|L×nj|V)I=(30×30×20) and (ni×nj|L×nj|V)II=(50×50×30). The t>0 solution are shown as curves C1 and C2 and are, respectively, obtained on grids that have: (ni×nj|L×nj|V)I×Δt=(30×30×20)×2.5 and (ni×nj|L×nj|V)II×Δt=(50×50×30)×5. At t>0, the number of grid lines (ni×nj|L×nj|V) changes somewhat from their value at t=0.
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For the vertical plate situation specified in Table 1 with α=90 deg, xe=48 and Lc=0.004 m, Curve 1 is a plot of the analytical solution of δ(x) as given in Nusselt 5. Curve 2 is the computational solution under the Nusselt assumption for stagnant vapor and zero liquid inertia. Curve 3 is the computational solution under the assumptions of stagnant vapor while allowing for liquid inertia. Curve 4 is the computational solution that allows vapor motion and liquid inertia (the vapor/liquid velocity profiles are shown only for this case). Though not shown above, vapor velocity tends to zero as y→∞.
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The above is a plot of natural values of Ze|Na for different xe values for a representative flow situation specified in Table 1 with α=90 deg and xe=48. The “Increased Rein” case just changes Rein to a new value of 1300. The “Increased Ja” case just changes Ja to a new value of 0.0443 (i.e., ΔT=65°C).
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For the flow situations described in Fig. 16 and xe=25.0, the above figure reports the representative wall heat flux values q̄w(x), in W, as a function of x with 0≤x≤xe
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For the flow situations described in Figs. 1617 and xe=30.0, the above figure reports the values of δsteady(x)

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