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Research Papers: Evaporation, Boiling, and Condensation

Forced Flow of Vapor Condensing Over a Horizontal Plate (Problem of Cess and Koh): Steady and Unsteady Solutions of the Full 2D Problem

[+] Author and Article Information
S. Kulkarni, S. Mitra

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

A. Narain

Department of Mechanical Engineering-Engineering Mechanics, Michigan Technological University, Houghton, MI 49931narain@mtu.edu

J. Heat Transfer 132(10), 101502 (Jul 29, 2010) (18 pages) doi:10.1115/1.4001636 History: Received December 10, 2009; Revised March 13, 2010; Published July 29, 2010; Online July 29, 2010

Accurate steady and unsteady numerical solutions of the full 2D governing equations—which model the forced film condensation flow of saturated vapor over a semi-infinite horizontal plate (the problem of Cess and Koh)—are obtained over a range of flow parameters. The results presented here are used to better understand the limitations of the well-known similarity solutions given by Koh. It is found that steady/quasisteady filmwise solution exists only if the inlet speed is above a certain threshold value. Above this threshold speed, steady/quasisteady film condensation solutions exist and their film thickness variations are approximately the same as the similarity solution given by Koh. However, these steady solutions differ from the Koh solution regarding pressure variations and associated effects in the leading part of the plate. Besides results based on the solutions of the full steady governing equations, this paper also presents unsteady solutions that characterize the steady solutions’ attainability, stability (response to initial disturbances), and their response to ever-present minuscule noise on the condensing-surface. For this shear-driven flow, the paper finds that if the uniform vapor speed is above a threshold value, an unsteady solution that begins with any reasonable initial-guess is attracted in time to a steady solution. This long time limiting solution is the same—within computational errors—as the solution of the steady problem. The reported unsteady solutions that yield the steady solution in the long time limit also yield “attraction rates” for nonlinear stability analysis of the steady solutions. The attraction rates are found to diminish gradually with increasing distance from the leading edge and with decreasing inlet vapor speed. These steady solutions are generally found to be stable to initial disturbances on the interface as well as in any flow variable in the interior of the flow domain. The results for low vapor speeds below the threshold value indicate that the unsteady solutions exhibit nonexistence of any steady limit of filmwise flow in the aft portion of the solution. Even when a steady solution exists, the flow attainability is also shown to be difficult (because of waviness and other sensitivities) at large downstream distances.

Copyright © 2010 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

The figure shows a schematic for a typical finite computational domain for a film condensation flow over a horizontal plate due to a forced uniform vapor flow at the left inlet

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Figure 2

For flow of R113 vapor with U∞=2 m/s, ΔT=5°C, p∞=1 atm, xe=45, gy=0, and Ye=0.004 m, this figure compares nondimensional film thickness (δ) values for the steady solution (obtained from solving the steady governing equations) with those obtained from Koh’s similarity solution (2)

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Figure 3

For the steady solution of Fig. 2, this figure shows, at a fixed y=0.8 location, computationally obtained variation of nondimensional pressure π2 with nondimensional distance x

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Figure 4

For the steady solution of Fig. 2, Fig. 4 shows computationally obtained y-directional variation of nondimensional pressure π across different cross sections along the domain. The pressure in vapor domain above condensate film is represented by π2 and pressure in liquid domain below the interface is represented by (ρ1/ρ2)π1.

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Figure 5

The figure shows the streamline pattern for the case in Fig. 2. The pattern is obtained from the reported simulation technique. The contour on the background represents the magnitude of uI velocities.

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Figure 6

For the steady solution of Fig. 2, this figure compares nondimensional values of x-directional liquid velocities at the interface (u1i) as obtained from the similarity solution (2) with those obtained from the computational solution

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Figure 7

For flow of R113 vapor with U∞=2 m/s, ΔT=5°C, p∞=1 atm, xe=50, and gy=0, this figure compares nondimensional values of film thickness δ obtained from simulations for the same steady conditions under two different choices of grids (ni×nj) and domain heights Ye. Grid 1 corresponds to the grid size of 30×50 with Ye1=0.004 m and Grid 2 corresponds to the grid size of 35×70 with Ye2=0.008 m. Nondimensional values of δ and x for Grid 2 are converted and compared in terms of Grid 1 (by multiplying them by Ye2/Ye1).

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Figure 8

For flow of R113 vapor with U∞=1.7 m/s, ΔT=5°C, and gy=0, this figure shows nondimensional film thickness values at different nondimensional times given by the unsteady solution of the problem. An initial-guess given at time t=0 (about 16% below the final long-term solution) is seen to get attracted to the long-term steady solution at different rates. The markings, at different times, demarcate the zones that have “nearly” converged to the steady solution from the zones that have not.

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Figure 9

For flow of R113 with U∞=1.7 m/s, ΔT=5°C, and gy=0, this figure shows different rates of attraction versus time—as indicated by different representative deceleration rates—for different x values along the length of the plate. The value of the initial attraction rate ∂Δ/∂t(x,0)≡∂Δ/∂t∣init, as well as the strength of the attractors (as marked by the representative magnitude of deceleration rates associated with the slopes of the lines AB, A′B′, etc.), decreases with increasing x. The initial-guess of δ(x,0) for the unsteady solution was 16% below the long-term steady solution.

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Figure 10

For flow of R113 with ΔT=5°C, gy=0, and x=30, the figure shows different rates of attraction versus time—as indicated by different representative magnitudes of deceleration rates associated with the slopes of the lines AB, A′B′, etc.—at different vapor speeds. The initial-guess of δ(x,0) for the unsteady solution was 16% below the long-term steady solution. The figure demonstrates higher rates of attraction for higher speeds.

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Figure 11

This figure shows computationally obtained curves depicting the rate of change in film thickness at x=23 when the unsteady solutions approach the same long-term steady attractor from three different initial-guesses for flow of R113 vapor at U∞=2 m/s, gy=0, and ΔT=5°C. The three initial-guesses 1, 2, and 3 are, respectively, 2%, 5%, and 7% away from the unique long-term steady attractor. The subsequent time duration (marked τRep) over which a nearly constant deceleration rate (∂2Δ/∂t2) exists is marked by nearly equal constant decelerating slopes of lines AB, A′B′, and A″B″ on curves X, Y, and Z. This shows that for a given vapor speed, the above characterized attraction rates over τRep are associated with the long-term steady solutions rather than the values of the initial-guesses.

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Figure 12

For flow of R113 vapor with U∞=3 m/s, xe=50, ΔT=5°C, and gy=0, this figure shows the stable response of the long-term steady solution to the rather large initial disturbance given at time t=0. The nondimensional disturbance is given as δ(x,0)=δsteady(x)[1+εoδ′(x,0)], where δ′(x,0)≡sin(2πx/λo), εo=0.35, and λo=5. The disturbance dies out, almost completely, by the time t=1500.

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Figure 13

This figure depicts the definition of the type of miniscule ever-present noise given to the condensing-surface to investigate the sensitivity of condensing flows to persistent disturbances. The inset shows the displacement profile of the condensing-surface at two out-of-phase instants associated with a mode of the standing wave.

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Figure 14

For R113 vapor with U∞=3 m/s, xe=48, ΔT=5°C, and gy=0, this figure shows unsteady response of the flow to the typical condensing-surface noise with λ(≡λp/Ye)=10, Tw(≡U∞/(fp⋅Ye))=240, and εw(≡vmax/U∞)=3×10−6. The noise given to the condensing-surface is represented as v1(x,0,t)=vmax sin(2πx/λp)⋅sin(2πfpt), where v1(x,0,t) is condensing-surface velocity. The figure shows nondimensional film thickness δ(x,t) plotted versus x at two different nondimensional times t=140 and t=260. The steady film thickness values δ(x)steady are shown as an initial solution at time t=0.

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Figure 15

For different ranges of non-dimensional condensing-surface noise parameters, namely: time period Tw=Tb, wavelength λ, and condensing-surface velocity vibration amplitude εb=εw, and different vapor speeds U∞; this figure plots computationally obtained nondimensional a/Dmax values (amplitude of interfacial waves divided by amplitude of bottom wall displacement waves) against dimensional values of x(=x⋅Ye). The range of R113 vapor (with gy=0) flows considered here is described in the inset.

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Figure 16

The figure plots two different sets (for U∞=0.2 m/s and U∞=0.08 m/s) of long-term film thickness values δ(x,t) with x at large nondimensional times t=22 s and 34 s. The flows are of R113 vapor at ΔT=5°C and initial conditions (not shown) for each of these cases were the Koh similarity solution (2). For U∞<0.2 m/s, the aft portions of these curves suggest nonexistence of limt→∞ δ(x,t).

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Figure 17

The figure plots long-term steadiness measure |∂Δ/∂t| (estimated at x=40) with freestream speed U∞. The flows are of R113 vapor at ΔT=5°C. The values of |∂Δ/∂t| for U∞>0.2 m/s is considered effectively zero within computational error. This suggests existence of a long time steady solution. However, the rising positive values of |∂Δ/∂t| for U∞<0.2 m/s suggest nonexistence of a long time steady solution.

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Figure 18

For flow of R113 vapor with ΔT=5°C, Xe=0.2 m, and gy=0, this figure plots normalized viscous dissipation rates φ/φref (see Eq. 13 in the Appendix for the definition of φ) obtained from steady and unsteady (long-term) steady solutions in a representative control volume given by 0<x<40 and 0<y<0.5. As the vapor speed U∞ reduces below 0.2 m/s, dissipation rates can be seen becoming effectively equal to the zero value associated with U∞=0.

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