Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979
d,, diameter of percolating particles ( L ) d,, diameter .~ of largest particle that can percolate spontaneously ( L )
E,, bulk particle diffusion coefficient ( L 2 T 1 ) kf,k,,kh,k$,, constants Nb, number of bulk particles in system N,, number of bulk particles having separation in region q p , probability q, region R , largest distance between surface of bulk particle and extremity of cage containing bulk particle (L) R*, critical value of R which, if exceeded, will lead to percolation ( L ) 8, mean value of R ( L ) R value of R in region g ( L ) L f i , number of ways of distributing the lengths R, Greek Letters
a1,cy2,Lagrangian
27
multipliers
i.,rate of shear strain (Tl) Literature Cited Bridgwater, J., Sharpe, N. W., Stocker, D.C., Trans. Inst. Chem. Eng., 47, T114 (1969).
Bridgwater, J., Cooke, M. H., Scott, A. M., Trans. Inst. Chem. Eng., 56, 157 (1978).
Campbell, A. P., Bridgwater, J., Trans. Inst. Chem. Eng., 51, 72 (1973). Cohen, M. H., Turnbull, D., J. Chem. Phys., 31, 1164 (1959).
Roscoe, K. H., Geotechnique, 20, 129 (1970). Scott, A. M., Bridgwater, J., Ind. Eng. Chem. Fundam., 14, 22 (1975). Willemse, Th. W.. Chem. Weekbl., 57, 377 (1961).
Received for review February 8, 1978 Accepted September 6, 1978
The work was made possible by the provision of an S.R.C. research grant and by a studentship from the S.R.C. and Thorn Lighting Ltd.
Experimental Behavior of Falling Liquid Films at High Surface Tension Numbers Myron A. Hoffman* and Winston W. Potts Department of Mechanical Engineering, University of California, Davis, Davis, California 956 76
The application of thin, falling liquid films of lithium to cool the inner walls of future laser fusion reactors has been investigated. In order to extend the existing experimental results on falling films to the higher surface tension numbers characberistic of alkali metals such as lithium, experiments using hot water on a vertical plate were run, primarily in the wavy laminar flow regime. Results show that very low minimum flow rates can fully wet the surface provided that the surface is carefully cleaned and preconditioned. Both the wave inception distance and the equilibrium wave arnplitude decrease as the surface tension number increases for a given flow Reynolds number. Based on these results, the scaling laws for the minimum wetting rate, the wave inception distance, and the equilibrium wave amplitude have been extended up to surface tension numbers of about 10 000.
Introduction This research was motivated by the possible application of falling liquid lithium films to protect the inner walls of a future laser fusion reactor from the micro-explosion products. Detailed studies of the effects of short pulses of photons and ions impinging on solid first walls made by Hovingh (1976) revealed some severe limitations. The energy deposition depth in first walls for the fusion micro-explosion products of a small pellet of deuterium and tritium is typically on the order of a few hundred micrometers for the CY particles, energetic deuterium and tritium ions, and the X-rays. Furthermore, the particle energies are deposited on a time scale of the order of a microsecond, while the X-rays are deposited over an even shorter time. This implies that very high pressure and temperature peaks may be reached in the front layers of the first wall material, which can severely limit the permissible micro-explosion energy yield if spalling is to be avoided. The use of thin films of liquid lithium to protect the first wall was proposed in a pioneering study by Booth (1973) at LASL (Los Alamos Scientific Laboratory). The present research program grew out of a modified version of the LASL wetted-first-wall concept proposed a t LLL (Lawrence Livermore Laboratory) and referred to as the suppressed-ablation coincept (Hovingh et al., 1974). Uniformity of the liquid coating is very important for these possible fusion reactor applications. If the coating 0019-7874/79/1018-0027$01.00/0
is locally too thin, too large a fraction of the ions and radiation will penetrate the liquid film and possibly cause spalling and/or sputtering of the solid first wall material. If the liquid coating is locally too thick, then excessive peak liquid temperatures can be reached in these regions, resulting in excessive ablation of the liquid lithium and larger pressure pulses to the substructure. (In addition, it is well known that if the liquid film is nonisothermal, dry spots can develop due to gradients in the surface tension coefficient.) In the first phase of this research reported here, we have concentrated on the hydrodynamics of the falling liquid films in the wavy laminar flow regime under near-isothermal conditions (Hoffman and Munir, 1976; Hoffman, 1977). Review of Thin Falling Liquid Flows Characteristics of Various Liquids. The basic properties of the more common liquids used in falling film research a t typical experiment temperatures are summarized in the upper part of Table I. The properties of some liquid metals (for which no experimental falling film data have been found) are given in the bottom part of the table. The key nondimensional parameters governing the isothermal falling film flow of pure liquids when heat and mass transfer effects are absent are the Reynolds number, N R E , the surface tension number No, defined as follows 4iib N Re = -4riz = - =4-r (1) -
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PW
0 1979 American Chemical Society
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Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979
Table I. Typical Liquid Property Data liquid
temp, “ C
94% glycerine Chevron no. 15 oil isopropyl alcohol 45% glycerine 27% glycerine cool water room temperature water hot tap water hot water potassium sodium-potassium alloy (Na % 22) sodium ga11ium mercury lithium
density, g/cm3
viscosity, glcm-s
surface tension, dyn/cm
No
25 25 20 22 25 10 20 50 90 100 100
0.868 0.788 1.112 1.063 1.00 1.00 0.99 0.970 0.819 0.847
1.46 0.0301 0.0440 0.019 3 0.0129 0.01006 0.0055 0.003 2 0.00458 0.00468
30.8 27.6 70.4 72.2 75.0 73.4 68.0 61.7 86.0 120.0
1.7 2.2 372.0 600.0 1794.0 3 140.0 4 280.0 8 850.0 16 400.0 13 420.0 18 400.0
100 50 20 500
0.928 6.09 13.5 0.484
0.006 86 0.0189 0.0155 0.0034
206.0 735.0 465.0 349.0
19 550.0 33 800.0 36 340.0 67 900.0
where m is the mass flow rate, r is the mass flow rate per unit width, w ,of the film, u is the average film velocity, 6 is the average film thickness, and v is the kinematic viscosity, and
105
1
I
I
I I I I I I
1
1
I
I I 1 1 1 1
![ 1 1
113
N,E P
v4g
where u is the surface tension coefficient. Two other nondimensional parameters, a Weber number and a Froude number, are sometimes used, but these parameters can be shown to be functions of NReand N , in the low Reynolds number flow regimes (below about 400) of most interest here (e.g., Fulford, 1964). The surface tension numbers for the various liquids commonly used in experiments to date span the range from about 2 for 95% glycerine solutions and heavy oils to about 4900 for room temperature water a t 25 “C. For the liquid metals, the values of the surface tension number are appreciably higher than 4900. In particular, lithium has a surface tension number of almost 68 000 a t 500 OC, the average temperature of the lithium film in the LLL reference design (Hovingh et al., 1974). In order to verify the falling film scaling laws for liquid lithium, it was initially felt that experiments with some liquid metal, perhaps lithium itself, might be necessary. However, a simpler alternative approach is to use hot water, since the surface tension number of ordinary water increases substantially with temperature, due primarily to the reduction in viscosity. The curves in Figure 1show this trend and indicate that ordinary 57 “C hot tap water has a surface tension number of over 10000. It should be noted that pressurized hot water could be used to extend the data to very high values of the surface tension number if desired, as indicated in Figure 1. Falling Film Flow Regimes. As a liquid flows from a well-designed, disturbance-free distributor down a vertical plate or cylinder wall, a relatively smooth entrance region is observed. The length of this visually smooth entrance region depends on NReand N , as well as on the inlet geometry (Cerro and Whitaker, 1971). Then waves begin to appear on the liquid surface. These waves may be almost two-dimensional even far down the surface in the so-called wavy-laminar film flow regimes, while they may become highly irregular and three-dimensional in the turbulent film flow regimes. The wavy laminar flow regime is estimated to extend up to Reynolds numbers of 300 to 400 (e.g., see Ishigai et al., 1972). We have chosen to study the film flows primarily in the wavy laminar regime for the
“
Y
lo]
tI
10
1 I
1
I
50
I
1 I , , ,
I
im
I
1
I 1 1111
5m
1I
lwo
TEMPERATURE. *C
Figure 1. Variation of the surface tension number with temperature for liquid water and lithium.
fusion reactor wetted-wall application. The choice of this flow regime is dictated by a desire t o achieve both reasonably low flow rates and reasonably long smooth inlet flow lengths (Hoffman, 1977). It is now felt by most researchers in the field that the liquid film flow on a vertical plate is unstable a t all Reynolds numbers (e.g., see Anshus and Goren, 1966). However, because of entrance effects and the finite spatial growth rate of the waves, it often appears to the naked eye (and even to most of the detectors normally employed in these experiments) that there is an initially smooth region of the film surface near the flow entrance. As a result of this entrance phenomenon, the concept of a visual wave inception line (or distance) has been introduced by researchers in this field. It can be seen immediately that the very concept of unique wave inception line is a somewhat ambiguously defined parameter, since it depends on the resolution of the detector used and on the choice of wave amplitude used to define “wave inceRtion”. In the wavy-laminar flow regimes, NReS 300-400, the first waves to appear tend to have a regular frequency and tend to grow exponentially until nonlinear effects cause a saturation and a so-called “equilibrium” wave amplitude is reached. The large waves continue to change shape somewhat as the liquid flows down the plate and some waves coalesce so that far down the plate, the frequency of the large waves becomes irregular. However, there is evidence that certain statistical properties of these waves remain constant once a so-called fully developed state is reached far down a long enough plate or cylinder (Telles
Ind. Eng. Chem. Fundam., Vol. 18, No. 1,
and Dukler, 1970; Chu and Dukler, 1974, 1975). For,the wavy laminar flow regime, the asymptotic film thickness in the absence of waves for flows down a vertical surface has been experimentally verified to be the value predicted by Nusselt for laminar flow (e.g., see Fulford, 1964)
I Os
-- -
Hartley and Yvplroyd Thai
1979 29
/
Hoblor and CzaiLI Empirical Ep
(3)
The mean film thickness in the presence of waves, 6, differs in general from the 'Nusselt film thickness in a rather complex way (e.g., Chu and Dukler, 1975; Portalski and Clegg, 1972). However, in wavy laminar flow, it is probably within 10% of b,; for simplicity, the Nusselt film thickness has,been used in place of 6 in nondimensionalizing our results. Minimum Wetting Rate Phenomena. It has been demonstrated experimentally that a minimum flow rate is required to fully wet or coat a surface with a falling liquid film. This so-called miinimum wetting rate depends on the liquid properties, the contact angle between the liquid and solid plate material, the liquid feed port geometry, the plate inclination angle, disturbance levels, and so forth. In actuality, the minimum wetting rate can have a t least three different values for a given liquid-solid substrate combination depending on the initial plate conditions in the experiment (Hobler and Czajka, 1968): case 0, initially unwetted plate with the flow increasing from zero, I'(mm)O; case 1, previously wetted plate with the flow increasing from zero, r(m,njl, i.e., the rewetting case; and case 2, fully wetted plate with the flow decreasing from a high value, r ( m l n p The various initial surface conditions obviously determine the effective contact angle. The minimum flow rates found experimentally follow the rule r(mm)o
'
r(min)l
'
r(min)z
all other things being equal, (Le., same distributor, plate angle, etc.). The following scaling law for the rewetting of an isothermal vertical plate was derived by Hartley and Murgatroyd, (1967) (4) where 0 is the contact angle. A similar equation has been obtained by Ponter et al. (1967). An empirical scaling law for case 0 has been proposed by Hobler and Czajka (1968) based on extensive experiments with water and glycerine-water mixtures. Their proposed equation is
where fl is the plate inclination angle from the horizontal. The fit of this equation to their experimental data shown in Figure 2 can be seen to be reasonably good as far as the overall N , dependence is concerned. However, the scatter in their water data is on the order of &loo%, which may imply that their 0 and/or p dependencies are not quite correct or that some external disturbances were present. Description of Expeiriments The water films were produced on a polished aluminum plate, 15 cm wide and about 50 cm long. This length was considered more than adequate for the study of the smooth inlet region of the flow. The plate was fed from a reservoir equipped with turbulence suppression screens and the water flowed over a 45" knife edge at the top of the plate.
Figure 2. Comparison of the minimum wetting rate data with two proposed equations (initially dry, vertical surface).
The entire test stand was shock mounted on water-filled inner tubes in order to provide attenuation of the external disturbances. An accelerometer mounted on the apparatus was used to monitor the quality of the vibration isolation; this is discussed in more detail by Potts (1978). Both the room temperature water and the hot water employed came directly from the tap in the series of experiments to be described, and the water was used for only one pass over the plate. The plate surface was carefully cleaned with alcohol and then with freon TF before each run series. The plate was then immersed in a large container of pure water for a t least 24 h before use. It was discovered empirically that this preconditioning produced a surface with an adsorbed layer of water on it which greatly reduced the contact angle between the falling water film and the plate. The water flow rate was measured with a rotameter-type flow meter. The flow meter was calibrated for several different water temperatures. The water temperature was measured with a glass mercury thermometer in the top reservoir. The water was allowed to flow down the plate for a sufficient time before taking data such that the plate surface temperature was very close to the water temperature. The entire flow and plate surface were estimated to be isothermal to within about f 4 "C for the hot water and within f l "C for the room temperature water (Potts, 1978). In order to obtain an accurate visual estimate of the wave inception distance, the frequency of a stroboscope was increased very slowly from the value which "froze" the waves. This made the waves appear to move up the plate toward their point of "birth" and was a very sensitive method for determining the visual wave inception point. Many photographs were also taken, but the above method was felt to be more accurate than the photographs where the uncertainty in photographing the first visible wave can easily be on the order of f h / 2 , i.e., about half a wavelength. Measurements of the maximum film thickness variation along the plate were made using a micrometer probe technique. The micrometer probe was inserted in an electrical circuit as was done by Brauer (1956), Ishigai et al. (1972) and others in hopes of obtaining the maximum sensitivity. However, both electrical contact detection and visual contact detection were used particularly when waves of different amplitudes were present. Based on repeated measurements a t the same flow conditions on several different days, the overall scatter in the bMAXdata falls
30
Ind. Eng. Chern. Fundarn., Vol. 18, No. 1, 1979 4 *Watsr ( 2 5 " C ) , Nu = 4920 Water ( 2 7 ° C ) . Nu = UW OWater (51°C t , N u = 9QW
T 3 0 +
