values the transition occurs. The best graphical estimate one can make from this set of data appears to be Re, ‘v 1960, xrhich is subject to considerable error because of the paucity of data. The theoretical value of 1900 is included within the uncertainty of the data. Rothfus et al. ( 7 7 ) proposed empirically that the product Rec(zl,,,/D) is a constant equal to 4200. The ratio t ~ ~ is easily shown to be a function of R independent of Re. Consequently, the curve of lie, us. R can simply be converted into a curve of Re,(u,,,,,lo) us. R. Table I contains the theoretical values of Kec(umaT,’D),a s suggested by those authors, for the points tabulated in their paper. This parameter is clearly not a constant as suggested by Rothfus t t a / . ( 7 7 ) . Obviously, if the experimental va1ue.s of Re,, which agree well with the theoretical curve of Re, us. R, were to be multiplied by the theoretical ratio Z : ~ , ~ ~ the / U , same nonconstant behavior would be observed. The existence of the minimum in the product Rec(LNm:,x!D) as a function of R is indicated even in the tabulation of Kothfus et al. ( T 7 ) , although not to quite the degree shown by the theoretical results tabulated in Table I. However, as a n approximation? their parameter does succeed in removing much of the. variation of Re, with R. Conclusions
-4theoretical curve of the lower critical Reynolds number (defined in terms of hydraulic diameter) hai been calculated as a function of the aspect ratio, a / b , for the steady isothermal rectilinear flow of Newtonian fluids in straight ducts of constant rectangular cross section,. Data reported in the literature by seven independent investigators obtained \\ ith ten different systems were compared \$ith the theoretical calculations of this study and found to be in good agreement The present study is thought to represent the first systematic theoretical calcula-
Table 1.
Theoretical Values of Re,(v,,,/V) Aspect Ratios
Aspect Ratio, R 1. o
2.04 2.36 2.92 3.92 >10
hle,
1I n d o
2060 1900
2.10 1.97 1.97 1.86 1.79 1.50
1060
2085 231 5 2800
for Selected
ReAvmax16) 4326 3743 3861 3878 4144 4200
~
tion of the lower critical Keynolds number for this geometry, and is believed to be more reliable than available empirical correlations. Further experimentation, particularly \vith reference to the spatial location of the laminar-turbulent transition within the flow field, is desirable. The theoretical curve presented in Figure 2 is recommended ~ for / use 6 in determining the lotver critical Reynolds number for the laminar-turbulent transition in rectangular ducts. As a practical rule of thumb, when the duct aspect ratio exceeds about IO, the critical value of R e is approximately 2800 if defined in terms of the hydraulic diameter. Critical values of any Keynolds number defined in terms of an arbitrary characteristic length, L. may be obtained from Figure 2 by multiplying Re, from that graph by the ratio L/’D,,. Acknowledgment
The authors thank the Brigham Young Universiry Computer Center for making its facilities available for the calculations. literature Cited (1) Allen. J.: Grundbcrg. N. D.. Phil. M q . 23, 490 ( 1 937). ( 2 ) Bird, R. B., Stewart. I V . E.. Lightfoot, E. N..“Transport Phenomena.” \Viley. New York. 1962. (3) Cornish. R. J., Proc. Roy. Soc. (London) 120A, 691 11928). (4) Davies, S. J.. \Vhite, C. hi..Ibid.. 119A, 92 (1928). (5) E>ckert, E. R. G.. Irvine. T. F.. Jr.. Proceedings of Fifth Midwest Conference on Fluid Mechanics, p. 122. University
of Michigan Press, ;inn Arbor, Mich.. 1957. (6) Hanks. R. \V., A . I . Ch. E. ,J. 9 (1). 45 (1963); “Generalized Criterion for the Laminar-Turbulent Transition in the Flow of Fluids,” Oak Ridge Gaseous Diffusion Plant. Union Carbide Nuclear Co.. Oak Ridge. Tenn.. U.S. Atomic Enerpv Cornsn. Rent. K-1531 ih’ov. 19. 1962). 17) Hanks. R. \
t
14c1 120.
100 SOLUTION
v
Eaob.
60 40 0
20 -
100
OO
200
300
400
5 00
10 0 .
r:
:
?-
The average error in the use of Equation 21 is about ~ k 2 0 7 ~ , ivhich is somev, hat greater than that resulting from Equation 19. Hoivever, Equation 21 does not contain the molecular diffusivity. D , M hich is advantageous in estimating drop vaporization times for situations ivhere molecular diffusion may be negligible.
IC
0
0
(3,
. .
ClLO
Y
In
W
N
.
*.WATER A-ETHYL ALCOHOL m - B E N Z EN E f.CARBON T E T R A C H L O R I D E
0
o < 0 0.10 10
Figure 8.
1.0
Correlation of experimental d a t a
and (C,p/k) and (p/p,D) approximately unity. Borishansky ( 4 ) presents a dimensionless correlation of his data for spheroidal drop vaporization times which includes the specific heat, thermal conductivity, and interfacial tension of the liquid. However, for (Clk,AT,’kiX) < 0.10; C1 and k i drop out of the correlation (7), resulting in simply
Lee (8) has successfully correlated his data for spheroidal droplets as follows:
Conclusions
The Leidenfrost phenomenon is caused by a drop of liquid being supported above a hot surface by its own vapor. The vapor is generated primarily by heat conduction across the vapor cushion ; homever. radiation is also an important heat source for ordinary fluids Mass transfer over the top surface of the drop can contribute significantly to the rate of vaporization. The analytical model presented here, based upon simultaneous conduction and radiation of heat and molecular diffusion, results in theoretically predicted drop vaporization times in satisfactory agreement 1% ith the corresponding experimental data over a wide range of plate temperatures and physical properties. The experimental data have been successfully correlated by dimensional analysis, most of the data exhibiting less than a lOyGdeviation. T h e Leidenfrost point (the plate-to-drop temperature difference corresponding to the maximum vaporization time Tor a given size drop) for water occurs a t a temperature difference of about 190’ C., for the organic liquids at a temperature difference somewhat less than 120’ C. These results are reasonably consistent with the observations of other investigators. VOL. 5
NO. 4
NOVEMBER 1 9 6 6
567
Appendix A.
Nomenclature
Radiation View Factors
T h e rate of radiative heat transfer between the plate and the drop, neglecting any absorption of radiation by the vapor, can be approximated by the following expression from McAdams (70) :
Q
=
4irrO2o5(Tp4- Tb4)
(22)
where
The geometric view factor, F, can be thought of as the fraction of radiation leaving the drop which strikes the plate. I n this case, F is approximately ‘/z. Since A , 4ar,2, the last term in Equation 23 is negligible, and the radiative heat transfer rate becomes
>>
The radiative heat transfer rate can alternatively be written
A , = heat transfer area of plate, sq. cm. C = concentration of vapor in air, g./cc. CI = specific heat of liquid, cal./(g.)(” C.) Cp = specific heat of vapor, cal./(g.)(” C.) D = molecular diffusivity, sq. cm./’sec. F = geometric view factor 5 = radiation factor 51 = radiation factcr for lower drop surface 5 2 = radiation factor for upper drop surface g = gravitational acceleration, cm./sec.2 k = thermal conductivity of vapor, cal./(cm.) (set.)(' C.) k i = thermal conductivity of liquid, cal.,/(cm.) (sec.) (” C.) M = molecular w e i ~ h of t vaDor Pp* ”= excess pressurqbeneath drop, dynedsq. cm. vapor pressure of drop, dynes/sq. cm. Q = radiative heat transfer rate to drop, cal./sec. q = heat flux to lower drop surface, cal./(sq. cm.)(sec.) q‘ = heat flux to upper drop surface, cal./(sq. cm.)(sec.) R = ideal gas constant, (dyne) (cm.)/(mole) (” K.) r = radial space variable, cm. r, = drop radius, cm. Tb = boiling point temperature of drop, K. T p = plate temperature, ” K. t = time variable, sec. u = mean radial vapor velocity beneath drop, cm./sec. drop volume, cc. = initial drop volume, cc. u = radial vapor velocity beneath drop, cm./sec. le’ = mass Aux at lower drop surface, g./(sq. cm.)(sec.) == mass flux at upper drop surface, g./(sq. cm.) (sec.) t = axial space variable, cm. Y = interfacial tension of liquid, dynes/cm. 6 = thickness of vapor cushion, cm. E = thermal emissivity of liquid Ep = thermal emissivity of plate A = heat of vaporization cf drop, cal./g. @ = viscosity of vapor, g./(cm.) (sec.) p i = density of drop, g./cc. PO = density of vapor, g./cc. u = Stefan-Boltzmann constant, cal./(sq. cm.) (sec.) (” K.4) = drop vaporization time, sec. T
v = v,
as
where 51 and 52 refer to the bottom and top drop surfaces, respectively. Again assuming the bottom drop surface to be parallel to the plate, as shown in Figure 2, the geometric view factor, F1, becomes approximately unity and 51 = E. Combining Equations 24 and 25 and solving for 52 gives
‘(3
5 2
- E)
+ E)
= ____
3(1
Appendix B. Analytical Solution for Drop Vaporization Time Neglecting Thermal Radiation and Molecular Diffusion
Neglecting thermal radiation and molecular diffusion, and replacing r, with (3 V/4n)ll3,Equations 15 and 16 become
Literature Cited (1 ) Baumeister, K. J., “Heat Transfer to JVater Droplets on a Flat
Plate in the Film Boiling Regime,” Ph.D. thesis, University of Florida, 1964. ( 2 ) Baumeister. K. J.. Hamill. T. D.. “CreeDinc Flow Solution of the Leidenfrbst Phenomenon,” Natl. Adro;. Space Admin., NASA T N D-3133 (December 1965). ( 3 ) Baumeister, K. J., Hamill, T. D., Schwartz, F. L., Schoessow, G. J., “Film Boiling Heat Transfer to Water Droplets on a Flat Plate,” NASA TMX-52103 (August 1965). ( 4 ) Borishansky, V. M., “Heat Transfer to a Liquid Freely Flowing over a Surface Heated to a Temperature above the Boiling Point,” “Problems of Heat Transfer during a Change of State (U.S.S.R.),” S. S. Kutateladze, ed., U. S. At. Energy Cornm. Translation Series, AEC-TR-3405 (1953). (5) Godleski, E. S., Bell, K. J., “The Leidenfrost Phenomenon for Binary Liquid Solutions,” Proceedings of 3rd International Heat Transfer Conference, Vol. IV, Chicago, August 1966. ( 6 ) Gottfried, B. S., “Evaporation of Small Drops on a Flat Plate in the Film Boiling Regime,” Ph.D. thesis, Case Institute of Technology, 1962. (7) Gottfried, B. S.,Lee, C. J., Bell, K. J., “The Leidenfrost Phenomenon. Film Boiling of Liquid Droplets on a Flat Plate,” Intern. J . Heat M a s s Transfer, in press. (8) Lee, C. J., Ph.D. thesis, Oklahoma State University, 1965. ( 9 ) Leidenfrost, J. G., ‘‘De Aquae Communis Nonnullis Qualitatibus Tractatus,” ( A Tract about Some Qualitites of Common Water), Duisburg, 17.56; translation of portions to appear in Intern. J . Heat M a s s Transfer. (10) McAdams, W. H., “Heat Transmission,” 3rd ed., McGrawHill, New York, 1954. (11) Patel, B. M., Bell, K. J., “The Leidenfrost Phenomenon for Extended Liquid Masses,” 8th National Heat Transfer Conference, Los Angeles, August 1965. (12) Pleteneva, N. A , , Rebinder, P. A., Phys. Chem. 961, 973 (1946): Bull. USSR h a d . Sci., Div. Tech. Sci. 12 (1946). RECEIVED for review February 21, 1966 ACCEPTED June 15, 1966 Presented in part at 42nd annual meeting, American Institute of Chemical Engineers, New York, N. Y., December 1961. \ - I
where
4T
=
T p - TO
Combining Equations 27 and 28 to eliminate 6, m d replacing p V ) with P I , gives
(PI
-
dt Integrating, with V = V, when t = 0, results in
Since
T
is the time required for V to vanish,
which is the desired result. 568
l&EC FUNDAMENTALS
