able aid to us in the analysis of our results and programming the computer. literature Cited
(1) Dow Chemical Co., “JANAF Thermochemical Tables,” 1960, 1961. (2) Errede, L. A., Peterson, W. R., U. S. Patent 2,979,539 (April 11, 1961). (3) Frey, H. M., Progr. Reaction Kinetics 2, 155 (1964). (4) Gozzo, F., Patrick, C. R., Nature 202, 80 (1964). (5) Hudlicky, M., “Chemie der Organischen Fluorverbindungen,” pp. 235, 305, Deutscher Verlag der Wissenschaften, Berlin, 1960. (6) Mahler, TY., Znorg. Chem. 2, 230 (1963).
(7) Mann, D. E., Acquista, N., Plyler, E. K., J . Res. Natl. Bur. Stand. 52. 67 (19541. (8) Norton; F. j . , RGrig. Eng. 65, 33 (1957). (9) Park, J. D., Benning, A. F., Downing, F. B., Laucius, J. F., McHarness, R. C., Znd. Ene. Chem. 39. 354 (19471. (10) Potter, R. L., J . Chem. Phys. 31, 1100 (1959): (11) Semeluk, G. P., Bernstein, R. B., J . Am. Chem. SOG.76, 3793 f 19541. (li)Se;pinet, J., Chim. Anal. 41, 146 (1959). (13) Simons, J. P., Yarwood, A. J., Nature 192,943 (1961). (14) Trotman-Dickenson. A. F., “Gas Kinetics.” u. 113. Butter‘ worths. London. 1955.’ (15) Weissman, H. B., Meister, A. G., Cleveland, F. G., J . Chem. Phys. 29, 72 ( 1 958). I
.
RECEIVED for review December 28, 1964 ACCEPTEDMay 4, 1965
FILMS ADHERING T O LARGE WIRES UPON WITHDRAWAL FROM LIQUID BATHS J O H N A. T A L L M A D G E , R O L A N D A . L A B I N E , l A N D B E N J A M I N H. WOOD* Yale Unioersity, N e w Haven, Conn.
The Goucher (Go) and capillary (Ca) numbers are suggested for describing flow regimes in the prediction of flow rate and thickness of laminar, liquid films adhering to cylinders, which are withdrawn from liquid baths. These numbers are dimensionless cylinder radius and dimensionless withdrawal speed, respectively. Experimental measurements were made with large wires of G o -- 0.1 1 and 0.25 and compared with literature relationships for small wires (low Go)and large cylinders (high Go). The resultant disagreement indicated that an intermediate region exists where these special-case literature relationships do not hold; this region was found to b e at least 0.1 1 Go 0.25 and no more than 0.04 Go 1.8. The relationship between film thickness and flow rate is derived for the case of continuous withdrawal of cylinders.
1.8 and flat plates of Go = m . T h e thicknesses found (experimentally were substantially less than those predicted by flat plate relationships; thicknesses were only 30 to 6091, of that predicted for large radii cylinders; somewhat greater diffrrence was noted for the small wire. I t was concluded that, for 0.2 < C a < 1.2, the relationship based on large cylinders does not predict the film thicknesses in
Figure 10. (Ca, Go)
Estimated graphical form of
7
=
7
the intermediate Goucher region. The difference between data and prediction decreased as wire radius-Le., Goucher number-increased. Proposed Graphical Relationship
For a description of effect of flow regimes as well as for design purposes, it is desirable to express thickness for all cylinders on a common plot. As the dimensionless thickness (AIR,) is not suitable for large cylinders, T was chosen; thus a suitable plot is one of us. Ca a t lines of constant Go. The equivalent form of the Goucher and Ward correlation for small wires, Equation 11, when written in terms of T,is: = 6.8 Ca*I2Go
(16)
Figure 9 is a comparison, on a T-Ca plot, of the data of this work with extrapolations of this small wire correlation. As indicated in Figure 7 , the data in Figure 9 have thicknesses below those predicted by extrapolations of Equations 11 and 16. Based on the comparisons shown in Figures 8 and 9, neither the relationships for low or high Goucher number describe film thicknesses for Go of 0.1 1 to 0.25, where Ca is between 0.2 and 1.2. I t is concluded that the data reported herein are in an intermediate Goucher region and therefore that this intermediate region starts from within 0.04 < Go < 0.1 1 and extends to within 0.25 < Go < 1.8. That film thicknesses in this intermediate region are less than both low and high Goucher number predictions seems, a t first, a paradox. This apparent paradox is due to invalid extrapolations of empirical correlations; it may be resolved by showing both predictions on a T-Ca plot together with an estimate of intermediate relationships. I t is believed that, for each Go, the shape of the T-Ca relationship is similar to that of the large cylinder Equation 14 and that If values approach that of Equation 14 as an upper limit. Figure IO, constructed using this assumption, indicates that thicknesses in the intermediate region (0.04 < Go < 1.8) probably do lie above the upper limit of those described by Equation 16 for small wires and below the values described by Equation 14 for large cylinders. VOL. 4
NO. 4
NOVEMBER 1 9 6 5
405
Further work is necessary to determine general predictive methods for film thicknesses and flow rates in this intermediate region. Whether or not there is a secondary effect of Ca number is not known. Some attempts have been made to correlate the data of this work with analytical expressions in order to obtain a method for predicting film thicknesses in the intermediate region. Correlations based on linear log-log plots have been achieved. They are not reported here, since it is felt that general relationships cannot be represented as simple power functions, as indicated by the nonlinearity of the constant Go curves in Figure 10. Discussion of Parameter Choice
In general, there are apparently four parameters in continuous withdrawal of cylinders : gravity, surface tension, viscosity, and cylinder radius. These can also be expressed as four characteristic lengths if gravity is shifted from thickness to other terms: Film thickness, h Capillary or static length, a Cylinder radius, R, Dynamic length, d where d = (2 pu,/pg)'12. These four parameters may be expressed by four dimensionless groups, one of which is the dependent variable and another of which is redundant:
hlR,, thickness/radius Ca, viscous/surface tension T , gravity/viscous ( K is related) De, gravity/surface tension (Go is related) where the Derjaguin number, De, is defined as the dimensionless film thickness based on surface tension, or h/a ( 5 ) . Related to T and De are two dimensionless groups, K and Go, respectively, which use cylinder radius as the characteristic length for the gravitational force in place of film thickness. The functional relationships can be expressed in several combinations of these dimensionless groups. However, to describe a function general for all regimes for design purposes, it is useful to have film thickness appear only in the dependent variable and to avoid use of cylinder radius in this dependent variable and in more than one independent variable; these criteria reduce the combinations to only T or De as the dependent variable and to only (Ca, Go) or (Ca, K ) as independent variables. If we drop consideration of T = T (Ca, K ) and De = De (Ca, Go) because each form includes one force in all three groups, the two best forms are:
T h gravity VlSCOUS
T (Ca, Go) viscous
P-
or
(17A)
R, gravity
- T (surface tension' surface tension
De = De (Ca, K )
and or
=
h gravity = De surface tension
(
viscous surface tension'
(18A)
R, gravity)
(18B)
406
6.8
Call2 Go
I&EC FUNDAMENTALS
(19)
Conclusions
As shown in Figures 7 and 8, film thicknesses measured with large wires (intermediate Go) differed substantially when compared with those predicted by either of two special solutions, which have been developed in the literature for small wires (low Go) and for large cylinders (high Go). Thus an intermediate region does exist for cylindrical bodies in withdrawal and little information is available for predicting film thicknesses in this region. Special cases for cylinders do not describe the intermediate Go region as well as special cases for flat plates happen to describe the intermediate Ca region. Some estimate of the magnitude of the film thicknesses in the intermediate Go region for cylinders is given by the data in Figure 10. A less reliable, but the only available, estimate of the effect of fluid properties, withdrawal speed, and cylinder radius in this region is given by the dashed lines a t constant Go on the log-log plot of T us. Ca (Figure 10). A quantitative criterion for this intermediate region is, as shown in Figure 6, a t least from Go of 0.11 to 0.25 and may be as large as Go from 0.04 to 1.8. The relationship between film thickness and flow rate adhering to cylinders is given by Equation 9. Acknowledgment
Interpretation of the data and preparation of this manuscript were supported by National Science Foundation Grant 19820. Nomenclature = capillary length = (2 u/pg)'/2, cm. Ca = capillary number = d / a = pu,/u d = dynamic length = (2 p ~ , / p g ) ' ' ~ ,cm. De = Derjaguin number = h / a = h ( p g / 2 u)llZ F = function of (h,/R,) given by Equation 9 g = acceleration of gravity, cm./sec.2 Go = Goucher number = RJa = R, (pg/2 u)l/2 h = film thickness, cm. K = dimensionless cylinder radius = (R,/d)2 = R,'Jpg/2 puVr Q = cylinder flux, volume flow per unit wire circumference, sq. cm./sec. r = radius to any given point = rP/Rw rp = radius to any given point, cm. R, = cylinder or wire radius, cm. u = vertical fluid velocity a t any given point, cm./sec. u, = withdrawal velocity, cm./sec. v = vertical fluid velocity = u / u , V = volume flow of adhering liquid, cc./sec. T = dimensionless film thickness = h (pg/pu,)1/2 = h m a
GREEKLETTERS
(16)
4- (h/R,) surface tension, dyne/cm. = viscosity, poise or g./cm.-sec. = 3.14159 = density, g./cc.
cy
= 1
u
=
p ?r
p
SUBSCRIPT = at liquid-gas interface in Region I, constant thickness region
o =
Ca31zK112
viscous
The form of Equation 17 was chosen in this work, since T is much less sensitive than De to changes in Ca over the entire flow regime. This phenomenon is demonstrable for large cylinders by comparison of plots of data in both T (70) and De (5) form. For small wires, this difference in sensitivity is shown by comparison of the power on Ca for the T and De forms of Equation 11, which are:
T
%e = 4.8
It is possible that other combinations of the four parameters present may be useful for analysis of experimental results where film thicknesses are known and may lead to relatively simple analytical expressions, but none is known at this time. Such forms would be less convenient for design purposes, since iterative methods would be needed to solve for film thickness.
SUPERSCRIPT - = based on mean film thickness or flow rate, given by Equation G literature Cited (1) Adam, N. K., “Physics and Chemistry of Surfaces,” 2nd ed.,
pp. 363-5, Clarendon Press, Oxford, 1938. (2) Boucher, D. F., Alvcs, G. E., Chem. Engr. Prog. 59, No. 8 , 75 (1963). (3) Brownell, L. E., Ibid., 43, 537, 601, 703 (1947); 46, 415 (1950). (4) Coulson, J. M., Richardson, J. F., “Chemical Engineering,” Vol. 2, p. 404, Pergamon Press, New York, 1955. (5) Derjaguin, B. V., Titiyevskava, A. S., Dokl. Akad. Nauk. SSSR 50, 307 (1945).
(6) Goucher, F. S., Ward, H., Phil.Mag. 44, (6), 1002 (1922). (7) Gutfinger, C., Tallmadge, J. A., A.I.CI1.E. J . 10,774 (1964). (8) Harris, N. L., Hirst Research Centre, General Electric Co., Wembley, England, personal communication, Dec. 10, 1963. (9) Landau, L. D., Lifshitz, E. M., “Fluid Mechanics,” p. 232, Addison Wesley, Reading, Mass., 1959. (10) Van Rossum, J. J., Appl. Sci. Res. A7, 121 (1958). (11) White, D. A., Tallmadge, J. A., Chem. Engrg. Sci. 20, 33 (1965). (12) Wood, B. H., Labine, R. A., Dept. Chem. Engrg., Yale University, unpublished report, May 1957. RECEIVED for review August 11, 1964 MARCH 29, 1965 Division of Water and Waste Chemistry, (in part) 148th Meeting, ACS, Chicago, Ill., September 1964.
HEAT TRANSFER I N LIQUID-FILM FLOW T. 0 . PENMAN AND R . W.
F. T A l T
Chemical Engineering Department, The University of Adelaide, Adelaide, South Australia
Measurements have been made of local heat transfer coefficients to liquid-vapor mixtures flowing upward inside vertical tubes at atmospheric pressure. Four tube sizes and six liquids were tested with a 10 to 1 range in feed rates. The heat transfer coefficient has been found to depend mainly on the vapor velocity when liquid-film (generally known as “climbing film”) conditions have been established, For any one liquid the heat transfer in the liquid-film region is dependent solely on the vapor velocity and independent of the liquid feed rate. An equation has been derived which correlates our data and those of other authors for liquid-film conditions, irrespective of whether the film b e moving upward or downward in vertical tubes, or on the inside of a horizontal tube.
N RECENT
years, considerable attention has been paid to the
I determination of heat transfer coefficients for liquids boiling
inside tubes. Several excellent reviews (5, 74, 78) are available, although few papers-for example, ( 7 , 9, 27)-relevant to this work have been published recently. In many cases [including (4, 6, 7, 77, 79, 20, 27)] authors have used average values for the heat transfer coefficient for the whole length of a given tube. We believe that the use of a n average value is fundamentally unsound, in that it tends to obscure the marked chamges in boiling conditions which occur as a boiling liquid flows through the inside of a heated tube. T h e various zones of boiling have been adequately described (73) and may be summarized as:
A. A zone where bubbles are relatively uniformly dispersed throughout the liquid (1:he homogeneous zone) B. A zone where slug-type flow predominates (the slug zone) C. A zone in which the action of the vapor forces the liquid u p the wall in the form of a thin film (the liquid-film zone) D. A zone where the liquid film is discontinuous (the drywall zone) Zone C, the liquid-film zone, is commonly referred to as the climbing-film zone. This term is undesirable, in that it gives a misleading impression of the way in which the thin liquid film is formed, and also gives the impression that it is fundamentally different from other forms of film flow. This we have not found to be the case. There are two possible reasons for the formation of a discontinuous liquid film (zone D): There is insufficient liquid present to wet the wall properly or vapor is being formed so fast that it blankets the wall in a manner analogous to that found in what is commonly referred to as “film boiling.” We have found no evidence that the latter type of discontinuity occurs with the heat fluxes we used.
Despite the extensive literature, the amount of concrete information available on the effect of such factors as surface tension, the nature of the tube surface, the nature of the boiling action, and liquid and vapor velocity is disappointing. Most authors have failed to appreciate the need for concentrating on local conditions in order to obtain a satisfactory description of the mechanism of heat transfer to liquids boiling inside tubes. Even splitting the action within the tube into zones of homogeneous flow, slugging, and liquid film flow is not enough, as within such zones it is difficult to separate from one another the effects of such variables as temperature difference, heat flux, vapor velocity, feed rate, and geometry of the heating surface. T h e apparatus described below was designed to enable us to study local conditions. The results obtained for zones A and B have not proved amenable to correlation by any simple equation, but those for zone C can be expressed by means of a n equation involving only two dimensionless groups. Apparatus and Materials
It was clear from the work of all previous investigators that the apparatus should consist of a heated tube, preheater, and condenser, plus a circulating pump and metering devices. A diagram of the apparatus is shown in Figure 1 and the main features of the design are discussed below. Tube. Four sizes of tube were investigated, having dimensions as follows: Nominal Size, Inch 3/a ‘/2 ”4
1
Heated Length, Inches 72 68 75 75 VOL. 4
I.D., Inch 0.375 0.500 0,747 0,996
0.D., Inches
0.500 0.751 1.006 1.249
NO. 4 N O V E M B E R 1 9 6 5 407
