Ind. Eng. Chem. Fundam. 1982, 21, 333-336
333
ARTICLES
Thickness of the Film on a Vertically Rotating Disk Partially Immersed in a Newtonian Liquid K. Vljayraghvan and Jal P. Gupla" ChemiC8l Engineering &p8rtfnent, Indian Institute of Technology, Kanpur, Kanpur-2080 16, Indis
Centrifugal, viscous, inertial, surface tension, and gravitational forces determine the shape and thickness of film formed on a vertical disk partially immersed in a Newtonian liquid and rotated about its horizontal axis. Film thickness has been measured over a wide range of experimental conditions with a pointed probe with an accuracy of 1 X m. A total of 486 measurements have been correlated by regression analysis to obtain an expression for the film thickness as a function of the system variables and liquid properties. The expression obtained has been integrated to calculate the entrainment of the liquid on the rotating disk. Further, the flat plate withdrawal theory has been modified to predict the entralnment under certain conditions. The entrainments predicted by both techniques have been compared with experiments. Knowledge of the film thickness, shape, and entrainment will be useful in mass transfer studies on vertically rotating disk gas-liquid contactors and in solving the equations of motion in such a system.
Introduction Vertically rotating disk-type gas-liquid contactors are used in blood oxygenerators, tissue culture, wastewater treatment, polycondensation reactors, etc. A number of disks, usually mounted on one horizontal shaft, pick up a film of liquid from the pool in which they are partially immersed. This liquid film is exposed to the desired gaseous mixture in the enclosing cylinder as the disks rotate. Mass transfer theories for such systems have neglected the actual shape of the film on the disk (White, 1969;Yamane and Yoshida, 1972; Lightfoot, 1974). A more rigorous mass transfer theory would benefit from a knowledge of the film thickness, shape, and liquid entrainment or holdup on the disks. A number of studies are available on the thickness of the film formed when a flat plate is withdrawn vertically (Groenveld, 1970; Spiers et al., 1974; Lee and Tallmadge, 1976) and on a horizontal disk rotating about a vertical shaft (Matsumoto et al., l972,1973a,b). Murakami et al. (1972) in their classical study on vertical disks measured holdup of very high viscosity polymer solutions (50 to 5000 P). The polymer solutions had completely fded the spaces between the disks and hence no question of measuring the film thickness arose in their study. In our studies reported here, we have measured the thickness of the film formed on a partially immersed vertical disk rotating around a horizontal axis. Newtonian liquids of varying viscosity, density, and surface tension were used and the disk was rotated at different speeds. The data obtained have been correlated to get an expression for the film thickness as a function of the system variables and the liquid properties. Further, the expression Visiting Professor, Chemical Engineering Department, The University of Michigan, Ann Arbor, MI 48104.
has been integrated to obtain the liquid holdup, which has been compared with the modified theory of flat plate withdrawal (Spiers et al., 1970) and with our experimental results. Preliminary observations on film formation in the system under study were done using colored water and vacuum-pump oil. In the latter, miniature bubbles of oxygen were entrapped and their motion in the oil film on the disk was followed by a stroboscope. These qualitative experiments showed that the film thickness was far from uniform and that the flow field was very complex. Because of these complexities, a theoretical prediction of the film thickness and liquid holdup can hardly be obtained. Murakami et al. (1972) have also made similar comments about the prediction of holdup in their system. Experimental Setup The experimental setup is shown in Figure 1. A Plexiglas disk of 0.145 m radius, precision milled to a smooth, flat surface of uniform thickness (runout less than 1 X m) was mounted on the horizontal shaft of a l/*-hp motor. The shaft diameter was 0.01 m. The alignment of the shaft and the disk was carefully checked to be horizontal and vertical, respectively. The motor was mounted on a jack so that the depth to which the disk was immersed in the liquid could be easily varied. The liquid was contained in a bath 0.35 X 0.21 X 0.35 m in depth. The wall effects were negligible. A pointed probe with a needle point was mounted on a travelling micrometer which could move both in the horizontal and vertical directions and had an accuracy of 1X m in either direction. The micrometer reading with the probe touching the stationary, dry disk was noted at several locations to get the base reading. The probe was retracted and the motor started. The disk rotated and picked up a liquid film. The probe was moved forward
0196-4313/82/1021-0333$01.25/00 1982 American Chemical Society
334
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982 01 I
Figure 1. Experimental setup.
5
2
1 Icd
3L
2
1
distance
3
'
~
2
12 m
Figure 3. Film thickness for vacuum-pump oil at various angular and radial positions: (-) 52 rpm; (---) 90 rpm; (- - -) 116 rpm.
Figure 2. Thickness measurements done along dotted lines. Table I. Physical Properties of Liquids Used
liquid
surface tension, N/m
water glycerol vacuum-pump oil
0.070 0.067 0.032
viscosity, kg m-l s''
density, kg/m3
0.0008937
1000 1164 910
0.0131 0.7826
till it just touched the film. This could be judged by carefully watching the reflection of the pointed probe in the liquid film on the disk. When the probe and its image just touched, the forward motion of the probe was stopped and the reading was noted. Any further forward motion pressed the probe into the liquid film, which produced a streak on the film surface that could easily be noticed. The readings were repeated several times and found to be reproducible. (A capacitance method of measuring the film thickness had posed a number of technical problems and finally had to be discarded in favor of the pointed probe.) Groenveld and Van Dortmond (1970) have also used a pointed probe in one of their related studies. Studies were done using water, glycerol, and vacuumpump oil at room temperature. Their physical properties are noted in Table I. The viscosities were measured in a rotary Rheotest viscometer. The surface tension values were obtained by the drop-weight method which had been pretested using standard liquids. Three different rpm values, between 50 and 180, were used for each liquid. These values were measured by a stroboscope. The highest value for a particular liquid was the one above which the film was thrown off the disk. The thickness of the film
Figure 4. Film thickness for water at various angular and radial positions: (-) 60 rpm; (---) 104 rpm; ( - - - ) 180 rpm.
was measured at every 30" angle interval and at a radial interval of 0.01 m (Figure 2). Closer to the disk edge, the radial interval was reduced. Representative sets of film profiles for water and vacuum-pump oil at 0" (horizontal), 60°,120°,and 180" and various radial locations are shown in Figures 3 and 4 at three different rpm values each. While no definite trend with respect to rpm is observable, one can say that, in general, the film thickness first increases as the rpm increases but thereafter it decreases. This is expected because of the complicated interaction among the various forces acting on the film. Figures 3 and 4 also show that the film thickness is far from uniform; rather, it varies by order of magnitude as one moves toward the outer edge.
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982 335
Hence it should be taken into account in mass transfer theories. No attempt was made to smooth the data since ripples were actually observed on the film surface. A total of 486 data points were obtained from the experiments on the three liquids. These points were correlated to get an expression for the film thickness in a dimensionless form. Dimensional Analysis The film thickness, h, is dependent on a number of parameters. A functional relationship can be written as h = F(r, 8, Ro, R, P , P, 6, Q) (1) Equation 1has nine variables, including the dependent variable, h. There are three dimensions. Hence, we need six independent dimensionless groups to represent the system adequately. The following groups were chosen: (i) the dimensionless film thickness
T = h(
-$)”
(ii) The ratio of the viscous to the surface tension forces, known as the capillary number PQr
Ca = rJ
(3)
(iii) A dimensionless surface tension number dependent only on fluid properties as defined by Esmail and Hummel (1975) (4)
(iv) Observations showed a noticeable dependence of the film shape on R. To take this into account, a modified “capillary number” based upon R was proposed as PQR
Cas = U
I
/
\
A
Figure 5. Half-immersed disk viewed as the withdrawal of a series of flat plates.
an overhead tank to maintain a constant level in the bath. The volume collected was divided by the time taken to obtain the entrainment per unit time or the volumetric flow rate. This was measured for three different rpm values with a 0.14-m radius disk dipped to a depth of 0.125 m in glycerol. The results were compared with those obtained by numerically integrating eq 9 from R to Ro over the total exposed surface of the film. This gave the holdup on the disk. It was divided by the average time for which the liquid film on the disk is exposed outside the bath at the specific speed to obtain the volumetric flow rate or the liquid pickup per unit time. Comparison was also made with the flat-plate withdrawal theory as modified below for the circular disk. For a flat plate, Spiers et al. (1970) gave the volumetric flow rate per unit width to be q = uh (1
-
$)
(5)
(v) To account for the dependence of the film shape on the depth to which the disk is immersed, a dimensionless number is defined as
If the flat plate withdrawal velocity u is replaced by the rotational velocity of the disk surface Qr, eq 10 can be written as q = Qrh( 1-
(vi) To account for the variation in the radial component of the acceleration due to gavity, a dimensionless number is defined as Q2Ro sin 8 (7)
g)
In dimensionless form, eq 11can be written as Q = Ca3/’T (1 - P / 3 )
(12)
where
x=T--
A relationship of the form T = ACaa~bCa~cY2d~e (8) was assumed, logarithms of both the sides were taken, and a regression analysis using the 486 data points was performed. The values of the exponents were obtained to two significant figures giving a correlation for the dimensionless film thickness as 7.ggCa2.93,,0.15~5.23
T=
Cas3.09x0.024
(9)
Entrainment of Liquid Entrainment of liquid by the disk was measured by wiping the film from the surface of the disk before the disk reentered the liquid bath. The liquid wiped off was collected in a receptacle outside the bath for a measured length of time and fresh liquid was fed to the bath from
By straightforward integration of eq 11or 12 from R to Ro we get the volumetric rate at which the disk withdraws liquid from the pool. The flat-plate withdrawal theory holds true when the disk is half-immersed, up to the axis, so that it can be assumed to be like the vertical Withdrawal of a very large number of infinitesimally small thin flat plates with withdrawal velocity of zero at the center and ORoat Ro (Figure 5). It was for this reason that the plate was dipped to the maximum extent possible in the experiments without spilling the liquid. The maximum depth, H, was 0.125 m for a disk of 0.145 m radius. The comparison of the experimental results with the values obtained by integrating eq 9 and eq 12 is shown in Table I1 for the entrainment of glycerol. The comparison is good when one considers the complexity of the phenomena and the possible experimental errors. The slightly
336 Ind. Eng. Chem. Fundam., Vol.
21, No. 4, 1982
Table 11. Volumetric Entrainment per Unit Time of Glycerol on Disk of Radius 0.145 m Dipped to a Depth of 0.125 m flat-
exptl,
correl, eq 9,
plate theory,
rad/s
m3/s
10m3/s
eq 1 2 ,
6.283 8.377 11.518
20.0 39.0 57.2
21.26 37.89 56.66
m3/s
21.5 40.0 59.2
higher value given by the flat-plate withdrawal theory is due to the fact that it assumes the disk to be half immersed ( R = 0), giving a wider film than was actually the case since R = 0.02 m in our experiments. The correlation, eq 9, is valid for disks immersed to any depth. Because of the complexity of phenomena and measurements, slight deviations from the experimental values are expected. One can therefore use eq 9 to predict the thickness of the film and can integrate eq 9 over the film surface to obtain the liquid holdup. Conclusions An empirical correlation has been obtained for the film thickness formed on a vertical disk partially immersed in a bath of Newtonian liquid and rotated about a horizontal shaft. It has been found to predict the liquid holdup with good accuracy when compared with the experiments and with the flat-plate withdrawal theory. It proves that the measurement of film thickness by a pointed probe was quite accurate. The correlation can be used to predict the film thickness and the liquid pickup for other experimental conditions within f7% as the comparison in Table I1 indicates. The only restriction is that the rpm should not be so high as to throw the film off the disk. These will aid in mass transfer studies as discussed in the beginning and also in predicting the cumulative volume of all the drops thrown off the disk at higher speeds. Photographic technique may have to be used to find the size of the individual drops. A preliminary study on the disintegration of a liquid film by a vertically rotating disk has been reported by Gupta et al. (1978). One point should be mentioned about the correlation obtained, eq 9. The exponents on Ca and Cas are not too far apart and probably the ratio of CalCas could be replaced by rlR, using eq 3 and 5. However, we resisted the temptation to do so because Ca does play an important role in the flat-plate withdrawal theory. Further, and more importantly, our experiments are continuing with other systems, such as wire-mesh disks and nonwetting disks and
we did not wish to throw out these dimensionless groups without a more comprehensive study. Nomenclature A = constant in eq 8 a, b, c = Exponents in eq 8 Ca = capillary number defined in eq 3, dimensionless Cas = modified capillary number defined in eq 5, dimensionless d, e = exponential in eq 8 F = functionality in eq 1 g = gravitational acceleration, m/sz H = depth of disk immersed in liquid bath, m h = film thickness, m Q = dimensionless volumetric flow rate per unit width defied in eq 13 q = volumetric flow rate per unit width defined in eq 10 for flat-plate withdrawal, mz/s Ro = disk radius, m R = radius of disk above the liquid surface = Ro - H , m R = dimensionless number defined in eq 6 r = radial coordinate, m rpm = revolutions per minute T = dimensionless film thickness defined in eq 2 u = flat-plate withdrawal velocity, m/s 9 = dimensionless number defined in eq 4 x = dimensionless number defined in eq 7 0 = angle with respect to horizontal and measured counterclockwise, deg R = velocity, rad/s p = density of liquid, kg/m3 I.L = viscosity of liquid, kg/ms u = surface tension of liquid, N/m Literature Cited Esmail, M. N.; Hummel, R. L. AIChE J . 1975, 2 1 , 958-965. Grovenveld, P. Chem. Eng. Scl. 1970, 2 5 , 1267-1273. Grovenveld, P.; Van Dortmond, R. A. Chem. Eng. Scl. 1970, 2 5 , 1571-1578. Gupta, J. P.; Vijayraghvan, K.; Mohan, R. "Proceedings, 1st International Conference on Liquid Atomization and Spray Systems"; Tokyo, 1978; pp 139-144. Lee, C. Y.; Tallmadge, J. A. Ind. Eng. Chem. Fundam. 1976, 15, 258-266. Lightfoot, E. N. "Transport Phenomena and Living Systems"; Wiley: New York, 1974; pp 380-387. Matsumoto, S.; Saito, K.; Takashima, Y. Bull. Tokyo Inst. Techno/. 1972. No. 109, 69-85. Matsumoto, S.; Saito, K.; Takashima, Y. Bull. Tokyo Inst. Techno/. 1973a, No. 116, 85-89. Matsumoto, S.;Saito, K.; Takashima, Y. J . Chem. Eng. Jpn. 1973b, 6 , 503-508. - .- - .- .
Murakami, Y.; Fujimoto, K.; Kaklmoto, S.; Sekino, M. J . Chem. Eng. Jpn. 1972, 5 , 257-263. Spiers, R. P.; Subbaraman, C. V.; Wilkinson, W. L. Chem. Eng. Scl. 1974. 2 9 . 389-396. White,'D. A. Chem. Eng. Sci. 1969, 2 4 , 369-376. Yamane, T.; Yoshida, F. J . Chem. Eng. Jpn. 1972, 5 , 381-386.
Received for review November 18, 1980 Revised manuscript received May 12, 1982 Accepted July 1, 1982
