In the least favorable case of an ideal backmixing reactor the mean value of the distribution for a cutoff time 8, = 4 was 774 smaller than the values calculated for e, = Q). Moreover, the values of the variance of the distribution were about 25% smaller. Although the absolute value of the error is large, no abnormal dispersion was observed, because the errors are all of the same sign and affect all the values equally. Therefore, it is possible to use the mean and variance values obtained experimentally in a qualitative way. However, unless larger cutoff times are used, the absolute value of the mean and variance of the distribution must be considered to have between 10 and 20% error. Nomenclature
d
D F(e) Z(e) L N NR,
= = = = = = =
diameter of blades of agitator, cm diameter of tank, cm cummulative distribution function, min residence distribution function, min height of tank, cm agitator speed, rpm Reynolds number (see Equation 11)
dimensionless time variable macroscopic residence time, min cutoff time, dimensionless mean value of distribution, min mean value calculated using finite cutoff time viscosity of fluid, g/cm sec density of fluid, g/ml variance or second central moment of distribution variance calculated using finite cutoff time literature Cited
Ambwami, D. S., Adler, R. J., A.Z.Ch.E. J . 12,612 (1966). Danckwerts, P. V., Chem. Eng. Sci., 2 , 1 (1953). Levenspiel, O., “Chemical Reaction Engineering,” pp. 244-9, Wiley, New York, 1962. Wolf, D., Resnick, W., IND.ENG.CHEM.FUNDAMENTALS 2, 287 (1963).
Facultad de Zngenieria Quimica Santa Fe, Argentina
RAMON L. CERROI JOSE M . PARERA
for review March 6, 1969 RECEIVED ACCEPTEDSeptember 29, 1969
GREEKLETTERS e, 7 = parameters of Wolf and Resnick’s model
1 Present address, University of California. Davis, Calif. 95616
Film Boiling from a Sphere during Forced Convection of Subcooled Water Forced convection film boiling from a sphere was investigated. The water was subcooled; a vapor film existed in which no net formation of vapor occurred. The resulting rates of heat transfer were considerably greater than that predicted for a saturated fluid or for forced convection from the constant temperature sphere formed by the liquid-vapor interface. If it is assumed that the film does not exist, a poor correlation exists between the experimental and theoretical results; this assumption cannot be justified, since a film was observed in all cases.
and Clark (1963) treated the case of a saturated free convection film boiling from a sphere. Using a boundary layer analysis they found that REDERKING
Witte (1968) presented an analysis for saturated film boiling from a sphere during forced convection. By assuming that the velocity profile is linear and that the interfacial velocity can be determined from potential flow theory, he found that
Sideman (1966) used potential flow theory and assumed slip a t the wall to develop the following equation for nonboiling heat transfer during forced convection flow past a sphere. q = 1.13[--] U,(PkC,)
2
(T - Te)
(3)
Subcooled forced convection film boiling from a sphere has been investigated experimentally by Witte (1968) and Witte et al. (1968). They concluded from a study of liquid
sodium that a vapor film does not exist for large degrees of subcooling of the liquid, and that the heat transfer rate may, therefore, be described by Sideman’s relationship (1966). However, Witte (1968) suggested that further experiments be performed to verify the absence of a vapor film. It is important to obtain heat transfer data for forced convection film boiling on a sphere under conditions in which the film may be observed. Apparatus
A Lapel high frequency 5-kw induction heater operating between 2.5 and 5.0 MHz was used to heat the sphere shown in Figure 1. This differs from the experiments of Witte et al. (Witte, 1967; Witte et al., 1968) in that the sphere was heated continuously. The liquid used was water and the sphere was steel. A constant fluid velocity was maintained by means of a constant head tank and the flow rate was measured by means of a rotameter. The glass wool was placed in the tube to remove any lateral temperature gradients, and the Lavite disk was used t o prevent the hot sphere from touching the glass tube. The hot sphere was supported by the liquid flow itself and was not in contact with the Lavite disk. The temperature of the sphere was determined with a Pyro microoptical pyrometer. This apparatus permitted film boiling of subcooled liquids and simultaneous visual observation of the film. VOL. 9 NO. 1 FEBRUARY 1970
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FUNDAMENTALS
183
Results
Subcooling of the water to between 10' and 50°C resulted in a stable vapor film and no bubbles left the film a t flow rates of between 0.4 and 2.7 cc per second. The cool liquid prevented the emission of vapor from the film due to buoyant forces. This greatly simplified the energy balance, since no vapor was present downstream. The resulting rates of heat transfer were much greater than predicted by Equation 2 for a saturated liquid (Figure 2). This is in qualitative agreement with the theory developed by Cess and Sparrow (1961) for film boiling of a subcooled liquid on a vertical flat plate. Since no net formation of vapor occurs and the vapor-liquid interface is approximately spherical, Sideman's Equation 3 could be expected to apply if T is set equal to T,. However, the experimentally measured
T=T, 0
5
IO
qexp. GLASS WOOL
~
w
~
TUBE
THERMOCOUPLE INLET
Figure 1 .
iq
0
Film boiling apparatus
'd
T, =12.2'C
A T, =50.6'C
a V
TB=23.3"C
W
A
A T~'50.6"C
15 20 25 CAL/SEC-CM~
30
values are an order of magnitude greater than predicted (Figure 3). Witte et al. of liquid \ (1968) concluded~ from their study ~ sodium that no film was present, since their data correlated with Sideman's Equation 3 with T set equal to T,. If these assumptions are made here, a not unreasonable correlation exists, as also shown in Figure 3. However, since a film is visible, this assumption is not justifiable in this case. In all the cases noted above, the film was visually observed to surround the sphere completely and intermittent liquidsolid contact was not visually apparent. However, by greatly increasing the rate of liquid flow past the sphere, a condition of simultaneous nucleate and film boiling could be made to exist. The nucleate boiling occurred on the downstream side of the sphere, accompanied by a significantly lower surface temperature in the nucleate boiling region of the sphere. Conclusions
15
8
0IO
4
5-A
0.04
0.05
0.698(h'fg pv k, U,/O(T,
I
I
I
0.06
0.07
0.0s
-TS))"'(Tw
-Ts),
CAL/CM'-SEC
Figure 2. Experimentally measured rates of heat transfer for subcooled water compared to that theoretically predicted for saturated water 184
0 T~=12.2"C
0
Figure 3. Experimentally measured rate of heat transfer for subcooled water compared to that theoretically predicted by Sideman's Equation 3
1/2" DIA STEEL SPHERE 7/8" GLASS I D
T=T,
I&EC FUNDAMENTALS
VOL. 9 NO. 1 FEBRUARY 1970
Forced convection film boiling of water subcooled below its boiling point may result in a vapor film in which no net formation of vapor occurs. The resulting rates of heat transfer are considerably greater than that predicted for a saturated liquid and for forced convection from the constant temperature sphere formed by the liquid-vapor interface. If it is assumed that the film does not exist, correlation between experiment and theory is poor; however, this assumption is not justified, since a film was visually observed. A relationship for predicting rates of heat transfer from a sphere to a subcooled liquid during forced convection film boiling remains to be found.
~
Nomenclature
C, D g lo
k q
T 11 p I.(
specific heat, cal/g-"C diameter of sphere, cm = acceleration of gravity, cm/sec2 = latent heat of vaporization, cal/g = h, 0.68.C, ( T , - T J ,effective latent heat of vaporization, cal/g = thermal conductivity, cal/sec-cm-'C = heat transfer rate, cal/sec-cm2 = temperature, "C = bulk flow rate/annular area between sphere and glass tube, cm/sec = density, g/cm2 = viscosity, g/cm-sec = =
+
SUBSCRIPTS B = bulk exp = experimental I = liquid
v
= saturation = vapor
w
=
s
wall
literature Cited
Cess, R. D., Sparrow, E. M., Heat Transfer (Trans. A S M E ) 90, 394 (1961). Frederking, T. H. K., Clark, J. A., Advan. Cryog. Eng. 8 , 501 (1963). Sideman, S., Ind. Eng. Chem. 58 (2), 54 (1966). Witte, L. C., "Heat Transfer from a Sphere to a Liquid during Forced Convection," Ph.D. thesis, Oklahoma State University, 1967. ENQ.CHEM.FUNDAMENTALS 7, 517 (1968). Witte, L. C., IND. Witte, L. C., Baker, L., Haworth, R. R., Heat Transfer (Trans. A S M E ) 90, 394 (1968). R. N. JACOBSON F. H. SHAIR California Institute of Technology Pasadena, Calif. 91109 RECEIVED for review April 11, 1969 ACCEPTEDNovember 28, 1969
Settling of Spheres in Drag-Reducing Polymer Solutions The fall velocities of spheres in drag-reducing polymer solutions were determined at 20" and 40°C. Drag reduction occurs at Reynolds numbers below the limit lo4specified by Sanders, and the value of Reynolds number alone is not an adequate indicator of whether the phenomenon is present or not.
SEVERAL reports on the settling of rigid spheres in dragreducing dilute polymer solutions have appeared in the literature (Ruszczycky, 1965; Sanders, 1967; White, 1966). The problem is of interest from both the practical and experimental viewpoints. Aside from the fact that this particular geometry is well defined, it is particularly attractive from the experimental viewpoint because one need not subject the test solution to the continuous shearing action of the flow, as is generally inevitable in pipe flow, for example. Consequently, complications arising from possible shear degradation of the solution, to which these drag-reducing polymers are highly susceptible, are largely avoided. However, even for Newtonian fluids (Maxworthy, 1965; Pruppacher and Steinberger, 1968) there are a number of difficulties associated with obtaining accurate and reproducible data. Temperature gradients in the test fluid, effects of the wall of the fluid container, eccentricities of the fall path from the container axis, vibrations, and other extraneous effects are often responsible for these difficulties. Whereas the motivation for obtaining accurate falling sphere data in Xewtonian fluids arose from the need to test the validity of the asymptotic solutions of Stokes, Oseen, etc., the motivation here resides in determining the conditions under which the drag reduction phenomenon begins to manifest itself. We report here the results of a large number of sphere fall velocities, in which considerable care was taken to avoid and/or account for most of the experimental problems mentioned. The falling sphere setup used consisted of four jacketed 6-foot-long Plexiglas cylinders with diameters of 20.32, 13.36, 6 35, and 3.18 em. Water from a 60-gallon thermostated bath was circulated in the outer jacket of each cylinder, using individual pumps. The time of fall was determined by means of a set of four photocells, each pair of which was connected to a separate millisecond electronic timer. The photocells were mounted vertically, so that the time of fall in two 30-em lengths of the cylinder could be determined. The two timing sections were separated by a distance of 30 cm. The distance from the top fluid surface to the beginning of the first timing
Table 1.
Sphere Diameters
Sphere
Ruby Tungsten carbide Hastelloy C Teflon Steel
Inch
1/8, 3/16, 1/4, 5/16, 1/8, 3/16, 1/4, 5/16, 1/8, 3/16, 1/4, 5/16, 3/16, 1/4, 5/16,
3/8 3/8 3/8 3/8, 1/2
1/2
section was 64 cm, and that from the lower boundary of the bottom timing section to the bottom of the cylinder was 35 cm. The spheres used consisted of five diameters of ruby, five of tungsten carbide, five of Hastelloy C, one of steel, and five of Teflon (Table I). Each sphere was dropped five times in the test fluid, effectively resulting in ten velocity determinations which were averaged. Three test fluids-distilled water and 50 and 100 ppm of Union Carbide Polyox WSR301-were investigated a t 20" and 40°C. Altogether, the total number of sphere velocities determined for this report was 5040. The sphere velocity data from the two timing sections were used as a test on attainment of terminal velocity. The average value of the velocities in the upper timing section agreed with that in the lower timing section within 1% for all spheres except the l/s-inch steel sphere and the 3/8-in~h tungsten carbide. For the l/z-inch steel sphere a t 2OoC the average velocities in the upper and lower timing sections were, respectively, 153.3 and 154.3 cm per second for the 100-ppm Polyox and 155.8 and 159.3 ern per second for the 50-ppm Polyox. Thus, the difference between the two values for the 50-ppm Polyox was less than 2.3%. The corresponding values for a/8-inch tungsten carbide were 190.3 and 193.6 cm per second for the 100-ppni Polyox and 195.2 and 199.2 cm per second for the 50-ppm Polyox. Replication in the velocity measurement was necessary to assess the inherent experimental scatter in this type of data. The difference of individual velocities from the average value did not exceed 30j0, and this is mainly attributable to the fact that, a t the VOL. 9 NO. 1 FEBRUARY 1970
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185
