is then made, under a helium atmosphere, using the level indicator. Approximately 15min after transfer has occurred, the xenon in the stripper is frozen out by adding liquid nitrogen to the cold trap in the stripper section, and the diffusion pumpToepler pump combination is diverted to evacuate the inner container of the stripper through the cold trap. During this period the annulus formed by the inner and outer stripper compartments is not pumped. The gas collected from the stripper is combined with that previously obtained from the annulus in the gas buret, and the amount is determined. Pumping of the annulus is then resumed, whereas the cold trap is gently heated and the xenon is again allowed to circulate through the salt. After about a 2-hr period, during which the hydrogen is alternately collected from the inner stripper compartment and the annulus, recovery of the saturating gas is complete. The total amount of gas collected in the buret is determined and samples are taken for mass spectrometric analysis. Completion of the experiment then simply entails the retransfer of the salt to the saturator section by adjusting the xenon and helium pressures, thawing the freeze valve, and, once a xenon breakthrough is noticed on the saturator side, again freezing a plug of salt a t F. Typical Results
A test of the applicability of Henry’s law in describing the solubilities of helium and hydrogen in a LiF-BeF2 eutectic a t 600°C is provided by the data which are graphically displayed
in Figure 2. The solid lines in the figure represent the corresponding Henry’s law constants for the two solute-solvent systems at this temperature; the constants which were derived from the data shown in Figure 2 are (8.40 f 0.16) x 10-8 mole of gas/atm-cni3 salt for helium and (4.34 f 0.20) x 10-8 mole of gas/at,m-cm3 salt for hydrogen. Further studies with these two gases and the LiF-BeF2 eutectic over the temperature range 500-800°C are currently in progress. Acknowledgment
We are grateful to Warren R. Grimes for calling our attention to the previously classified work of R. F. Newton and for several helpful discussions which facilitated this effort. Similar discussions with the late George SI. Watson, and the skill of William P. Teichert and Wiley Jennings, Jr., likewise proved valuable during the design and construction of the apparatus. We are also very grateful to Loness Guinn for the mass spectrometric analyses of the gas samples. literature Cited
Cleaver, B., Mather, D. E., Trans. Faradav SOC.66, 2469 (1970). Grimes, W. R., Smith, N. V., Watson, G. AT.,J . Phys. Chem. 62, 862 (1958). Newton, R. F., Hill, D. G., U.S. Atomic Energy Commission, Report ORNL-1771, p 70, 1954. RECEIVED for review March 22, 1972 ACCEPTEDJuly 13, 1972 This research was supported by the U.S. Atomic Energy Commission under contract with Union Carbide Corporation.
COMMUNICATIONS
laminar Natural Convection Heat Transfer in Dilute Aqueous Polymer Solutions An experimental investigation was conducted of laminar natural convection heat transfer from a horizonta I heat cylinder to a dilute aqueous solution of polyethylene oxide. Concentrations of Polyox up to 1000 ppm were investigated. Similar measurements were taken in pure water for comparison. The addition of Polyox results in reduction of the heat transfer rate in laminar natural convection.
T h e addition of small quantities (less than 1000 ppm) of high molecular weight polymers to water has been shown to cause a reduction of the friction coefficient. A review of the dragreducing effect resulting from the addition of such polymers to water flow is given by Lumley (1969). He reports that polymer concentrations in the range of 50 ppm give substantial drag reduction. This friction reducing capability of these polymer additives created potential application in various engineering problems. An important area of interest for application of such polymers is fluid flow through heat transfer equipment. There have been several investigations of the effect on heat transfer of these polymer additives. Gupta, et al. (1967), analyzed the turbulent heat transfer characteristics of flow in a pipe with a heated wall. Pressure-drop data were also 586 Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 4, 1972
reported with this work. Further work on turbulent pipe-flow heat transfer was presented by Wells (1968), who was able to use the friction factor correlation of hleyer (1966) with the heat transfer analogy and to predict heat transfer rates from pressure drop and flow measurements. A similar investigation is presented by Poreh and Paz (1968) for turbulent heat transfer in pipe flow. The area to which this paper addresses itself is heat transfer in laminar flow of dilute polymer solutions. It was the objective of this research to develop a relationship between the laminar heat transfer characteristic of dilute polymer solutions and that of pure water. In this way established correlations defining the laminar heat transfer characteristic for pure water can be used t o estimate the heat transfer characteristic
TEST SECTION f ' 0 , 0. I ,049. WALL
\
"ONEL SPRING
.00s. TUlCK
COPPER ELECTRODE CHROME PLATED
MICA SHIELD
\
\
I
WOO
8,OW
. 6.000 . 7.000
Figure 1. Tes
for polymer solutions. The problem of specific interest is laminar natural convection heat transfer from a horizontal heated tuhe immersed in a dilute polymer solution.
-
-
5,000
N
c 4,000 .
Experimental Apparatus and Procedul
The high molecular weight polymer additive used was polyethylene oxide (w 301), produced by Union Carbide Co., which will be referred to hereafter as Polyox. Polyox is a water-soluble resin with very high molecular weight (ea. 106). Specific concentrations (50, 100, 300, and 1000 ppm) of Polyox were introduced into a distilled water bath containing the tube test assembly (Figure 1).The tank contained 20 gal of the water and Polyox additive. Located a t the hottom of the tank were two 220-V flat-plate heaters used for solution temperature control. A mechanical agitator was used in the tank prior to each test to assure uniform bulk solution temperature. The test section consisted of a 7 in. long Monel tuhe of commercial finish, nominally 8/n-in. 0.d. by 0.049-in. wall with the ends silver-soldered to copper electrodes. A motorgenerator do power source connected to the electrodes was used for electrical resistance heating of the test section. A cross-sectional view of the test section is shown in Figure 2. Direct current supplied to the tube caused it to act as a res& ance heater. The current flow was measured with an ammeter. Temperature measurements of the inside, adiabatic wall were made by 12 calibrated chromel-constantan thermocouples with glass and high-temperature-varnish insulation. A t each position shown in Figure 2 there were four thermocouples, oriented a t 0, 90,180, and 270'. A sheet of mica 0.003 in. in thickness was inserted in the tube before the thermocouple assembly was installed. This provided electrical insulation of the thermocouples from the tube, therehy eliminating de interference to the thermocouple emf measurement. Each thermocouple junction was silver-soldered and held in position by a ceramic head fastened by cement to an Inconel spring. This spring forced the thermocouple junction against the mica sheet with a force of about 1 lh. All thermocouple leads from the test section left the tube through a 1/&. hole in one of the test section supports above the Polyox solution. Although the tube thermocouples were in physical contact with the mica sheet rather than the tube wall, tube surface temperature measurements made with an external probe verified that the inner thermocouples provided a satisfactory measurement of temperature of the surface of the
I
"P
1,000 10
20
30
40
50
60
70 80
AT f"F)
Figure 3. Heat transfer results of laminar natural convection from a heated horizontal cylinder to an aqueous Polyox solution
tube. The hulk-solution temperature was determined by placement of several thermocouple prohes into the Polyox solution a t several locations. To initiate a test run, the bulk solution temperature was first adjusted to the desired value, the mechanical agitator being used to eliminate stratification and to maintain a uniform Polyox solution temperature. The agitator was removed and the bulk-solution heater current was removed, thus eliminating any artscial convective currents from being superimposed on the natural currents around the test sections. The de current for heating the test section was adjusted. With the entire system in equilibrium, test data were recorded. Measurements of the tube and hulk-solution thermocouple emf and the current flow to the test section were recorded. To reduce any error associated with a slight shift from equilibrium during observations, deviations in the testsection heating current were averaged during data reduction. When deviations in the current readings were greater than 10.1% the entire test was repeated. This procedure was followed for several bulk- and tube-temperature cornhinations for Polyox concentrations of 0, 50, 100,300, and 1000 ppm. Results and Conclusions
The heat transfer results of the distilled water and waterPolyox experiments are presented in Figure 3. A correlation given by McAdams (1954) for a large sampling of experiInd. Eng. Chom. Fundom., Vol. 11, No. 4, 1972
587
,4t
‘-0 100 200 300 400 300 600 700 800 900 1000
tion must be a t rest a t the start of the test. Some changes in concentration could have occurred during this period for some of the tests and this could explain the increased scatter a t the higher concentrations. The effect of various concentrations of Polyox additive in natural convection heat transfer is best presented by a cross plot of the results, as shown in Figure 4.Addition of polymer causes a decrease in the heat flux. Additions of relatively small quantities of polymer cause significant changes in heat transfer. The addition of 1000 ppm of the polymer reduces the heat flux approximately 40%.
CONCENTRATION ( pprn 1
Figure 4. Effect of Polyox concentration on laminar natural convection heat transfer
mental and analytical results is presented for the distilled water. The comparison of the experimental data to the correlation of McAdams indicates relatively good agreement. The agreement obtained for pure water indicates the ability of the apparatus to give reliable results. The data for the Polyox additive are also presented in Figure 3. The results for 50 and 100 ppm indicate little deviation from a line drawn parallel to the McAdams correlation for pure water. The 300 and 1000 ppm data scatter more. The scatter may result from poor dispersion of the Polyox a t these higher concentrations for some of the test batches. Every effort was made to assure that dispersion was complete, but owing to the nature of natural convection, the bulk solu-
Literature Cited
Gupta, hl. K., Metzner, A. B., Hartnett, J. P., Int. J . Heat Mass Transfer 10, 1211 (1967). Lumley, J. L., Annu. Rev. Fluid Mech. 1, 367 (1969). McAdams, W. H., “Heat Transmission,” p 177, McGraw-Hill, NPWYork. N - .Y - . . 1CL54. ---Meyer, W:A., A.I.Ch.E. J . 12 (3), 522 (1966). Poreh, At., Paz, V., Int. J . Heat Mass Transfer 11, 803 (1968). Wells, C. S.,Jr., A.Z.Ch.E. J . 14 (3), 406 (1968). I
I
DONALD W. LYONS* JAMES W. WHITE JOHX D. HATCHER Departments of Textiles and Mechanical Engineering Clemson University Clemson, S. C. 29631
RECEIVED for review September 7, 1971 ACCEPTED June 5. 1972
Size Distribution for Crystallization with Continuous Growth and Breakage The unsteady-state differential equation i s formulated for uniform crystal growth and breakage. The model assumes that all crystals have the same linear growth rate, that each has an equal chance of breaking, that breakage occurs randomly at any plane in the crystal, and that the fragment volumes sum to the volume of the parent crystal. The moments of the size distribution are given as a set of ordinary linear differential equations. The steady-state crystal size distribution i s found in closed form for a stirred continuous crystallizer and i s compared with the size distribution that would occur if new particles were formed either by microscopic nucleation or from chipping of the edges of parent crystals.
T h e crystal size distribution (CSD) obtained in a crystallizer depends on the dynamic interaction of crystal growth, breakage, nucleation, and the inflow and outflow of crystals. A simple model of crystallizer performance has been given by Saeman (1956) and extended by Randolph and Larson (1962) for the case where crystallization occurs in a mixed suspension mixed product removal (MSMPR) crystallizer. Their treatment assumes that: (1) no breakage occurs; (2) nucleation occurs spontaneously and continuously within the crystallizer; (3) no crystals enter in the feed; (4) the McCabe AL law holds, which requires that all crystal diameters increase by the same increment in a unit of time, regardless of the crystal size. An extension of this model by Randolph (1969) investigated the effect of crystal breakage. For simplicity it was assumed that crystal breakage produced fragments whose diameters summed to the diameter of the parent crystal. The probability 588
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 4, 1972
of a crystal breaking was assumed proportional to some positive power of its diameter. Numerical solutions were given for the case when breakage occurred in the middle of the crystal and these solutions showed a narrowing of the distribution as well as a reduction in the dominant size. This paper presents a simple model of simultaneous crystal growth and breakage and gives solutions of the general integro-differential equation for the model. For the unsteadystate case a set of moment equations is found while for the steady behavior of an XLISMPR crystallizer a very simple form of the CSD is derived. Uniform Breakage Model
We will now formulate the general differential equation for simultaneous growth and breakage in a crystallizer. In this model we make the following assumptions. (1) ,4 crystal has
