Hutzenlaub, A., Kunststofe, 57, 1963 (1967). Langlois, W. E., I B M J . Res. Develop., 7, 112 (1963). Licht, L., J . Lubric. Technol., 90, 199 (1968). Ma, J. T. S., J . Basic Eng., 87, 837 (1965). Patel, B. J., Cameron, A., Proc. Con{. Lubric. Wear, 1957, Paper 73. Pfeiffer, J. D., T a p p i , 49, 342 (1966). Pfeiffer, J. D., "The Mechanics of a Rolling Nip in Paper Webs," 52nd Annual TAPPI Meeting, New York, February 19, 1967.
Rienau, J. H., Pap. Film Foil Converter, 1967, 42. Sakiadis, B. C., A I C h E J., 7, 221 (1961). Walowit, J. A., Battelle Memorial Institute, Columbus, Ohio, private communication, 1966. Wildman, M., Wright, A,, ASME Paper No. 64-Lub-20, October, 1964. RECEIVED for review July 10, 1969 ACCEPTED September 28, 1970
Heat Transfer to a Two-Phase liquid System in an Agitatedvessel with Coils Steady-State Determination Pramod J. Lavingia and Philip F. Dickson' Colorado School of Mines, Golden, Colo. 80401 The rate of heat transfer from coils to two-phase liquid mixtures in an unbaffled agitated tank was studied. Heat-transfer determinations were performed at steady state, with a continuous flow of fluid streams. Boundary layer properties at the coil surface are required for the correlation from dimensional analysis. Bulk, volume-average properties of the two-phase mixtures gave the following correlation with the least for the impeller Reynolds number standard deviation: NNu = 4.04 NRZ.4' Np~JJ(~/p,,,)U"4, range 3.7 x i o 4 5 N~~ 5 1.1 x lo6.
A g i t a t e d kettles, equipped with cooling or heating systems, are in common use in industry today. Heat-transfer rates have been studied previously in agitated vessels equipped with a jacket or coils. Most of these investigations have been carried out with single-phase liquids. Little work has been done for the rate of heat transfer t o twophase liquid mixtures in agitated vessels. The purpose of this study was t o determine and correlate the rate of heat transfer for two-phase liquid mixtures in an agitated vessel. The study was carried out under steady-state conditions with a vessel equipped with coils. Two-phase systems used for this investigation were 50% water-50'; toluene, 50% ~ a t e r - 5 0kerosine, ~~ and 75% water-255 kerosine. Chapman and Holland (1965) and Chilton et al. (1944) have studied the rate of heat transfer to single-phase agitated liquids. The results were expressed, generally:
NsU = CNheNNbp,Visd
Cummings and West (1950) measured heat-transfer rates for water-mineral oil mixtures in an agitated vessel equipped with coils under batch conditions. Because of the limited amount of data, correlation for the rate of heat transfer was not obtained. Bodman and Cortez (1967) measured heat-transfer rates for water-toluene and water-lube oil mixtures under transient conditions in an agitated vessel equipped with a jacket. Heat-transfer rates were correlated in a manner similar to that used for single-phase results (Equation 1). Over the Reynolds number range studied (6.2 x 10' to 5.21 x lo'), their results indicated that for estimating heat-transfer rates for two-phase liquid mixtures in jacketed kettles, correlations developed specifically for single-phase liquids may be used satisfactorily if the bulk, volume-average properties of the two-phase liquid mixtures are used.
(1)
There is a good agreement as to the value of the exponent on the viscosity ratio (Vis), which has been generally accepted as 0.14. The values of the other exponents and the constant, C , vary significantly with the shape and size of the impellers and the geometry of the vessel. The rate of heat transfer is also affected by the baffles in the system (Brooks and Su, 1959), and depends on whether coils or a jacket are used as a cooling or heating media for the system (Oldshue and Gretton, 1954).
Theory
T o determine the rate of heat transfer to a liquid being heated in an agitated tank equipped with coils, an energy balance is written relating to the contents of the vessel. For steady-state conditions, the energy balance may be expressed as follows:
in '
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Tco- Tr
THERMOCOUPLES
i
~
Jf#--7rT, (TOLUENE (KEROSINE)
n
was placed at the entrance and another at the exit of the coils to record the inlet and the outlet temperatures of the water. A thermocouple was placed a t the outlet of the vessel to record the exit temperature of the twophase liquid mixture. Another thermocouple was placed in the vessel about two-thirds of the way down to measure the temperature of the agitated two-phase liquid mixture and to compare with the temperature of the mixture leaving the vessel. These temperatures were recorded continuously on a Hewlett-Packard Moseley Dual Channel strip chart recorder (Model 7100B) until steady state was reached. The water entering the coils was preheated to the desired temperature when it passed through a constanttemperature bath. Another constant-temperature bath was used to lower the temperature of the two-phase liquid mixture to room temperature when it left the vessel and before it reentered the feed tank. One rotameter was used to regulate the flow rate of water entering the coils, and the other two rotameters were used to regulate the flow rate of water and toluene (and water and kerosine) to the vessel. Calibration of rotameters was done by timed mass delivery of fluid. An electronic Hewlett-Packard Frequency Meter (Model 500C) was connected to the shaft of the impeller. Speed of the impeller was continuously read on the frequency meter and could be read accurately to f10 rpm.
k-JUU&j"-l CONSTPNT TEMP B1TH
WLiTER TANK
HEA
EXCHANGE VESSEL
FEED T&N"
Figure 1. Flow diagram for the two-phase heat-transfer system
All the terms, except U, in Equation 2 are determined experimentally; hence, U can be calculated. The total resistance to heat transfer for cylindrical tubes may be expressed as the sum of individual resistances (Bird et al., 1960):
1 In (R,lRJ u=-[---+-R, R,h, K, 1
~
1
j'
R,h,
(3)
The equation for heat transfer at the inner wall of straight tubes for laminar flow is given by the modified Sieder-Tate equation (Bird et al., 1960):
h,D,/K = 1.86 (ReNp,D,/L)' ' (
~ / p ~ ) ~ ' (4) '
For flow inside helical coils with a Reynolds number less than 10,000, substitution of the term (DC/DL)"for ( L / D , ) in Equation 4 for straight tubes gives the appropriate inside heat-transfer coefficient, h,, for coils (Perry, 1963). Having determined U and h, from Equations 2 and 4, respectively, and knowing the dimensions of the coils and the thermal conductivity, one can use Equation 3 to evaluate the outside heat-transfer coefficient, h,. The calculated value of h, is then used to correlate the data in dimensionless form, as given by Equation 1. Apparatus
The flow diagram for the steady-state continuous heattransfer experiment is shown in Figure 1. Steady-state heat-transfer experiments were conducted in a two-gallon, dished-bottom steel vessel. The unbaffled kettle, as shown in Figure 2, was equipped with coils as well as a jacket. Actual capacity of the vessel was 1.5 gal. The wall of the vessel was made of >= specific heat, Btu/lb F c,,, = specific heat of the liquid mixture, Btu/lb F C P k C, P Z = specific heat of two phases, X and 2,Btu/ lb OF D, = diameter of the turn of the coil, in. D , = inside diameter of coil, in. D , = diameter of the impeller, in.
D r = diameter of the vessel, in. h, = inside (water side) heat-transfer coefficient, B t u / f t 2hr OF h,, = outside heat-transfer coefficient, Btuift’ hr xOF K = thermal conductivity of the fluid, Btujft’ hr x F/ft K , = thermal conductivity of copper, Btu/ft2 hr x°F/ft K , = thermal conductivity of the liquid mixture, B t u / f t Lhr “ F / f t K , , K , = thermal conductivity of two phases, X and 2,Btu/ft’ hr O F / f t L = total length of the coils, in. m = mass flow rate of water into coils, lb/hr N = impeller speed, rpm NNu = Nusselt number, h,DT/K Np, = Prandtl number, C,p/K NRe = Reynolds number for the tank, p D f N / p Re = Reynolds number for flow in coil, p D , V / p R, = inside radius of the coil, in. R , = outside radius of the coil. in. T,, = temperature of water entering coils, F T,, = temperature of water leaving coils, F T r = temperature of the bulk in the tank, ’F TlV = temperature of the wall of the coil, ’F U = overall heat-transfer coefficient, Btu/ft2 hr x OF V = velocity of water in the coil, ft/ hr X , 2 = volume fraction of two-phase mixture p = viscosity of the fluid, CP p m = viscosity of the liquid mixture, CP F,, = viscosity of fluid a t wall temperature, CP fir, p z = viscosities of two phases, X and 2 , CP p = density, lb/ft3 Literature Cited
Bird, R. B., Stewart, W. E., Lightfoot, E. N., “Transport Phenomena,” Wiley: New York, 1960. Bodman, S. W., Cortez, D. H., I n d . Eng. Chem. Process Des. Develop., 6, 127-33 (1967). Brooks, G., Su, G. J., Chem. Eng. Progr., 55, 54-7 (1959). Chapman, F. S.,Holland, F . A., Chem. Eng., 72, 1538 (1965). Chilton, T. H., Drew, T. B., Jebens, R. H., Ind. Eng. Chem., 36, 510-16 (1944). Cummings, G. H., West, A. S., ibid., 42, 2303-13 (1950). McCabe: W . L., Smith, J. C., “Unit Operations of Chemical Engineering,” p p 444-5, McGraw-Hill. New York, 1956. Oldshue, J. Y., Gretton, A. T., Chem. Eng. Progr., 50, 615-21 (1954). Olney, R. B., Carlson, G. J., ibid., 43, 473-80 (1947). Perry, J. H., “Chemical Engineers Handbook,” 4th ed., pp 10-24, McGraw-Hill, New York, 1963. Pratt, N. H., Trans. Inst. Chem. Eng.. 25, 163-80 (1947).
RECEIVED for review April 29, 1970 ACCEPTED November 6, 1970 The financial support of the Colorado School of Mines Foundation, Inc., is gratefully acknowledged.
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971
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