Dayton, R. LV., Fawcett, S.L., Grirnble, R. E., Sealander, C. E., Battelle Memorial Institute, Columbus, Ohio, Rept. BMI-747 (1952). De Acetis, J., Thodos, G., Ind. Eng. Chem. 52, 1003 (1960). Eichorn, J., \t’hite, R. R., Chem. Eng. Progr. Symp. Ser. 48, No. 4, 11 11952). Gamson, B. L\’,, Thod’as, G., Hougen, 0. A., Trans. A m . Inst. Chern. Engrs. 39, 1 (1943). Glaser, M. B., Thodos, G., A.Z.Ch.E. J . 4, 63 (1958). Hamming, K. \V., “iYi1merical Methods for Scientists and Engineers,” p. 362, McGraw-Hill, New York, 1962. Kim, D. S., Ph.D. dissertation, Ohio State University, 1965. Krupiczka, R., Intern. C h m . Eng. 7 , 122 (1967). Kunii, D., Smith, J. M., A.Z.Ch.E. J . 6, 71 (1960). Kunii, D., Smith, J. M., A.I.Ch.E. J . 7 , 29 (1961). Lapidus, L., “Digital Computation for Chemical Engineers,” McGraw-Hill, New York, 1962. Littman, H., Barile, R. G., Chem. Eng. Progr. Symp. Ser. 62, No. 67, 10 (1966). Littman, H., Stone, A. I>., Chem. Eng. Progr. Symp. Ser. 62, No. 62, 47 (1966). McConnachie, J. T. L., Thodos, G., A.Z.CI1.E. J . 9, 60 (1963). McHenry, K. it7.,Jr., \%.ilhelIn,R. H., A.Z.Ch.E. J . 3, 83 (1957).
Masamune. S., Smith, J. M., IND.ENG.CHEM.FUKDAMENTALS 2, 136 (1963). Meek, k. M’. G., International Heat Transfer Congress, ASME, New York, p. 770, 1961. Preston, F. IV., Ph.D. dissertation, The Pennsylvania State University. 1957. Pulsifer,’ A . H., Ph.D. dissertation, Syracuse University, 1965. Schuniann, T. E. IV., J . Frankltn Inst. 208, 405 (1929). Shearer, C. J., Nat. Eng. Lab., East Kilbridge, Glasgow, N.E.L. Rept. 5 5 (1962). \Vilhelm, R. H., Pure Appl. Chem. 5 , 403 (1962). \Vilke, C. R., Hougen, 0. A., Trans. A m . Inst. Chem. En,gr~.41, 445 (1945). It’ilson, E. J., Geankoplis, C. J., IND.ENG.CHEM.FUNDAMENTALS 5 , 9 (1966). RECEIVED for review October 16, 1967 ACCEPTED February 26, 1968 \York supported by the U. S. Atomic Energy Commission under Contract AT (30-1)-3639 and the Xational Science Foundation under grants GP-2994 and GK-174. Experimental work performed at Syracuse Cniversity.
PARTICLE-TO-FLUID NUCLEATE BOILING HEAT TRANSFER IN A WATER-FLUIDIZED SYSTEM F. M . YOUNG
Lamar State College, Beaumont, Tex. J . P. H O L M A N Southern Methodist Unicersity, Dallas, T e x .
An experimental study was conducted of nucleate boiling heat transfer from stainless steel balls fluidized in a water system. The steel balls were heated b y a radiofrequency induction heater. Regimes of particle, liquid, and vapor motion were defined and observed. The heat flux and saturation temperature difference could b e correlated using a form of Rohsenow’s nucleate boiling equation. Cavitation observed as a result of flow through the fluidized medium is postulated to b e the source of initial nucleation sites. vigorous nucleate boiling was observed at saturation temperature excesses as low as 0.7” F.
of attention has been given to the study of heat transfer in fluidixecl media. These studies may be grouped into three classifications, concerned primarily with the energy exchange (1) between the particles and the fluid, (2) between the particles and the walls of the containar, and (3) between the container walls and the fluid. These studies have been conducted with both liquid and gas as the fluid. There is, however, a notable absence of investigations Concerned with two-phase fluids, especially with phase change resulting from energy exchange with the fluid. This experimental study is concerned with the determination of the nucleate boiling heat transfer characteristics between a particle phase of stainless steel spheres and saturated Jvater.
A
GREAT DEAL
Previous Work Although there is a notable absence of work in the specific area of this investi?ation, the related areas of convection heat transfer in fluidized media and boiling heat transfer were examined for their applicability to the present investigation. T h e most pertinent study of convection heat transfer in fluidized media is that of Holman et al. (1955). This study is of particular interest because a slightly modified version of the experimental equipment was used in the present investigation.
As a result,
I n the previous investigation heat transfer coefficients were measured for the exchange of energy between an inductionheated medium of spherical particles and water. The heat transfer coefficient was defined as
where T b is the bed-average particle temperature and T , is the bed-average water temperature. The average particle temperature was obtained from a transient analysis; the average water temperature was determined from a n assumed linear water temperature distribution and was simply the average of the temperatures into and out of the test section. Heat transfer data were correlated by the equation
where pci is a reference viscosity (taken a t 80” F.) and F, is a velocity correction factor defined by
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T h e average deviation observed for the correlation of Equation 2 was 20.6%. Bowers and Reintjes (1961) and Frantz (1961) present surveys of investigations concerned primarily with energy exchange between the suspended particles and the fluid. An extensive listing of these investigations is given by Young (1 967). Of the data presented in these investigations, none is directly applicable to the present investigation. However, previous investigations of boiling heat transfer might be expected to identify a t least some of the parameters significant in the present study. As the temperature of a surface in contact with a liquid is raised above the saturation temperature of that liquid, there is some temperature at which phase change is observed. This initial phase change takes the form of small bubbles a t a few preferential sites. As the temperature is increased, more of these preferential sites are activated and more vapor is formed. As long as the phase change is in the form of discrete bubbles, boiling is called nucleate. Since this investigation has been restricted to nucleate boiling, other types of boiling are not discussed. An interesting aspect of nucleate boiling is the disagreement among investigators as to the heat transfer mechanism that results in such large values of heat transfer coefficient. There are two prevalent views, referred to as induced convection and phase change. The induced convection theory attributes the large heat transfer coefficients to microconvective currents created by vapor motion a t the heated surface. For example, consider a single vapor bubble attached to a surface. The bubble may either detach from the surface and rise, or collapse. I n either case liquid must rush in to fill the volume previously occupied by the vapor. The motion of the liquid constitutes the microconvective currents. The effect of these microconvective currents is that superheated liquid very near the surface is stirred with cooler liquid from a location farther from the surface. Rohsenow (1952), one of the advocates of the above theory, assumed that the heat transfer data could be correlated with the general form of the forced convection equation with appropriate interpretation of the parameters to reflect the influence of bubble motion. H e proposed the general form,
Fritz (1935) and Wark (1933) give the bubble diameter upon breaking from the surface as
Substituting Equations 6 to 10 into Equation 5, Rohsenow obtained
where
cz
,=
C[C, p d 2 y
Rohsenow found the value of m to be 0.33 for pool boiling; n varied between 0.8 and 2.0, depending upon the condition of the surface. The constant Cz is a function of the surface and liquid materials; its value has been observed to vary from 0.0025 for isopropyl alcohol-copper (Piret and Isbin, 13.54) to 0.01 5 for n-pentane-chromium (Chichelli and Bonilla, 1945). Rohsenow (1952) showed that a correlation of the form of Equation 11 fitted selected data for pool boiling of water with a deviation of =t20%,. An investigation by Vachon et al. (1967) suggests that the value of m is not constant but is a function of the surface and liquid materials, as is Cs. Forster and Zuber (19.54), although advocating the same basic theory as Rohsenow, suggested that the important influence of bubble motion on heat transfer was while the bubble was growing and still attached to the surface. They, therefore, developed the following expressions for characteristic bubble radius, Reynolds number, and Nusselt number based on bubble growth.
R = (
HfQP"
The correlation equation given by Forster and Zuber is
NU,, = 0.0015 ReBu0.62Prf113 Rohsenow rearranged the above equation for ease of data correlation and empirically arrived a t the form shown below. Re,,Pr
=
C RegunPrfm
NUBlr
(5)
The dimensionless numbers were defined as follows,
=
CPfPf f
k (7)
Re,,
(15)
The theory of phase change attributes the increase in heat transfer coefficient to latent heat transport of bubbles. I n contradiction to this theory, Tong (196.5) performed an order of magnitude analysis on data taken by Gunther (1951) and showed that the latent heat transport accounted for only 2y0 of the total heat flux. However, the total heat flux would be expected to be proportional to the latent heat flux, since the latent heat flux is a measure of the agitation of the liquid layer very close to the surface. Forster and Greif (1959), in advocating the phase-change theory, developed the following correlation for pool boiling data.
PcvdBu
= __
Pf
where V is the flow velocity parallel to the surface. The boiling number is defined as the ratio of vapor velocity away from the heated surface to flow velocity parallel to the surface and is given by Tong (1965) as
9" Bo = -
HfOPVV 562
l&EC FUNDAMENTALS
For the present study the groupings of variables in Equations 11, 15, and 16 are of particular interest. However, since Equation 16 was developed from the phase change theory, it might not be as significant in a study of boiling within a fluidized medium as Equations 11 and 15, because in a fluidized medium
STEAM
PACKING WATER
I ,
? DEIONIZER PUMP
PYREX TUBE NYLON SCREEN
SECTION
INJECTION SECTION
Figure 1.
y PRESSURE TAP
Experimental setup
-
-1"
COPPER TUBE
BALL SEPARATOR
-TEST
SECTION STAND
-2"
COPPER TUBE
COUPLINQ
-INJECTION -WITH
SECTION
PYREX TEST SECTION HEATINQ COIL
r - 1 1 2 ' COPPER TUBE
Figure 3.
Temperature-measurement system
3 JUNCTION THERMOPILE
WIRES - - - -- THERMOCOUPLE HYDRAULIC LINES
--
a-
_ ---
1 I b-63-
DETAIL OF INJECTION SECTION
E-
HEISE OAUQL
----
'-.--COUPLING LOWER FLANQE FLOW STRAIQHTENER
,--
Figure 2.
COUPLINQ
Test section
Figure 4.
the liquid near the surface of a particle is also agitated by the wakes of other particles and by collisions of particles. Since the phase change theory describes agitation only indirectly, modification to include wake and collision effects might be difficult. Experimental Program
An apparatus \vas constructed to measure the heat flux and saturation temperatlure difference for a fluidized system of spherical stainless steel particles \vhich were inductionheated to cause nucleate boiling. A 30-kw. radiofrequency induction heater \vas used to heat the balls. The experimental apparatus consisted of a borosilicate glass test section encircled by a n induction heating coil and a flow system for circulating ivater. Drawings of the flow system, the test section installation, and test section assembly are sho\vn in Figures 1, 2, and 3. A sketch of the pressure and temperature instrumentation is sho\vn in Figure 4. T h e pressure a t the outlet of the test section was measured with a Heise gauge, while the pressure drop across the test section was measured Ivith a manom'eter. T h e temperatures of the flow into and out of the test section along tvith the temperatures of the flow into the injection manifold Isere measured with iron-
Instrumentation
constantan thermocouples and recorded with a Honeywell multipoint recorder. The temperature difference from the inlet of the test section to the outlet of the mixing section was measured with a three-junction, copper-constantan thermopile and recorded with a Honeywell strip recorder. The purpose of the injected flow should be explained. Consider the system enclosed by the outer dashed line shown in Figure 5 . Suppose that vapor is being generated within the test section and that there is no injected flow. A steady-state energy balance can be written \vith a liquid inlet flow as
+
+
~'IHI, Q = ~ 3 ( H 3 f (17) A problem immediately arises in determining X:the outlet quality. I n this investigation suficient cold ivater to condense all ofthe vapor and to subcool the flow from the mixing section about IOo F. is injected just downstream from the test section. A steady-state energy balance results in
+
+
X'IHI, W S H ~ , Q = ~3H3f
(1 8 )
where, from continuity, 26'3
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=
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f
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LIQUID OUT
r I- -- -- - - - -1 0 I I I I
-1-0
!-I--------
:i Ptt
I
I
LIQUID4- VAPOR
I
'I
I I
I I I
I
I
I
-17
FLUIDIZED MEDIUM
000 0
INLET FLOW
.ii,
(LIOUIDI
Figure 5. Control volume for analysis of test section
T h e average particle temperature is determined in this investigation by recording the output of a n iron-constantan thermocouple attached directly to a ball. The junction was made by welding one lead of the thermocouple to each side of a ball. The ball was held within the fluidized medium by a cord and pulley arrangement as shown in Figure 3. Objection may be raised to this method of measurement on the basis that the instrumented ball is constrained by the cord and thus does not represent the particle phase, the thermocouple wire is subjected to internal energy generation which may further make the temperature of the ball unrepresentative, and a thermocouple may be unreliable when placed in an oscillating magnetic field. These objections can be answered to some extent by tests that were carried out in an attempt to determine the accuracy of this method. I n several tests balls coated with Tempilaq, a temperature-indicating coating, were introduced into the fluidized bed, the water flow was rapidly started, and then the output of the induction heater was increased until the Tempilaq coating gave its temperature indication by changing appearance. The output of the ball thermocouple indicated temperatures within 0.5' F. of the calibrated ratings of the Tempilaq. The Tempilaq measurement technique was calibrated in a separate experiment, so that it was believed to be reliable. Since the ball thermocouple agreed with the Tempilaq readings, this technique was adopted for the primary ball temperature measurements in the heat transfer experiments. Another method considered for measuring ball temperature was the measurement of the change of microhardness of special alloys available commercially as Templugs. This method was discarded because at the temperatures of interest the time required for a significant change in the microhardness was prohibitively long. A transient technique similar to that used by Holman et al. (1 965), for determining the average particle temperature, was unsuccessful because of the uncertainty in the vapor distribution within the test section. An order-of-magnitude analysis indicated that the error in the ball temperature might be about 31' F. Doubt of this magnitude caused us to undertake an experimental program of over 200 runs. The data indicated 564
l&EC FUNDAMENTALS
Table 1. Summary of Range of Experimental Variables Porosity, E 0.611 to 0.862 0.00521 to 0.0156 ft. Particle diameter, db 2229 to 6295 lb.m/hr. Primary mass flow rate, W I Saturation temperature, Teat 216.6' to 241.9" F. 4439 to 88,619 B.t.u./ Heat flux, Q'' (hr.)(sq.ft.) Saturation temperature difference, eb 1 . 5 ' to 21 .6" F.
an actual uncertainty in ball temperature of about 90' F . ; as a result, the transient technique was not employed in the final measurements. The measurement of particle temperature with a thermocouple welded to a ball, while not representing an ideal technique, seems to be a reasonable approach. Traverses with the instrumented ball in both the axial and radial direction were made within the fluidized medium; the temperature was observed to be constant with respect to position, but ball temperature varied with time. The amplitude of these fluctuations was approximately 10 to 307, of the saturation temperature difference. The time average of the temperatures, pb, measured with the instrumented ball was used. The primary flow rate was measured with a turbine-type flowmeter. The temperature of the primary flow into the test section was controlled by adjusting the steam flow to the preheater. Flow out of the mixing section was passed through an aftercooler before being returned to the supply tank. Injection flow was taken a t the temperature of the supply tank. The flow rate of the injection flow was measured with a flowrator. Calibrations for the flowmeters are presented by Young (1 967). Results of Experiments
The heat losses from the test section due to free convection and radiation were estimated to be 325 B.t.u. per hour in comparison to the minimum magnitude of heat transfer to the water of 10,000 B.t.u. per hour; the loss was therefore neglected. The heat transfer was then calculated from Equation 18 and the saturation temperature difference defined as
was taken from the appropriate measurements. Table I gives the range of parameters significant to this investigation. A complete tabulation of experimental data is given by Young (1967). Correlation of the test data was attempted, utilizing the correlation variables determined by Rohsenow as in Equation 11, by Forster and Zuber as in Equation 15, and by Forster and Greif as in Equation 16. No trend could be established utilizing the parameters of either Forster and Zuber or Forster and Greif. However, Figure 6 shows that a trend could be established utilizing exactly the same variable groupings and plotting form as Rohsenow (1952). Figure 6 also shows Rohsenow's correlation for nucleate pool boiling of water on nickel. Although the test data are of the same order of magnitude as Rohsenow's correlation, it is significant that the slopes of the correlation and the trends in the data differ. A phenomenon observed in the experimental program was thought at first to be inconsequential. When the temperature of the primary flow into the test section was raised within 1' to 2' F. of the saturation temperature, with no energy input from the induction heater, small bubbles of vapor were observed in the flow out of the fluidized medium. A careful check of the temperature and pressure instruments indicated
FROM ROHSENOW FOR WATER-NICKEL
Id2
I
t
l
l
L
IO !
IO-^
10'~
C f AT,,t H f g Pr'"
Figure 6. Correlation of experimental data using groupings of Rohsenow ( 1 952)
served for a saturation temperature excess as small as 0.7' to 1.0' F. This early onset of nucleate boiling is then attributed to the formation of nucleation sites by cavitation. O n these grounds it would be logical to include some factor that is indicative of the inertia of the liquid as it moves through the particles. Furthermore, the forces associated with the formation of vapor might be expected to be important. With the objective of including the two considerations mentioned above in the data correlation, an Euler number is defined as
SYMBOL BALL Ole
0
0.00521 0.00781'
0
0.0156'
0
where V is the superficial velocity and F, is the dimensionless velocity correction factor employed by Holman et al. (1965). I n accordance with the results of Figure 6, the ratio _ _TSat _
C+
~H f ,~
0.006
(L d, '?L PfHfLl
0.I
I
I
I
I I l l l l
Id2
Figure 7. Correlation' of Euler number modification
experimental data
Pf
-
-
_ )o'66
_
(22)
Pr1.7
Po)&
was plotted against the Euler number as defined by Equation 21. This ratio was found to vary with the Euler number to the 0.84 power. A plot of the data utilizing the observed variation with Euler number is shown in Figure 7. A curve was fitted to the data, resulting in the correlation:
utilizing
that this was not an instrumentation error and therefore must have been due to cavitation of the fluid within the particle medium. Furthermore, vigorous nucleate boiling was ob-
(23) Vachon et al. (1967) have suggested that the exponent of the term in parentheses above for pool boiling of water on polished stainless steel should be 0.52 and the coefficient should be VOL. 7
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0.0133 when used in the Rohsenow correlation-Le., m and C2 in Equation 11. The suggested exponent is in good agreement with the value determined in this investigation. The difference in coefficients can be a t least partially attributed to the early onset of nucleate boiling due to cavitation. An uncertainty analysis applied to the primary measurements of this investigation indicates that the total uncertainty in the grouping c 4 Tmt H,, Pr1.7
is approximately 0.00100. Inspecting Figure 7, it is seen that the vast majority of the data are consistent with the correlation given, within the limits of the uncertainty. The average deviation of the data is defined by
where A,{ is the observed value of the abscissa in Figure 7, A,, is the calculated value of the abscissa from Equation 23 a t the same ordinate, and n is the number of data points. The average deviation of the test data was calculated to be 23%. The heat transfer coefficient resulting from Equation 23 is independent of temperature excess and is given in nondimensional form as
T h e Clausius-Clapeyron equation in finite difference form for u, has been taken from Tong (1965)
u,
>>
1 sat
and substituted in the definition of Euler number to obtain Equation 24. Classification of Observed Patterns of Fluid and Particle Motion in Presence of Boiling
The descriptions of the patterns of fluid motion observed in this investigation have been classified by taking combinations of the descriptions of the patterns of the particle medium (Orr, 1966) and of the phases of the fluid (Tong, 1965). The motions observed in this investigation may be called: bubbly fluidized boiling, bubbly slug boiling, and slug boiling. Bubbly fluidized boiling is the formation of vapor bubbles within a uniformly distributed, fluidized, particle phase. The vapor bubbles are distributed uniformly across any cross section perpendicular to the flow. Bubbly slug boiling is the formation of vapor bubbles within a slugging particle phase. The vapor bubbles are also distributed uniformly across any cross section perpendicular to the flow. Slug boiling is the rapid formation of vapor within a slugging particle phase such that the vapor is also slugging. I t might be thought that a classification of a slugging fluid phase within a fluidized particle phase has been inadvertently omitted. However, within the range of the experimental program, slugging in the fluid phase always induced slugging in the particle phase. This observation is not surprising when the drag forces on the particles are considered. The liquid-vapor interface associated with the slugging fluid phase is a surface of discontinuity In the drag forces acting on the particles. This 566
l&EC FUNDAMENTALS
discontinuity in the drag forces is due to the density change associated with the slugging liquid-vapor interface. The presence of a particle phase destroys forces tending to create an annular fluid-vapor flow situation. For some cases of high exit quality, transition to annular flow occurs very rapidly as the fluid leaves the particle phase. Conclusions
Cavitation is believed to be the cause of the early onset of nucleate boiling in a fluidized system. The heat flux is correlated with the saturation temperature difference using the dimensionless groups that were developed by Rohsenow (1952) and the Euler number. The equation suggested by this investigation is
The constant 0.00125 may be anticipated to be afunction of the particular combination of liquid and solid surface. Test data are insufficient to indicate whether this constant is related to the analogous constant in Rohsenow's original correlation, or to determine whether the exponent of the Euler number is constant. The heat transfer coefficient is independent of the temperature excess and is given by
Acknowledgment
The authors gratefully acknowledge the support of the National Science Foundation for sponsorship of this work under grant KO.GK-1106. Nomenclature
A Bo c C d
=
value of abscissa
= boiling number
specific heat, B.t.u. (Ib.,,J(OF.) constant diameter, ft. Euler number, (pi. - psat)/pfVzaF> correction factor gravitational constant, ft. lb.m(hr.z)(lb.f) gravitational acceleration, ft./hr.2 heat transfer coefficient, B.t.u. (hr.)(ft.)(' F.) enthalpy B.t.u./lb., = thermal conductivity, B.t.u. (ft.)(" F.) = pressure, Ib.f/sq. ft. = heat flux, B.t.u. (hr.)(sq. ft.) = mass rate of flow, Ib.,/hr. = Nusselt number, hd/k = Prandtl number, pc/k = heat transfer, B.t.u./hr. = bubble radius, ft. = bubble radius velocity, ft./hr. = Reynolds number, p d V / p = Temperature, O F. velocity, ft./hr. quality
= = = ELI = F, = ge = g = h = H =
k
P
q" w
Nu Pr
Q
R
&
Re
T
v = x =
GREEKSYMBOLS a
fl fi p u
0
= thermal diffusivity, sq. ft./hr. = contact angle, radians
= dynamic viscosity, lb.f(hr.)(sq. ft.) = density, Ib.,/cu. ft. = surface tension, Ib.f/ft.
= temperature excess (Tb - T,,,),
F.
SUBSCRIPTS b = balls Bu = vapor bubble d = related to bubble diameter c = related to porosity f = liquid fg = liquid to vapor z = referenced a t 80” F. sat = saturation s = superficial velocity t = test section u = vapor w = wall 1 = inlet to test section 2 = outlet of test section 3 = outlet of mixing section 5 = injection water inlet
Forster, K., Grcif, R., J . Heat Transfer 81, 43-53 (1959). Forster, H. K., Zuber, N., J . Appl. Phys. 25, 474-88 (1954). Frantz. J. F.. Chem. Enp. ProPr. 57. f7). 35 f 1961). Fritz, W., PI&. Z.36, ?79-s”4 (1925).” Gunther, F. C., Trans. A S M E 73, 115-23 (1951). Holman, J. P., Moore, T. W.. Wone. V. M.. IND.ENG.CHEM. FUNDAMENTALS 4,21(1965). . -’ Orr, Clyde, Jr., “Fixed Moving, Spouted, and Fluidized Beds,” in “Particulate Technology,” Macmillan, New York, 1966. Piret, E. L., Isbin, H. S.,Chem. Eng. Progr. 50, 305-11 (1954). Rohsenow, W. M., Trans. A S M E 74, 969-76 (1952). Tong, L. S., “Hydrodynamics of Two-Phase Flow,” in “Boiling Heat Transfer and Two-Phase Flow,” pp. 48-55, Wiley, New York, 1965. Vachon, R. I., Nix, G. H., Tanger, G. E., “Evaluating of Constants for Rohsenow Pool-Boiling Correlation,” ASME Paper 67-HT-33, 9th National ASME-AIChE Conference, Seattle, Wash., Aug. 6-9, 1967; to be published in J . Heat Transfer. Wark, J. W., J . Phys. Chem. 37, 623-44 (1933). Young, F. M., “Particle to Fluid Nucleate Boiling Heat Transfer in a Water Fluidized Medium,” Ph.D. dissertation, Southern Methodist University, August 1967 ; University Microfilms, Inc. \
literature Cited
Bowers, T. G., Reintjes, H., Chem. En,?. Progr. Symp. Ser. 57, NO. 32, 69-74 (1961). Chichelli, M. ‘T.,Bonilra, C. F., Trans. A.I.Ch.E. 41, 755-87 (1945).
,
RECEIVED for review September 25, 1967 ACCEPTEDMarch 15, 1968
HEAT TRANSFER FROM A SPRAY-COOLED ISOTHERMAL CYLINDER J. W. HODGSON Mechanical and Aerospace Engineering Department, The University of Tennessee, Knoxville, Tetin.
37916
J. EDWARD SUNDERLAND Mechanical and Aerospace Engineering Department, North Caroltna State University, Raleigh, N . C. 27607 The local heat transfer coefficients around the periphery of an isothermal cylinder exposed to a crossflow consisting of a water-in-air spray have been investigated analytically. The analysis is valid over the leading 160” of the cylinder and considers integral forms of the continuity, momentum, and energy equations as applied to the liquid film which forms on the cylinder. All flow is considered incompressible and evaporation from the film is neglected. The theoretical results agree with experimental data.
HE
rate of heat tramfer between solid surfaces and a gas
T stream can be substantially increased by introducing water
droplets into the stream. Increases as large as a factor of 30 have been reported by Hodgson et al. (1968). The increase in the rate of heat transfer can be explained qualitatively by visualizing the boundary layer over the solid surface. The previously existing gas boundary layer will be replaced by a liquid boundary layer which will have a higher heat transfer coefficient. A survey of the literature reveals that although much effort has been expended in the area of internal two-component flow, very little has been done 10 investigate external two-component flow. Acrivos et al. (1964) made preliminary analytical studies which clearly showed the need of a more detailed analysis. Tifford (1964) presented an analytical study of spray-cooled surfaces which applied to flat plates. H e assumed that the velocity and temperature a t the free surface of the film are constants; it is the opinion of the writers that this assumption considerably limits the accuracy of the final solution. An analytical study of the spray-cooled cylinder was reported by Goldstein et al. (1 967). I n this work the differential forms of the continuity, momentum, and energy equations are solved numerically.
Smith (1966) investigated heat transfer from a spray-cooled cylinder. H e considered linear velocity and temperature profiles in the liquid film and included terms to account for the effects of evaporation. Despite an error in his final equation, good agreement is indicated between the theoretical and experimental heat transfer coefficients. An analytical and experimental investigation of the heat transfer from a spray-cooled wedge was conducted by Thomas (1967). Good agreement was found between the theoretical and experimental results. This investigation is concerned with finding the heat transfer which occurs when a heated, horizontal, circular cylinder is exposed to a crossflow consisting of an air stream containing water droplets. The theoretical analysis assumes the existence of a liquid film on the leading half of the cylinder and applies integral forms of the continuity, momentum, and energy equations to this film. When the flow is incompressible, evaporation is negligible, and all properties (except pressure and temperature) are constants, the continuity and momentum equations are not coupled to the energy equation. I t is therefore possible to obtain approximate closed form solutions for the film thickness and the velocity distribution directly from the momentum equation without coupling with the energy equation. Substituting VOL. 7
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