-
Group
Table VII. a
Halogen-Containing Groups b X lo2 G X lo4 d X lo0
-F
1.4382
0.3452
-0.0106
-C1
3 0660
0.2122
-0.0128
0.000276
-Br
2.7605
0 4731
-0,0455
0.001420
-1
3.2651
0.4901
-0
0539
-0.000034
0.001782
estimated deviation is less than 27,; and for hydrocarbons the average error is about l,5YG, while for nonhydrocarbons it is about 47,. Sample calculations for isopropyl alcohol and bromobenzene are shown in Table IX. Nomenclature
constants of heat capacity equation ideal heat capacity, cal. l o K./gram-mole contribution to heat capacity from stretching vibrations of bond i contribution to heat capacity from bending vibr tions of bond i number of single bonds about which internal rotation of groups can take place number of atoms in molecule any thermodynamic function total number of bonds in molecule gas constant, cal. / O K . /gram-mole temperature, O K. reduced temperature symmetry number literature Cited (1) Anderson, J. W., Beyer, G. H., Watson, K. M., Natl. Petrol. News 36, R 476 (1944). (2) Banerjee, S. C., Doraiswamy, L. K., Brit. Chem. Eng. 9, 311 ( 19 64) . (3) Barrow, G. M., Pitzer, K. S., Znd. Eng. Chem. 41, 2737 (1949). (4) Bigg, D. C., Banerjee, S. C., Doraiswamy, L. K., Brit. Chem. Eng., to be published. (5) Binder, J. L., Chem. Ph.ys. 17, 499 (1949). (6) Chermin, H. A. G., Petrol. Rej‘iner 38, No. 12, 117 (1959). (7) Zbtd., 40, No. 3, 181 (1961).
(8) Zbid., No. 4, 127 (1961). 19) Zbid.. No. 5. 234 11961). (lb) Zbzd., No. 6, 179‘(196?). (11) Zbid., No. 9, 261 (1961). (12) Dobratz, C. J., Znd. Eng. Chem. 33, 759 (1941). (131 Franklin. J. L.. Zbid.. 41. 1070 11949). (14) Fugassi, P., Ru’dy, C‘. E.; Jr., Zb;’d.,30, 1020 (1938). (15) Gelles. E., Pitzer, K. S., J . Am. Chem. SOC.75, 5259 (1953). (16) Godnev, I. N., Sverdlin, A. S.,Savojina, M. S., Zh. Fiz. Khim. 24, 807 (1950). (17) Green, J. H. S.,J . Chem. SOC. 1961, 2236. (18) Guthrie, G . B., Jr., Scott, D. W., Habbard, W. N., Katz, C., MaCullough, J. P., Gross, M. E., Williamson, K. D., Waddington, G., J . Am. Chem. SOL.74, 4662 (1952). (19) Hatton, W. E., Heldenbrand, D. L., Sinke, G. L., Stull, D. R., J . Chem. Eng. Data 7, 229 (1962). (20) Janz, G. J., J . Chem. Educ. 31, 72 (1954). (21) Kline, C. H., Jr., Turkevich, J., J . Chem. Phys. 12, 300 (1940). (22) Kobe, K. A., Harrison, R. H., Petrol. Rejner 30, No. 8, 119 (1951). (23) Zb;d.,33, No. 8, 109 (1954). (24) Zbid., 33, No. 11, 161 (1954). (25) Zbtd., 36, No. 10. 155 (1957). (26) Kobe, K.A , , Long, E.‘G., Ibid., 28, No. 7, 145 (1949). (27) Zbid, 29, No. 5, 89 (1950). 28) Kobe, K . A., Pennington, R. E., Zbid.,29, No. 9, 135 (1950). 29) Kothari, M. S., Doraiswamy, L. K., Zbid., 43, No. 3, 133 (1964). (30) McCullough, J. P., Scott, D. W., Pennington, R. E., Hossenlopp, I. A., Waddington, G., J . Am. Chem. Sot. 76, 4791 (1954). (31) Pitzer, K. S., J . Chem. Phys. 8, 711 (1940). (32) Rossini, F. D., associates, “Selected Values of Properties of Hydrocarbons,” Natl. Bur. Standards Circ. C461 (1 947). (33) Souders, M., Matthews, C. S., Hurd, C. O., Znd. Eng. Chem. 41, 1037, 1048 (1949). 34) Stull, 0. R., Mayfield, F. D., Zbid.,35, 639 (1943). 35) Valentin, F. H. H., J . Chem. Soc. 1950, 498. (36) van Krevelen, D. W., Chermin, H. A. G., Chem. Eng. Sci. 1, 66 (1951). (37) Waddington, G., Knowlton, J. W., Oliver, G. D., Todd, S.S., Hubbard, W. N., Smith, J. C., Huffmann, H. M., J . Am. Chem. SOC.71. 797 (1949). (38) Weltner, W. Jr.: Ibid:, 77, 3941 (1955). (39) Whiffen, D. H., J . Chem. Soc. 1956, 1350. (40) Williams, R. L., J . Chem. Phys. 25, 656 (1956).
I
I
RECEIVED for review December 11, 1963 ACCEPTEDJune 18, 1964 National Chemical Laboratory Communication 627.
PARTICLE-TO-FLUID HEAT TRANSFER IN WATER-FLUIDIZED SYSTEMS J . P. HOLMAN, T. W .
MOORE, AND V . M . WONG
Southern Methodtsl University, Dallas, Tex. Stainless steel and lead spheres were fluidized in water and heated by an induction heating field. Reynolds numbers based on particle diameter and superficial velocity ranged between 2 4 0 and 14,000. The heat transfer from the spherical particles to the water was correlated with: Nu = 1.28 X 1 O-5(ReF,)2.0Pr0.67(DT/correction factor, F,, is used to account for variations in porosity. The D p ) o ~ 5 ( p f / p p ) 2 ( ~ / ~ O ) The o ~ 8 3velocity , data of the present investigation compare favorably with previous particle-to-gas and particle-to-water heat transfer studies, although a different correlation from the one above is necessary for comparison, since porosity data were not available for the previous studies.
studies of fluidization heat transfer have been reMost of these studies are concerned with heat transfer from an exposed surface to a fluidized medium and suitable empirical correlations are available for predicting the heat transfer rates under conditions similar to those reported in the various investigations. Only a few studies consider the heat transfer from the fluidized particles to the fluidizing medium. It is necessary to know these heat transfer rates in order to design a fluidized nuclear reactor as proposed ( 7 7 , 72). ANY
M ported.
T h e experimental study described in this paper was initiated to determine the heat transfer coefficients for solid spheres fluidized in water and it is anticipated that the data will be applicable to natural-uranium-water fluidized nuclear reactors. Previous Work
Heat transfer in fluidized systems has been studied for both particulate and aggregative fluidization and for both gas and liquid systems, although the data for the liquid systems are VOL. 4
NO. 1
FEBRUARY 1965
21
meager. By far the greatest amount of data in the literature concern heat transfer from a heated wall to a gaseous fluidized system. The gas-fluidized investigations have been summarized by Zenz and Othmer (78), Wen and Leva (77), Leva and Grummer ( 9 ) , Leva (8),and Frantz ( 3 ) . T h e physical mechanism of wall-to-fluid heat transfer may be explained in the following way, consistent with various discussions presented in the literature. The primary resistance to heat transfer occurs in the thin layer of fluid at the wall of the heated column, and substantially all of the radial temperature drop between wall and fluid occurs in this region. I n the central portion of the fluidized column the radial temperature distribution is very uniform. The heat transfer coefficient for fluidized systems is substantially higher than for flow systems without the presence of fluidized solids, and the higher values may be explained by the fact that the solids scrub the wall and disturb the laminar sublayer to such an extent that its thickness is decreased, or a t least there is an injection of turbulence into the layer which decreases the thermal resistance and brings about a higher heat transfer rate. T h e specific heat of the solids has been found to affect the heat transfer rate from the heated wall to the fluidized system. Particles with high specific heats may transport energy from the fluid region near the wall more readily, and consequently bring about higher values of the heat transfer coefficient. As the heat transfer characteristics of fluidized systems are observed over a range of porosities, a maximum in the heat transfer coefficient is experienced. After the initial fluidization, the heat transfer coefficient increases with an increase in the mass velocity of the fluid (and a corresponding increase in porosity). T h e coefficient reaches a maximum and then decreases with a further increase in mass flow. The mass velocity corresponding to the maximum heat transfer coefficient has been designated as the dividing line between the so-called dense phase and dilute phase fluidization. Below this point the heat transfer is increased by the increased scouring action of the solid particles as the flow velocity is increased; but the concentration of the particles is steadily decreasing, and this decrease eventually overshadows the increased scouring action and decreases the heat transfer coefficient. An interesting effect in fluidized systems has been pointed out by Frantz (3)and others (5, 75). I n a heated vertical column containing the fluidized solids, the axial temperature profile of the fluid indicates that practically all of the temperature rise in the fluid occurs in a shallow region near the bottom of the column, and the temperature remains essentially constant above this point. This shallow region is called the active section. Lemlich and Caldas (7) report that this effect is more pronounced for high Reynolds numbers. A casual inspection of these results would lead to the conclusion that practically all of the heat transfer occurs in the region near the bottom of the column, even though the entire column is heated. This conclusion would be incorrect, however, because it does not take into account the secondary flow regions which may be present in such fluidized systems. Since the solid particles are maintained in suspension by the vertical fluid drag forces, the particle motion will vary with radial distance from the wall of the tube. In the central portion of the tube the velocity is largest, and hence the fluid exerts a force on the particles sufficiently strong to cause them to flow upxvard. S e a r the tube \Val1 the fluid velocities are smaller as a result of the viscous action at the boundary and are not large enough to maintain the particle suspension; consequently, they move downbvard in the column. Thus, there 22
I&EC FUNDAMENTALS
is an upward motion in the center of the column and a downflow near the wall. At the bottom of the column there is the so-called active section where the downward flow of particles is diverted upward again. Using the concept of an upflow in the center and a downflow near the wall, Lewis, Gilliland, and Girouard (70) were able to analyze experimental temperature profiles and determine the effects of particle diameter and flow parameters on an effective radial eddy thermal conductivity. A schematic of the secondary flow system is shown in Figure 1. The particles near the wall receive the energy from the wall and transport it downward into the active section, where it is distributed uniformly among the particles which flow upward. Thus, the temperature of the fluid measured a t the axis of the flow may, under certain conditions, show no substantial increase at points above the active section. This phenomenon is illustrated in Figure 1. The secondary flow system is important in fluidized systems, and as yet it has not been thoroughly investigated. The heat transfer from the solid particles to the fluid medium is of primary interest in the present investigation. Four significant articles relate to this problem (5, 73, 75,76), as well as the summary paper by Frantz ( 3 ) . Kettenring, Manderfield, and Smith (5) measured the heat transfer rates from fluidized silica gel and activated alumina to air. Heated air was used to fluidize the particles and evaporate water from them. T h e particles were introduced at a constant temperature, and the gas temperature was measured a t various positions along the fluidized column. The heat transfer from the air to the particles was computed by measuring the amount of water vapor picked up by the air and thus calculating the energy required to remove this amount of water from the particles I t was assumed that the particle temperature remained constant during this evaporation process. The assumption was made that the solids temperature was constant and equal to the gas temperature at the top of the bed. This was presumably a reasonable assumption, since the graphical plots of the gas temperature indicated that thermal equilibrium was attained a t the top of the column; however, the assumption of a constant solid temperature throughout the column is open to question. The total surface area of the particles was computed on the basis of an average spherical shape for the mesh sizes tested, and the data were correlated with:
where the Reynolds number is based on particle diameter and superficial velocity. Wamsley and Johanson (76) used a transient method to determine the heat transfer coefficient from various particles to air and carbon dioxide. Precooled particles were dropped into a heated gas stream, and the exit temperature of the gas leaving the fluidized column was recorded as a function of time. T o analyze the data the authors assumed that: (1) the solids temperature was uniform throughout the fluidized bed, (2) the heat losses from the column were negligible, (3) the gas in the column was perfectly mixed so that the gas temperature at any position is uniform and equal to the outlet temperature, and (4) the heat capacity of the gas is negligible in comparison to the heat capacity of the solid particles. (This assumption was not stated, but is implied from the analysis in the paper.) Based on these assumptions a transient energy analysis was made, which resulted in the equation:
Sunkoori and Kaparthi (73) performed heat transfer experiments ivith \sater, using granite and quartz particles in the range of 540 to 1100 microns. The unsteady-state mrthod used by \t’amsley and Johanson ( 7 6 ) \vas used to evaluate the heat transfer coefficients in the liquid system; however, two serious faults are present in this paper. First, a Beckman thermometer was used to measure Lvater temperature as a function of time. T h e total elapsed time for cooling the heated particles \vas of the order of 6 seconds. T h e time constant for the thermometer Lvould surely be large enough to cause one to question the absolute accuracy of the cooling curves: although the relative behavior might be consistent. Second. the neglect of the heat capacity of the fluidizing medium (water) cannot be justified for a liquid as it was for the gaseous systems of Lt’amsley and Johanson. [This assumption \vas implied from the analysis of Wamsley and Johanson (76), though not stated.] Consequently, the method of Sunkoori and Kaparthi appears invalid, since the heat capacity of the \sater in the column is of the same order of magnitude as that of the solids and hence cannot be neglected. T h e temperature-time plots of the paper indicate a n error in the analysis, because straight lines are not obtained as in (76). Frantz (3) in a survey article combines the liquid and gas data of (5: 7 7 , 76) by using the equation:
Figure 1 . Flow pattern and temperature distribution for fluidized beds (3, 5, 15)
Nu Lvhere TQLand T w dare the inlet and exit gas temperatures, respectively, and constants CY and p are given by:
(3)
0 = T,,- TQea t
7
= 0
A logarithmic plot of Equation 2 yields a straight line, the slope of Lvhich gives the parameter, a , which may in turn be used to determine the heat transfer coefficient, h. T h e most serious assumption of Wamsley and Johanson (76) is number 3, for the gas temperature obviously must vary throughout the column. Assumption 1 is also open to question because the particle temperature must vary along the length of the column in some manner similar to the variation of the gas temperature. ‘4 simple calculation shows that assumption 4 is reasonable. T h e assumptions used to derive Equation 2 are subject to questions, but the authors’ data d o plot as straight lines, and this fact tends to lend credibility to the method of analysis more than physical reasoning. IValton. Olson, and Levenspiel (75) measured heat transfer coefficients bet\veen air and crushed coal. T h e particle size ranged betLveen 360 and 730 microns. Gas temperatures rvere measured tvith suction thermocouples, and solids temperatures were estimated by using bare thermocouples. In accordance with the reasoning pertaining to the rapid axial temperature equalization in the so-called active section of fluidized beds, the authors made a n energy balance on a differential element in the active section. A constant solids temperature was assumed in order to obtain an expression for ihe heat transfer coefficient in terms of the axial gas temperature variations and the fluid properties. ‘4 graphical plot was then used to determine the heat transfer coefficient. This method tends to average and smooth the d a t a , but it is of dubious value because of the assumption of constant solids temperature in the fluidized column. T h e data were correlated with:
=
hDP ~
k
=
0.016 (Re)1.3(Pr)0.67
(5)
However, the data of Sunkoori and Kaparthi (73) d o not fit the correlation too well. Baumeister and Bennett ( 7 ) and Eichorn and LYhite (2) reported linear temperature distributions for packed beds with induction heating. Eichorn and White (2) further attempted to study heat transfer in fluidized beds, and assumed a bed temperature equal to the outlet gas temperature. Because they could not determine the difference between particle temperature and gas temperature in the fluidized bed, they were not able to calculate the heat transfer coefficients for the fluidized system. Heat transfer from single spheres has been reported by Kramers ( 6 ) . W e t and Leppert (73) have extended the data previously available and report the following relations for flow of water and oil over spheres in Reynolds numbers from 1 to 50,000 : N U P ~ - (~l .. ~~~ / p ) o=, ~1.2 j
+ 0.53 Re0.54
(6)
I t is suggested that this relation can be extrapolated to Reynolds numbers as high as 3 X 105.
Physical Mechanism
T h e heat transfer from fluidized particles to the surrounding medium depends primarily on the flow around the individual particles. In the case of the spherical particles a first approximation of the heat transfer coefficient might be made by using Equation 6 . Unfortunately, this relation cannot take into account the complex interaction of the solid particles which is present in the fluidized column. I n Equation 6 the Reynolds number is based on the particle diameter and the free stream velocity past the sphere. I n a fluidized system the particle diameter is kno\vn, but the velocity past the particles is generally not knoivn. Hoivever, by assuming that the particles are uniformly distributed throughout the fluidized column, such that each spherical particle is surrounded by a small cubical volume of fluid, the velocity through the minimum cross VOL. 4
NO. 1
FEBRUARY
1965
23
STF4M
M A I N CIRCULATION PUMP
B L ~ W - O Ur
VALVE
PElOWiZER PJMP T E S T SECTION
9
T O DRAIN
Figure 2.
Schematic of flow systems
section of flow area may be expressed in terms of the porosity of the system. Thus, r
1
where L ' ~is the velocity past the particle, uo is the superficial velocity in the tube (velocity a t 1007, porosity), a n d e is the porosity. F , is designated as the velocity correction factor. Particle interactions play a large role in determining the heat transfer. T h e frequency with which the particles strike each other, as well as their velocity. influences the degree to which boundary layers on the particles are disturbed. I t is not possible to predict the influence Ivhich this interaction \vi11 have on the heat transfer, except to indicate its functional dependence on certain parameters T h e interaction velocity is dependent on viscosity. while the frequency of interaction is dependent on particle density in the fluidized column. This particle density is in turn dependent on the density of the solid material a n d the particle diameter. These three parameters-viscosity, solid density, a n d particle diametermay be expressed in the dimensionless forms
where po is some suitable reference value of the viscosity. These three groups are useful in correlating the experimental data. Experimental Program
An apparatus \vas constructed to determine the heat transfer coefficients from heated metal spheres to water in a vertical fluidized column. An induction heater was used to generate heat internally in the balls in a n effort to simulate conditions which might be present in a fluidized nuclear reactor. T h e experimental apparatus consisted of a quartz test section enclosed by an induction heating coil and a flow system for circulating water through the test section. A schematic of the flo\v system is shown in Figure 2, and a schematic of the test section assembly is shown in Figure 3. A detail dralving of the test section is given in Figure 4. Pressure taps were attached to each end of the test section and provisions were made to insert thermocouples into the flow a t entrance a n d exit to the heated section. Thermocouples could not be used Lvithin the heated section because they would be heated by the induction heating field. T h e Lvater flow rate was measured \vith a turbine-type flow.meter. 4 manometer \vas used for the pressure drop measurements and facilities were provided for preheating the lvater 24
before entrance to the test section, so that the measurements could be made under varying temperature conditions. The entire test section assembly !vas enclosed in a 3- X 3- X 8foot ply\vood box which !vas lined \vith aluminum foil and grounded to reduce the effect of the induction heater field on auxiliary instrumentation in the laboratory. A thermopile arranegment consisting of three junctions of copper-constantan wire \vas used to record the temperature difference between inlet and outlet conditions to the test section. T h e thermocouple leads entered the plywood box through flexible grounded conduits and \vere totally enclosed by the conduits while in the presence of the induction field. T h e heating for the test section column came from a iyestinghouse induction heater which \\'as rated a t 30 k\v. a t 450 kc. T h e heating \vas accomplished by means of a ten-turn coil made of n,'l,-inch copper tubing wound about the quartz tube. Both stainless steel and lead spheres were used for the experiments, in order to determine the effect of particle density on the results.
I&EC FUNDAMENTALS
T o calculate the heat transfer coefficient it is necessary to know the temperature of the fluidized balls during the heating process as well as the water temperature and heat transfer rate. I t was assumed that the induction heater imposed a constant heat flux condition on the test section and the water bulk temperature was measured a t inlet and outlet. T h e heat transfer rate was thus computed from the measured water flow rate and steady-state temperature difference between inlet and exit conditions. I n order to determine the ball temperature a transient analysis of the test section column was made as shown below. For purposes of this analysis it was necessary to determine the transient cooling response of the heated column This transient response was determined by first allolving the induction heating and fluidization process to reach a stable condition. T h e induction heating power was then shut off a n d the temperature difference between inlet and outlet conditions was recorded as a function of time: using the thermopile arrangement mentioned above. The experimental cooling curve so obtained was then used to determine the average ball temperature before the induction heater was shut off. As a further aid to analysis of the data, a Barnes Model 4D-1 radiometer was used to scan the quartz test section columq and determine the axial water temperature distribution. T h e radiometer was equipped with a special quartz \vindo\v, so that only the radiation from the water inside the quartz tube was measured by the instrument. T h e radiometer was calibrated directly bvith the test section column operating under various preheat temperatures without the presence of the induction heating field. I n accordance n i t h other fluidization heat transfer investigations, the heat transfer was assumed to be primarily dependent on Reynolds number and the experimental program was established so that this would be the primary independent variable. Throughout the experiment the height of the fluidized bed was held constant a t 23 inches. This restriction was imposed to ensure a constant heating rate along the column by having the same height of balls above and below the heating coil and to simplify the calculations by making the expanded fluidized bed approximately fill the test section. iVith constant column height serving as a limitation on the experiment, the variables Tvhich influenced the Reynolds number lvere the particle diameter. water flow rate, and \\,eight and type of fluid particles. Consequently, tests were conducted n i t h different particle sizes a t different flow rates and tempera tures. A typical experimental run consisted of charging the test section lvith a kno\vn \\-eight of particles of a certain diameter, adjusting the flow rate to give a 23-inch column height. and then varying the inlet Lvater temperature. T h e Iveighr of the particles in the test section \vas then changed for the next run
I"DIA
2" COPPER TUBE
COPPER TUBE
/BALL TEST SECTION STAND
1 ~
REDUCING UNION
e /
rlNSTRUYENTATYWI FLINGE
SEPIRATOR
7 U
ZIDIA.COPPER
TUBE
I1
Q U A R T Z T E S T SECTION wirn UEATING COIL
Figure 5. Nomenclature for analysis of test section column
FLOW STRAIOUTENER
'L Figure 3.
FLOW
m
Test section assembly Figure 4.
to give a different flow rate. For each particle diameter the total \\.eight of the bails Lvhich could be used \vas influenced by the fact that the induction heater Lvorked best for a test secrion porosity between 55 and 85Yc. LVhen the porosity limit \vas reached for particles of one diameter, the test section \vas charged with another particle size, and the process was rrpeated.
A summary of the range of experimental variables is given in Table I and a detailed description of the experimental apparatus and procedure is given by Holman, Moore, and Wong
Detail of test section
LVhen the system has reached equilibrium after power is shut and the integral on the left side of Equation 9 off, Tu,= T,,>. will be zero. Designating the initial water and ball temperatures along the column as Tu, and T B z , respectively, the following relation involving the convection heat transfer coefficient may be written:
01
(4) '
Analysis of Tesi Section Column
Consider the test section system shown in Figure 5. T h e system is in a steady state? with energy being delivered to the balls by the induction heater, and this energy is dissipated into the xvater. T h e inlet and exit water temperatures, T,, and Tu!: are measured. LVhen the induction heater is turned off, the time variation of these temperatures is recorded. From this record the heat transfer coefficient from the balls to the water will be inferred. T h e following assumptions are made: T h e inlet water temperature remains constant, the height of the fluidized bed and hence the porosity remain constant, the convection heat transfer coefficient remains constant throughout the column and does not change during the cooling process, and the cooling time is so short that heat losses to the environment are negligible. 'The energy equation for this situation is:
where the terms on the right side of the equation represent the change in energy of the water and balls within the test section control volume with respect to time. If Equation 8 is integrated \vith respect to time:
where dA,/dx is the surface area of the particles per unit length of test section. If the ball and water temperatures can be determined, the heat transfer coefficient can be evaluated. After the cooling process is complete ( r + a),the water and ball temperatures will be constant and equal to T w j . Performing the integration on Equation 9 gives:
where mB and m, are the masses of the balls and water in the test section control volume. T h e left side of Equation 1 2 may be determined experimentally from cooling curves. Expressed in terms of porosity, Equation 1 2 becomes :
where B is the porosity and V is the total volume of the test section. If the initial water temperature distribution, T,=; is known, the integral
LL
TB, dx
VOL. 4
=
NO. 1
E FEBRUARY
1965
25
may be evaluated to determine the average initial ball temperature. This average temperature, in conjunction with average water temperature, may be used along with the measured heat flow to calculate the convection heat transfer coefficient from Equation 11. Expressed in terms of porosity, the average temperature difference becomes :
If a constant .heat flux is imposed on the test section, the water temperature would be expected to follow a linear variation along the length of the test section. If the heat transfer coefficient is also constant, the temperature difference, TB, T,,, would also be constant. I n this circumstance:
T,,
=
where .lrDT2,/4 dx is the elemental control volume. This equation assumes that only water enters and leaves the control volume and thus the transport of energy results only from this motion. I t is known that the particles are in a constant vertiginous motion, but the concentration a t an); spatial position should be essentially constant and equal to the average concentration in the entire test section. It is possible that secondary flow processes may act to transport energy due to ball motion in the axial direction, but the effects of these processes are neglected. Observing that:
TU.ZLd9 = T,,
+
~
Tm dx bX
Equation 20 may be written as:
Tu,
AT = TBz - T,, = constant for a given heat flux is the water temperature a t exit from the test where TWzi Also, Equation 11 becomes:
section before cooling.
-
q = hAAT
(18)
If the internal conductive resistance of the balls is considered small in comparison to the convection surface resistance, the ball temperature may be assumed as substantially uniform a t any instant. Then the heat transfer coefficient is the prime resistance to heat flow from the balls to the water during the cooling process. Thus,
Inserting Equations 16 and 17 in 15 gives: where A is now the surface area of each ball and V is now the volume of each ball. Substituting Equation 24 in Equation 21 gives : for a linear water temperature distribution. T h e integral is evaluated from experimental -cooling curves, so that the only unknown in Equation 15 is AT. T h e application of Equations 15 and 19 to the experimental data is discussed in a subsequent section. T h e analysis of the preceding section may be used to determine a n average value of the heat transfer coefficient. I t is worthwhile at this point to mention the orders of magnitude of some of the terms in Equation 19, in order to estimate the accuracy with which the heat transfer coefficient is determined. T h e temperature difference, T,,-- T,,, was always greater than 10' F., while the value of A T varied between 5" and 25' F. For most of the data, A T was greater than 10' F. Both terms in Equation 19 are of the same order of magnitude, so that neither can be neglected for any range of operation. For most of the data it is believed that the average heat transfer coefficient as defined by Equation 18 was determined within +5%. However, this method gives only a n average heat transfer coefficient for the entire fluidized column. I t is possible that such a n average value would not adequately describe secondary flow processes which may be present. Thus. as a check on the results of the above analysis and the assumptions involved, a more detailed analysis of the cooling characteristics of the test section may be performed as follows. Consider an elemental control volume of the test section of height dx. An energy balance on this element gives: 26
l&EC
FUNDAMENTALS
An integration of Equation 24 will give the transient variation of the water temperature along the length of the column. In general, this integration may be performed only by numerical methods using given initial distributions for the water and ball temperatures. Since the water temperature just prior to the entrance to the test section is known, a finite difference representation of Equation 25 may be expressed in terms of backward differences, and the numerical calculation may be performed by computer methods to obtain the exit water temperature as a function of time for different assumed values of the heat transfer coefficient. T h e value of h as obtained from Equations 15 and 18 may be inserted in Equation 25 for performing the numerical integration. If the calculated cooling curve is in essential agreement with the measured curve, the assumptions which were made to determine the heat transfer coefficient may be considered as reliable, and the results may be viewed with confidence. Experimental cooling curves are compared with theoretical curves obtained from the finite-difference analysis in connection with the analysis of experimental data. Results of Experiments
T h e particle-to-fluid heat transfer coefficients were calculated from Equation 18 using Equation 15 to obtain the average temperature difference between the metal balls and water T h e range of experimental variables is summarized in Table I
Table 1.
Summary of Range of Experimental Variables Variable Range Porosity, B 0,496 to 0.926 '/le, 3 / 3 2 , 3 / inch ~ ~ stainless steel Particle diameter, D, and no. 9 lead shot Mass flow rate, m 1665 to 10630 lb.m/hr. in 2-inchdiameter column Particle Reynolds number, 240 to 13,900 Re = D p p u O / ~ Mean water temperature 45" to 260" F. Heat rate, q / A 1185 to 67,100 B.t.u./hr.-sq. ft. (based on total surface aiea of particles) Heat transfer coefficient 100 to 3000 B.t.u. /hr.-sa. ft.- ' F. (based on total'surfack area of particles) Total of 164 experimental runs
T h e heat transfer coefficient was calculated from Equation 18 for two cases: (1) an assumed initial linear water temperature profile, and (2) an initial water temperature profile obtained from the radiometer measurements. T h e linear temperature profile gives for the average initial water temperature :
where the o subscript refers to the steady-state values of the inlet and outlet temperatures. T h e other method of calculating the mean water temperature was based on the water temperature distribution determined from the radiometer measurements. This temperature profile \vas obtained by converting the radiometer output readings to temperature using the calibration and by normalizing the temperature distributions using (T, - Ti)/( T I - Ti) as rhe normalized temperature parameter. At each column position the normalized data for 15 runs were averaged, and these averages were plotted against axial distance to obtain the temperature distribution used in the calculations. This temperature distribution is presented in Figure 6. T h e average water temperature determined from this distribution is: -
T,
=
T,,
+ 0.629 (T,? - T,,),
(27)
TWOvalues of the heat transfer coefficient were calculated for each experimental run : one from each of the two mean water temperatures used. T h e heat transfer coefficients calculated from the linear temperature distribution, with the mean water temperature defined in Equation 26, are hereafter called "linear heat transfer data" ; coefficients calculated from the nonlinear temperature distribution, with the mean water temperature defined by Equation 27, are called "nonlinear heat transfer Table II.
Correlation
;( )
Nu
=
3.66 X 10-6Re2.1Pr0.67
Nu
=
6.7
Nu
=
0.533 X 10-6(ReFc)2.1Pr0.67
Nu
=
0.291
X
1.83
10-6 Re'.j6Pro.67
X 10-6(ReFr)2.'2Pr0.67
-
AX1 AL DISTANCE INCHES
Figure 6. Nonlinear experimental water temperature profile
data." T h e heat transfer coefficients were then used to develop correlations in terms of dimensionless groups. T h e significant dimensionless groups which were used for correlating the experimental data were the particle Reynolds number, the Prandtl number, the velocity correction factor given by Equation 7, and the dimensionless groups /J
-
Po
P D, , P-, and - -
PI
DT
where the reference viscosity was taken as that a t 80" F. data were first correlated with the relation
The
Nu = C Rea PrO.67 (:>b
T h e exponent for the Prandtl number was chosen as 0.67 in accordance with the previous work of Frantz ( 3 ) . Next, the influence of porosity was investigated by applying the velocity correction factor as calculated from Equation 7 directly to the Reynolds number. The functional form of the correlation was thus :
(29)
Nu = C(ReF,)" PrO6.7
T h e values of exponent b were the same as those used with Equation 28.
Heat Transfer Correlations
Temperature Projle Used
Deviation, %
Nonlinear
58.0
Linear
42.0
Nonlinear
54.7
Linear
34.0
Nonlinear
30.4
Linear
20.6
VOL. 4
NO. 1
FEBRUARY
1965
27
Finally, the effects of the dimensionless diameter and density ratios were investigated. After several different correlations had been plotted, the dimensionless groups
were found to be of definite use in reducing the scatter of the data. The equations developed by considering all the groups were thus of the form:
Nu = C (ReF,)"
Pro.67
k)* y:(
(::>""
(31)
The specific correlations developed by considering all the factors are presented in Table 11, where there are two correlations for each set of factors: one developed from the linear heat transfer data, and the other developed from the nonlinear heat transfer data. These correlations are compared by means of the per cent deviation calculated for each case. A brief inspection of Table I1 reveals that the equations developed by considering all the factors had a much lower deviation than the other correlations. Thus the equations:
derived from the nonlinear heat transfer data, with a deviation of 30.4%, and
derived from the linear heat transfer data with a deviation of 20.670, were the best correlations for the data of the present work. These correlations are presented graphically in Figures I and 8. With the best correlations for the data of each temperature distribution developed and presented in Equations 32 and 33, it was decided to compare the values of the heat transfer coefficient predicted by these two equations. T o obtain an indication of the influence of the nonlinear temperature distribution on the predicted heat transfer coefficient, the nonlinear heat transfer data were replotted using the powers of modified Reynolds number and viscosity ratio which had been derived for the linear heat transfer data correlation in Equation 33. The resulting correlation is given in Equation 34 and shown in Figure 9.
(34) A comparison of Equations 33 and 34 shows that the use of the nonlinear water temperature distribution results in a heat transfer coefficient which is 50% higher than that calculated using the linear water temperature distribution. Comparison of Experimental and Calculated Cooling Curves
T h e analysis of the test section column and particularly Equation 25 showed how a finite difference analysis may be applied to the test section to determine the variation of the exit water temperature with time. To effect this analysis it is necessary to know the initial water and ball temperature
m
-n
-
X I p 102 N
3
