c
=
w
=
wall temperature of heat exchanger wall temperature of reactor
Literature Cited
Bilous, O., Amundson, N. R., A.I.Ch.E. J . 1, 513 (1955). Gilliland, E. R., Gould, L. A . , Boyle, T. J., JACC Proc. 5 , 140 ( 1964). Hamming, R. W., “Xumerical Methods for Scientists and Engineers,” RZcGraw-Hill, New York, N. Y., 1962. LandiL, J. G . , Ph.D. Thesis, University of Pennsylvania, 1970. Landis, J. G., Perlmutter, D. D., A.I.Ch.E. J . 18, 380 (1972): Razuniikhin. B. S..Ph.D. Thesis, Institut Medhaniki Akademiva Nauk USgR, 1958.
Razumikhin, B. S., A p p l . Math. Mech. 22, 215 (1959). Razumikhin, B. S., Automat. Remote Contr. 21, 515 (1960). J . G. LANDIS DANIEL D. P E R L N U T T E R * Department of Chemical and Biochemical Engineering L‘niversity of Pennsylvania Philadelphia, Pa. 192 74 RECIXVLD for review XTay 1.5, 1972 ACCEPTLD J d y 31, 1973 This research was supported by a grant from the Sational Science Foundation.
Effect on Heat Transfer Due to a Particle in Motion through Thermal Boundary layer over a Flat Plate A mechanism i s outlined to explain the heat transfer due to movement of a particle through the thermal boundary layer on a flat plate. It consists of the experimental determination of the improvement in heat transfer due to film scraping and b y particle convection. The heat picked up b y the particle was determined experimentally; a quantitative explanation of its variation i s given. The experimental values of the heat transfer coefficient for a moving spherical particle have been found to support literature relationships.
T h e study of the effect on heat transfer due t’o movement of a particle through the thermal boundary layer on a flat plate has a significant practical application in many fields of engineering. The most important of these is the field of fluidized bed heat transfer. The heat transfer coefficient in a fluidized bed is many times greater than the coefficient from wall to fluid alone. There are many reports in the literature explaining this fact (Botterill and Williams, 1963; Levenspiel and Kalton, 1954; Nickley and Fairbanks, 1955; Van Heerden, et al., 1951; TVicke and Fetting, 1966). According to Zabrodsky (1966), the thickness of the laminar sublayer is reduced anywhere from one-sixth of a particle diameter to a n arbitrary diameter due to particle movement; this increases the heat transfer coefficient. The particles themselves gain heat from their surroundings as they approach the heated surface, and when they leave the thermal region they give up their heat to the bulk. This is termed “particle convection heat transfer.” The effect of such transfer as a plausible mechanism was undertaken by Kashmund and Smith (1965), Ziegler and Brazelton (1964), and others. Though these authors have demonstrated the simultaneous presence of the two effects experimentally, their relative importance has not yet been investigated. It is the intention of this paper to determine experimentally the improvement in heat transfer due to film scraping and the heat picked up by a particle when it moves through the thermal sublayer. Many experimental studies have been undertaken with regard to the motion of a particle in a fluidized bed. Some research workers have made qualitative statenients about the motion being raiidom and others state that it is not random; both views are supported by experimental evidence. Handley’s (1957) photographic study with glass beads in a liquid fluidized bed shows that the particles move smoothly throughout the bed and as such they do not have a finite residence time a t the bed wall. .Is suggested by Mori aiid Wakhumara (19681,
when the particles are in downward motion along the wall, it is thought that’ they may pass through the fluid film adjacent to the wall. I n doing so, the particles pick up heat aiid also decrease the thickness of the thermal boundary layer. To isolate the effects of the particle heat conv’ection and the scraping of the film adjacent to the wall, the experimental setup we have used involves a single spherical particle moving in a circular path. During the course of its motion it passes through the boundary layer, adjacent to a heated platinum foil surface. Experimental Section
The details of the experimental setup are sketched in Figure 1. The particles, made of copper, were 0.4975, 0.3925, 0.2999,
0.29, and 0.25 cni iri diameter. A copper-constantan thermocouple was spot-welded to each of the spheres. The spheres themselves formed the ends of the thermocouple hot junction. The thermocouple was drawn through a groove in a silver steel rod of lja-iii. diameter and the copper-coristantaii wire was attached to two copper rings fitted in the rod. The rod rested in a bearing mounted in a pipe a t one end, and the other end was coupled to a worm and wheel arrangement. The worm could be rotated a t various speeds via a motor and set of reduction gears, thus achieving various velocities for the spherical particle. The platinum foil used as a heater was activated by a de electrical source. The position of the foil across which the particle moved measured 1.485 x 0.8275 x 0.00254 em. All the thermocouples were standardized a t the start of the experiment. The platinum foil was also calibrated to calculate its temperature from the resistance measurement. The voltage drop across the foil when there was no niovement of the particle was measured by a de microvoltmeter. Whenever the particle moved across the foil, its temperature fluctuations were recorded by measuring its voltage drop by an X-Y recorder. The particle temperature was measured by the thermoInd. Eng. Chem. Fundam., Vol. 12, No. 4, 1973
479
A
C
8
I'
K
Figure 1. Experimental arrangement: A, stabilizer; B, rectifier; C, X-Y recorder; D, Sargent recorder; E, motor and gear arrangement; F, copper ring; G, platinum foil; H, copper sphere; I, thermocouple wire; J, silver steel rod; K, angle iron frame; L, ball-bearing mounting
Fiaure 3. Somole outDut of X-Y recorder used in the cal< culation of heat loss from platinum foil
continuously and this temperature history is used in evaluating t h e particle convective heat transfer. It is calculated a s 8=8'dT Qpp = mcDJ - d0 = ~ C , [ T ] ~ = ~ , B = ~(1)' #=e1 do 0 = O1 denotes the time a t which the particle enters the thermal boundary layer and 0 = 0%is the time a t which it comes out of
the boundary layer. The output from the Sargent recorder is shown in Figure 2. Knowing the time for which the pastiole stays within the boundary layer, the temperatures a t 0 = O1 and 0 = e* can be read from the graph. The particle convective heat transfer coefficient is evaluated from
QPp = h'A
J8='*
(Txp - tb) d0
(2)
s=n1
Figure 2. Sample output of Sorgent recorder used in determining particle convective heat transfer coefficient
couple attached to it and was continuously recorded by a Sargent recorder. The distance separating the particle from the p!atinum surface aud its radius of motion were accurately measured by a cathetometer. Smce the thickness of the boundary layer varied across the platinum foil, from its lower to the upper edge, the plane of movement of the spherical particle was accurately determined. This was necessary so that the undisturbed boundary layer thickness could be known for use in the mathematical aualysis. A three-dimensional ternperature measurement enabled ns to know the thickness of the thermal layer a t the plane of the particle motion.
The quantity under the integraj is given hy calculating the area under the curve shown in Figure 2. These areas were accurately measured for 15 readings and are averaged. Knowing Q ,, from eq 1 and the surface area A , h' can be calculated from eq 2. b. Calculation of Improved H e a t Transfer D u e t o Film Scraping. This can be calculated from the X-Y recorder output. The output of the X-Y recorder is a continuous recording of the temperature of the platinum foil. The heat loss from the foil due t o the motion of the spherical particle has been calculated from Figure 3. The heat transfer coefficient during the particle's presence in the thermal region is high due t o the gain of heat by the particle when it enters the fluid film. The heat loss from the foil for the entire unsteady state is calculated as follows.
Qrn8= m'h' [area underthe cwve duringthe entire unsteady state]
Analysis of Results
The important contributions of the experimental analysis are threefold. First, it has enabled us to determine the heat picked up by the particle, known as the particle convection heat transfer. Secondly, it enables us to determine the heat transfer due to film scraping. Thirdly, we could verify the correlations of Rane and Marshall (1952) for the heat transfer coefficient of a spherical particle in motion. a. Calculation of Particle Convective Heat Transfer. The Sargent recorder records the particle temperature 480 Ind.
Eng. Chem. Fundarn., Vol. 12, No. 4, 1973
{e$ - 8,) Qm
+
(3)
The heat transfer coefficient htUsis calculated from the value of the heat loss . 'Q . QIU8
= h',.A'AT
(4)
The combined loss of heat due to particle convection and film scraping is given as total loss = (h'"*
- h..)A'(Oz - QAT
(5)
Table I Diameter
No.
of particle
h X lo3
1
0,0163
0.0 1.30 2.95 0.0 1.3 0.0 0.0 0.0 0.328 0.0 2.43 0.0
2 3 4 5 6 7 8 9 10
0.0128 0.0098 0.0095
11
12
0.0082
Time for foil to attain steady state
Velocity of particle
15.6 15.0 1.10 5.10 3.60 1.15 20.0 7.5 15.0 7.0 8.0
0.013 0,013 0,264 0,065 0.065 0.264 0.025 0.025 0.025 0.065 0.127 0.013
...
has
6.90 6.90 8.54 8.39 8.39 7.37 5.60 6.18 7.75 8.93 7.35 ...
Q~~
htUs
7.35 7.32 8.76 8.67 8.67 7.52 5.69 6.40 7.79 9.10 7.41
x
Total heat loss
x
103
1.89 1.69 0.835 0.282 0.282 0.072 0.357 0.270 0.218 2.64 0.108 ...
103
2.16 2.03 1.08 0.322 0.330 0.084 0.386 0.322 0.274 0,305 0.139
%
Particle convective transfer
87.5 84.5 77.0 87.5 85.5 85.5 92.5 83.5 79.5 84.0 77.5
\
\
I
Figure 5. Plot of Nusselt number vs. Reynolds number for particle heat transfer coefficient
I /
Equation 8 can be compared with the correlation given by Rane and Marshall (1952) Nu
Figure 4. , ,,eoretical model for determining convective heat transfer of a particle in circular motion
Knowing the total loss and the particle convective heat transfer the heat loss due to film scraping can be calculated. Table I shows some of the values obtained for Qpp, total loss, h,, and h'uB along with the diameter, velocity, and the distance separating the foil and the particle. The data included in the table give some representative values only. The omission of some entries in the bottom row indicates that the data were insignificant. The particle convective transfer has been calculated and the last column of the table gives the percentage of particle convective transfer to the total loss. c. Correlation for the Spherical Particle. T h e heat transfer coefficient for the spherical particle is correlated by t h e following form. Xu
=
A
+ B(Re)"
(6)
Using the least-squares technique, the constants A , B , and n have been found to be Nu Assuming A
=
=
2.15
+ 0.504(Re)0.55
(7)
2.0 the values of B and n are
Xu
=
2.0
+ 0.719(Re)0.4S(Pr)0.33
(8)
=
2.0
+ 0.6(Re)o.5(Pr)0.33
(9)
The coefficients in eq 8 and 9 seem to agree well within the experimental error. The particle convective heat transfer is also calculated using the equation which is derived from theoretical considerations (Figure 4). It is given by &pp
=
mcp(T.xp
-
tb)
(10)
where
where c = V / a . The comparison between the experimental values and the calculated values is made and the average percentage error is about 8.0%. The reason for the discrepancy may be due to the assumption of linear temperature variation within the thermal sublayer, made in the derivation of eq 11. -4plot of Nusselt number us. Reynolds number using Ran2 and Marshall's correlation and the least-squares fits (eq 7 and 8) is shown in Figure 5 . Conclusions
K e have experimentally isolated the effects of measured heat transfer coefficients due to particle convective transfer and film scraping. The results agree Kith the findings of Ziegler Ing. Eng. Chem. Fundam., Vol. 12, No. 4, 1973
48 1
a i d Brazelton (1964), who based their conclusions on evaporative cooliiig of a celite sphere. Nomenclature
p w
= = =
time, sec density of the particle, lb/fts angular velocity of the particle, radians/sec
literature Cited
radius of the circle in which the particle moves, ft A surface area of the particle, ft2 surface area of the platinum foil, f t 2 A’ boundary layer thickness, ft 6 specific heat of the particle, Btu/lb O F specific heat of platinum foil, B t u / l b O F d = diameter of the uarticle. ft h’ = particle heat t r a k f e r coefficient, Btu/hr f t 2 O F h,, = heat transfer coefficient under steady state conditions, B t u j h r it2 O F hIu, = heat transfer coefficient during the entire unsteady state, Btu/hr ft2 O F k = thermal conductivity of particle, B t u j h r ft OF K = h ’A / m C, m = mass of particle, lb m’ = mass of the platinum foil, lb Q = heat gained by particle, Btu Q,Y: = heat loss for the unsteady state, Le., till the foil regains the steady temperature, B t u tb = temperature of the fluid a t the thermal boundary, OF t, = temperature of the platinum foil, O F T = temperature of the particle, OF AT = temperature difference, OF T,, = temperature of the particle as it leaves the thermal region, O F velocitv of the uarticle. ft/sec extent bf penet;ation &thin the thermal region, ft a
e
= = = = = =
v = x =
GREEKLETTERS = total angle subtended by the arc length, deg X = distance separating the particle from the foil, ft
Botterill, J. S. &I., Williams, J. R., Trans. Inst. Chem. Eng. 41, 217 (1963).
Handley, D., Ph.D. Thesis, University of Leeds, Leeds, England, ,n-v
lYJ(.
Levenspiel, 0 Walton, J. S., Chem. Eng. Progr. Symp. Ser. 50 19). 1 119541.
Mickley: H. S., Fairbanks, D. F., A.I.Ch.E. J . 1, 374 (1955). Mori, Y., Wakhumara, K., J . Chem. Eng. Jap. 1 (a), 186 (1968). Ranz, W. E., Marshall, W. R., J . Chem. Eng. Progr. 48, 141 (14A2\ ,--~-,. Van Heerden, C., Nobel, A. P., Van Kravelen, D., Ind. Eng. Chem. 1, 151 (1951). Washmund, B., Smith, J. W., Can. J . Chem. Eng. 43, 246 (1965). Wicke, E., Fetting, F., Chem. Eng. Tech. 26, 301 (1966). Zabrodskii, S. S., “Hydrodynamics in Heat Transfer in Fluid Beds,” F. A. Zenz, Ed., M.I.T. Press, Cambridge, Mass., 1466 --..-.
Ziegler, E . N., Brazelton, W. T., IND.ENG.CHEM.,FUNDAM. 3, 94 (1964).
XARAYANAN S. SUBRAMANIAN’ D. PRAHALADA RAO* T. GOPICHAND2 Birla Institute of Technology and Science Pilani, Rajasthan, India
Present address, Department of Chemical Engineering, University of New Hampshire, Durham, N. H. 03824. Present address, Department of Chemical Engineering, I.I.T., Madras-36, India.
p
RECEIVED for review August 24, 1972 ACCEPTED April 12, 1973
Viscous Flow with Rectangular Cross Sections Methods are developed for calculating the viscous flow of liquid in rectangular open channels: (a) where the channel i s sloping and has liquid flowing at constant depth and (b) where the channel i s horizontal and has liquid flowing with varying depth. In case (b) it i s shown that there i s a limited length of channel allowable for given conditions, and means of calculating this are indicated.
T h e work here described arose in connection with experiments on the condensat’ion of vapor inside a rotating tube. The aim was to remove the condensate via small longitudinal grooves in the tube surface. The grooves being small, the flow in them was viscous. Ailsothe centrifugal force in the rotating tube constituted a n intense artificial “gravity” everywhere normal to the tube. Under these circumstances it was desired to calculate the carrying capacity of a groove. 1. The problem was approached by first studying the viscous steady incompressible flow iii a closed rectangular duct (see Figure 1). Let the cross section be as shown. Then for flow in the zdirection
(where the symbols have their usual meaning). A solution for eq 1 exists in the form of a n infinite series, which is not very useful for computation. We therefore pro482
Ind. Eng. Chern. Fundarn., Vol. 12, No.
4, 1973
ceed to normalize eq 1 preparatory to developing a usable numerical solution. Let y / a = Y, z / a = Z, b/a = a,and U
,$=-
a-2 b ( P P
+ rh)
bx
Thus eq 1 becomes
