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The pressure field around a two-dimensional gas bubble in a fluidized bed
Chemical Engineering Science, 1973,
Vol. 28, pp. 2231-2243.
Pergamon Press.
Printed in Great Britain
The pressure field around a two-dimensional fluidized bed
gas bubble in a
HOWARD LITTMAN and GEORGE A. J. HOMOLKAt Rensselaer Polytechnic Institute, Troy, N.Y., 12181, U.S.A.
(First received 16 August 1972; in revisedform
5 February
1973)
Abstract-Measurements of the entire pressure field around a two-dimensional bubble rising in an incipiently fluidized bed are reported. Above the center of pressure line, the field is Laplacian and reasonably well described by Davidson’s theory. The pressure wake is closed. It extends for 12 bubble radii below the stagnation point and four radii on either side of the vertical axis of symmetry. INTRODUCTION PRESSURE field measurements around an isolated bubble in an incipiently fluidized bed have been used previously for studying flow patterns near bubbles and for testing the theoretical models which describe those flows. ’ Reuter [ 161 measured the pressure field around a three dimensional half bubble rising along a plane wall and his data along the vertical axis of symmetry have been compared with theory by Stewart[ 171. Stewart’s analysis showed that in front of the bubble there was good agreement with Davidson’s theory [2, 33 and that the Murray [ 131 and Jackson [9] theories forecast dynamic pressures there which are much too low. Below the bubble Reuter’s dynamic pressures, although lower in magnitude than Davidson’s theory predicts, are in fair agreement with that theory. More recently, Littman and Homolka [ 111 measured the pressure field along the vertical axis of symmetry of a two dimensional bubble. Their results are in general agreement with the Davidson theory except below the bubble where the magnitude of the dynamic pressure exceeds that predicted. Those results were confirmed by Motamedi[12] who, in addition, found that the pressure field extended downward along the axis of symmetry for about 8 bubble radii below the floor. These two studies are in general agreement with Reuter except in the wake where apparently Reuters pressure field is modified by having the bubble rise along the wall. The magnitude of his
dynamic pressure and the extent of his bubble’s wake are both less than found by Littman and Homolka and by Motamedi. Measurements of the entire pressure field around a two dimensional bubble are reported for the first time in this paper. The data are compared with theory and discussed with regard to the nature of the pressure field and in relation to the forces which cause the local pressure gradient. The boundary of the entire detectable dynamic pressure field is shown. EXPERIMENTAL DETERMINATION OF THE NATURE OF THE PRESSURE FIELD ABOVE THE CENTER OF PRESSURE LINE
All experimental pressure field data above the center of pressure line of a rising bubble follow Davidson’s theory reasonably well. Since that theory predicts that the pressure field is Laplacian, studies were undertaken to test this prediction experimentally for a two dimensional bubble of arbitrary shape. Davidson’s pressure field is Laplacian because Darcy’s law is assumed to hold in the dense phase. The Jackson and Murray theories, on the other hand, are not Laplacian because the pressure gradient is related to the particle phase inertial force. Table 1 gives the pressure field equation for each theory. Davidson’s solution to Laplace’s equation is obtained analytically by considering an exterior problem whose boundary conditions are defined
tPresent address: Western Co. of North America, Forth Worth, Texas 76101, U.S.A.
223 1
H. LITTMAN
and G. A. J. HOMOLKA
Table 1. Comparison of expressions for the dynamic pressure due to the presence of a two-dimensional bubble in a fluidized bed
Theory
Field
Davidson
Laplacian
Pressure field equation
fi+,c,,
e=y
-&+scosp=~$
[
1-2sinzB-&
Field not Laplacian
Jackson
1 1
with iJb = i (ga) I’* Field not Laplacian
Murray
-&+scose=~
cos 28
Origin of the coordinates is the center of the circular bubble. 0 = 0 at the bubble stagnation point.
at the surface of the bubble and at infinity. The bubble is considered to be perfectly circular and the medium infinite in extent. Since experimental bubbles are irregularly shaped and the bed finite, testing the Laplacian nature of the pressure field
for arbitrarily shaped bubbles requires not only the measurement of the field but the formulation and numerical solution of a boundary value problem for Laplace’s equation. That boundary value problem is specified in Fig. 1.
_
Surface of bed
v2p,=o
Bubble
P‘
‘Pa
=P,t(l-_b)p*
Center
pressure 47 H*
of
llne
Fig. 1. Boundary value problem from which the theoretical pressure distribution around a bubble was calculated.
2232
The pressure field around a two-dimensional gas bubble Surface
At the upper surface of the bed, the pressure is atmospheric and motion pictures have shown that this surface remains flat as long as the bubble is more than three radii of curvature below the surface. The conditions along the vertical sides of the bed must be hydrostatic providing the bed is wide enough and this was easily verified experimentally by placing a transducer two inches from the outer edge of the bed. The pressure along the lower boundary of the region is constant if that boundary is composed of the center of pressure line and the bubble boundary. The distance of the center of pressure line below the surface of the bed is obtained from its definition. Thus
(1) The pressure inside the rising bubble, pb, required in Eq. (1) was measured using a single transducer positioned inside the bubble boundary at the instant the bubble was photographed (see Fig. 2a). This photograph gives the bubble size, shape and location in the bed and completes the specification of the boundary value problem. The pressure field can then be calculated and compared with the experimental one. The experimental field is obtained using four transducers set in a horizontal line one inch apart as shown in Fig. 2a. Their outputs together with
I--
-Time
Fig. 2a. Bubble and transducer locations for experimental measurement of the pressure field around a bubble.
that of the pressure transducer located inside the bubble were recorded continuously on a Honeywell 906C Visicorder from the time the bubble was injected. A sample of the transducer outputs appear in Fig. 2b. The photograph of the rising bubble and the Visicorder record must be synchronized in time and this is accomplished by recording on the chart the time interval over which the camera shutter is open (l/SO0 set). Experimentally this was done by placing two photo cells inside the
increasing
I
Carnero photocells
I
’ Time of bubble photograph
Fig. 2b. Sample of visicorder record-O gives the pressure at each transducer at the instant the bubble was photographed. Location of each transducer in field is shown in Fig. 2a.
2233
H. LI-ITMAN
and G. A. J. HOMOLKA
camera on either side of the photographic plate and recording the amplified signal from the photo cells on the Visicorder chart (top trace in Fig. 2b). The pressure at each location in the field at the time the photograph was taken is indicated by the circle on each pressure-time trace. The distance of the bubble from the various transducers at that time is easily obtained from the photograph. Repeating this procedure for many different bubbles varying the distance between the. bubble and the transducers gives the data required to test the field.
NUMERICAL
SOLUTION EQUATION
OF LAPLACE’S
The numerical solution of the boundary value problem is approximated by a relaxation method
PI. The method by which the relaxation is carried out was chosen on the basis of stability and the speed of convergence. It was found that an S.O.R. (Successive Over-Relaxation) iteration permitted the use of acceleration factors from l-6 to 1.9, depending on the size of the matrix, while maintaining its stability. This assured smooth and rapid convergence. The relaxation procedure gives only an approximate solution to the boundary value problem. An exact solution is obtained by means of Richardson’s extrapolation scheme which consists of solving the problem for several different grid spacings (h) and extrapolating to h = 0. pf (approximate)
= pr (exact) +f( h )
where f(h) = a,h2+a,h4+-
--
The advantage of this method is that, as the grid becomes finer, error due to the bubble shape approximation is diminished and extrapolation need only be done on those points which are of interest. Grid spacing of l/4 X l/4 in. and l/8 X l/8 in. were used in this study. The round off error was about 1.5 per lo6 per iteration. Details of the calculations are given by Homolka [7].
MAPPING
THE
ENTIRE
PRESSURE
FIELD
The experimental technique used for mapping the entire field is similar to that described for the pressure field above the center of pressure line except for the use of high speed motion pictures of the bubble instead of single photographs. Synchronization of the Visicorder traces and the film was achieved by placing a high speed clock in the field of view.
Equipment The two dimensional fluidized bed used was 57 in. high, 20 in. wide and l/2 in. thick. It was constructed from l/2 in. thick aluminum plates which were reinforced by steel structures to limit the maximum outward deflection of the plates due to hydrostatic pressure to O-002 in. In the front plate was a coated conductive glassPlexiglas window, 29 in. high and 9 in. wide, through which the bubbles were photographed. The air distributor was a sintered metal plate with a 20 p average pore size. The fluidizing air was humidified above 60 per cent relative humidity and this together with the grounded aluminum walls of the bed and conductive glass window prevented electrostatic charge buildup between the walls and the particles. The bubble injection system was similar to that of Davidson et al.[4]. The pressure transducers and the bubble injection tap were located in the rear plate. The field transducers were 12 in. above the injection tap, 3 in. above the bubble transducer and approximately 9 in. below the top of the bed. Two types of pressure transducers, an MKS Baratron pressure meter with a Type 77 head, and a Pitran (PT-3) transducer were used in this investigation. The Baratron is a secondary pressure standard and it was used to calibrate the Pitt-arts. The Baratron pressure head is a very sensitive capacitance transducer and the meter has a 5 place digital readout for 8 full scale pressure ranges from O-01 to 30 mm Hg. The instrument was calibrated against a deadweight tester to 0.01 percent of the actual pressure for any range used.
2234
The pressure field around a two-dimensional
The frequency response of the Baratron is greater than 20 Hz. The Pitran transducer consists of a planar silicon NPN transistor with a stress sensitive emitter-base junction which is connected to a diaphragm. It was mounted in a holder and placed in the wall of the bed. The positive pressure side of the transducer was covered with a 100 mesh screen to protect the diaphragm from direct particle contact because the transducer has a very high frequency response (150,000 Hz). The noise caused by the particle impact on the diaphragm in the absence of the screen would obscure the pressure signals. The Pitrans have a linear operating range from 0 to 26.5 mm Hg and a maximum output sensitivity of 143V/mm Hg using a 20 V power supply. Single photographs of the bubble were made with a 4 X 5 in. plate camera equipped with a Schneider Symmar lens and a Synchro-Compur shutter. The 4 X 5 in. negative was enlarged to make an 8 x 10 in. print for use in measuring the distances between the bubble nose and the transducers. Most of the motion picture data were obtained using a 16 mm Hycam high speed camera operating at a speed of 188.6 frameslsec. Two copies of the motion picture film were made. The original was projected and examined visually. The duplicate copy was cut up and examined frame by frame in a microfilm reader (enlargement factor = 25). The changes in the bubble’s size, shape, aed position which occurred during the run were obtained from every fifth frame of the mounted film strip. Details of the equipment are given by Homolka [7]. Bed and particle data are given in Table 2.
gas bubble Table 2. Bed and particle data Property
Shape Mesh size Particle diameter ( p) Density C,gm/c3 Experimental U,(ft/min) Voidage at U, Reat UN Bed height at U.&in.) Bed weight (lb)
RESULTS
The Laplacian character of the pressurefield above the center of pressure line
Eleven runs were made to test the nature of the pressure field and the results are given in Table 3. Distances from the stagnation point, R, were varied from O-79 to 3.91 in. for bubbles whose radius of curvature were of the order of two inches. The angle, * was varied from 0 to 77”.
round 30140 454 248 31.5 0.384 4.3 42.5 27.885
P,, the measured dimensionless interstitial fluid is defined as the total pressure pressure, (dynamic + hydrostatic) at a point in the field divided by the total pressure inside the bubble. PT is the analogous quantity to P, calculated assuming the pressure field is Laplacian. For the 44 data points listed in Table 3, the mean value of P,lP, is 0994 and the standard deviation, u, is 0.02 about the mean. The adequacy of this test of the field depends on the ratio of the dynamic to hydrostatic pressures since the dynamic pressure is not a large fraction of the total. The first and fourth points in Run 15 provide a representative test of the magnitude of the difference between a Laplacian and a Jacksonian field for the data presented. For those points, the dynamic pressures calculated from Jackson’s equation are O-270 and O-676 mm Hg respectively and this makes the corresponding values of P,/P, O-920 and O-880, well beyond the 3a limit or the maximum deviations found experimentally. Thus the pressure field above the center of pressure line is Laplacian.
(b) The entire pressure Davidson’s
(a)
Glass
field-
comparison
with
theory
The entire pressure field was examined for 5 radii of curvature vertically above and 15 radii below the center of the pressure line. It was also studied for 5 radii of curvature on either side of the vertical axis of symmetry of the bubble. The results of the investigation are presented in Figs. 3-7 as plots of the normalized dynamic pressure, pJp*ga, vs. distance from the stagnation point to a point along a line parallel to the
2235
H. LITTMAN
and G. A. J. HOMOLKA
Table 3. Comparison of the measured pressure above the bubble roof with a Laplacian pressure field- glass particles 454~
Run number 6
I
1OA
1lA
Superficial velocity Wmin) 32.5
Bubble height W 3.38
Bubble radius of curvature W 1.91
Transducer position rel. to bubble stag. point R ti (in) (de@
Dynamic pressure (mm. Hg)
(uu. H@
3.01-41.6 246-24.0 2.25 0.0 2.46 24.0 In bubble
22.544
2.338 2.530 2.598 2+&l 1404
24.882 25.074 25.142 25.204 31.067
0.801 0.807 0.809 0.811 l+OO
0.789 0.816 0.824 0.816 lJJO0
0996 1.011 1.019 lGO6
2 1.963
2.319 1.945 2~100 2.261 2.435
24.282 23908 24.063 24.224 31.718
0.766 0.754 0.759 0.764 lwl
0.742 0.753 0.759 0.758 lW0
0.969 0999 l+Oo 0.092
1.555 I.939 2.087 2.353 -0.119
23.555 23.939 24.087 24.353 29.214
0.806 0.819 0.834 0.834 lXlO0
0.812 0.832 0.846 0.844 1NNJ
1.007 1.016 1.014 1.012
1.737 2.781 3.551 3.412 - 2.598
23.764 24.808 25.578 25.439 26.573
0.894 0.937 0.963 0.957 lGO0
0.881 0.921 0.977 0995 IWO
0.985 0.983 I.015 lW_l
0.947 1Ml I.156 I.346 2.505
23.103 23.216 23.312 23.502 32.257
0.716 0.720 0.723 0.729 lM0
0.716 0.723 0.727 0.726 l+lOO
1xKm lGO4 1.006 0996
2.228 2.820 2.942 2.786 2.568
24.380 24.972 25.094 24.938 32.3 14
0.753 0.774 0.777 0.773 lJJO0
0.730 0.746 0.741 0.740 l+lOO
0.969 0964 0.954 0.957
1.385 I.609 1.542 1.419 3.054
23.534 23.758 23.691 23.568 32.804
0.717 0.724 0.722 0.718 l+OO
0.724 0.733 0.725 0.734 lGO0
I.010 I.012 I@4 1.022
2.738 3604 3.907 3.479 - 1.277
24.829 25.695 25.998 25.570 29.019
0.856 0.885 0.896 0.881 l+KlO
0.829 0.858 0.877 0.859 I+00
0.968 0.970 0.979 0.975
2.682 2.920 3.487 3.399 2.351
24.782 25.020 25.587 25.399 31.806
0.779 0.787 0.804 0.802 1GOCl
0.757 0.775 0.786 0.783 1.000
0.972 0.985 0.978 0.976
2.359 2.426 2605 2.495 2.655
24.447 24.514 24.693 24.538 32.115
0.761 0.763 0.769 0.765 lM0
0.754 0.768 0.773 0.768 I GO0
0991 1 Ml7 lMJ0 lNl4
2.335 3.022 3.445 4.083 0030
24.945 25.632 25.055 26.693 29.750
0.838 0.862 0.876 0.897 IWO
0,824 0.852 0.882 0.889 1NlO
0.983 0.988 1 xl07 0991
29663
32.9
2.38
1.67
3.72-42.3 3.13-28.6 2.80 - 10.3 2.80 10.3 In bubble
33 .o
3.08
2.33
3.40-54.0 266-41.2 2.14-20.6 2.02 7.1 In bubble
22J)OO
3.33-77.0 2.37-71.6 1.46-59.0 0.79 18.4 In bubble
22.027
32.1
3.67
2.50
29.283
29.333
29.171
13
33.4
3.22
2.08
3.91-39.8 3.35 - 26.6 3.049.5 3.04 9.5 In bubble
13A
33.4
2.86
1.72
3.75 -36.9 3.25 - 22.6 3.01-4.8 3.09 14.0 In bubble
22.152
3.35 - 26.6 30-9.5 3.04 9.5 3.35 26.6 In bubble
22.149
13AR
14
148
14BC
15
33.6
32.9
32.9
32.8
31.7
3.29
1.42
22.156
29.752
29.746
29.750
3.56
1.88
2.35 -58.0 160-38.7 1.25 0 1.60 38.7 In bubble
22.091
3.15
1.75
3.36-42.0 3.05-35.0 2.61- 16.7 5.7 2.51 In bubble
22.100
3.01-42.6 246 - 24.0 2.25 0 2.46 24.0 In bubble
22.088
3.35 -63.4 2.50-53.1 1.80-33.7 I.50 0 In bubble
22.610
3.21
3.79
1.88
2.09
Theoretical pressure PT
Hydrostatic pressure (mm. He)
30.296
29.455
29460
29.750
2236
Pfor
Pb
p.
pr p.
The pressure field around a two-dimensional gas bubble
0 Experimental
Davidson
Distance
data
PI/&
go
theory
from stagnation
point,radii
of curvature
Fig. 3. Normalized dynamic pressure along the vertical axis of symmetry (x/a = 0).
I
I
I
I,O-
$0.5’
0 Experimental
I
data P,/p,ga
I
I
I
I
I
1
A
0
0
OExperimental DavIdson
e -0
: 5
-0.4-
f n
-0.6-
P .I! 0 : c
-
thewy
I
0
a i
doto P&w
-0
2--
2-04-
-0.6
-10
-I.O-
L
0
-1.4 4
I 2 Distance
I 0
I -2
I
2
Distance
0
-1.2-
4
I
0 -4
I -6
I
0
I
-2
I
-4
from the stagnation pomt.mdli
I -6
-8
of curvature
Fig. 5. Normalized dynamic pressure along a line parallel to the vertical axis of symmetry (x/a = 1).
-6
fmm the stagnation polnt,radu of curvature
Fig. 4. Normalized dynamic pressure along a line parallel to the vertical axis of symmetry (da = 0.5).
bubble’s vertical axis of symmetry. These lines starting with the axis of symmetry (x = 0) are half a radius of curvature apart. On each plot, the data points are compared with the dynamic pressure (cos 0/s) predicted from Davidson’s theory. Pressures predicted from Jackson’s theory are not shown because they obviously do not fit the data.
Above the center of pressure line, the agreement between theory and experiment is good but the pressure rise in the nose is slightly less than predicted by theory. Below the center of pressure line, the agreement is much poorer becoming progressively worse as the distance from the axis of symmetry increases. The magnitude of the dynamic pressure below the center of pressure line is always greater than that predicted by the Davidson theory for about six radii below the stagnation point.
2237
H. LITTMAN I
Ei
I
I
and G. A. J. HOMOLKA
I
Vertical
I
$1.5
$
06-
9”
0.4 -
0 Experimental 0
of
0 4.0
pd/pbgu
data
axis
symmetry
I I
o"
E ‘C
-2.0
-
-4.0
-
0
4
I
I
2
0
I -2
I
I
-4
-6
I
: E z
Distance
from the stagnation
point, radii
of curvature
H v) Q) f
Fig. 6. Normalized dynamic pressure along a line parallel to the vertical axis of symmetry (x/a = 1.5).
g
$
I
I
I
E ~e
I
I +.2
04-
0
Q!
0 Experimental
data
pd/pbgo
J
8 -lO.O-
0
-0.4 -
0
6 j
-CO-
b := 0
E! 2 2 h .2 E e
-6.O-
a 2 S :
0 4
-06,-
2
0
of curvature from the vertical axis of symmetry
Radii
Distance
from
the stagnation
point, radii
of curvature
Fig. 7. Normalized dynamic pressure along a line parallel to the vertical axis of symmetry (x/a = 2).
It can be seen from the curves that the center of pressure for the bubble is less than one radius of curvature below the stagnation point in agreement with the findings of other investigators [ 11, 12,161. It can also be seen that the vertical distance between the bubble’s center of pressure and its stagnation point is approximately independent of the horizontal distance from its axis of symmetry. This is good evidence for the assumption that the center of pressure line is horizontal. In Fig. 8, the pressure field data are plotted in terms of radii of curvature from the stagnation point against radii from the axis of symmetry.
Fig. 8. Normalized dynamic pressure bubble -experimental.
field around
the
The pressure field above the center of pressure line shows dynamic pressures which are positive and the pressure gradient is largest near the bubble boundary. Below the center of pressure line, dynamic pressures are negative and there is a pressure minimum two bubble radii below the stagnation point and half a bubble radius off the vertical axis of symmetry. The presence of this minimum shows as expected that the pressure field in the wake cannot be Laplacian. It is interesting to note that the isobars follow the shape of the bubble floor for about three bubble radii of curvature below the floor. The pressure wake is much larger than the particle wake extending 12 radii below the stag-
2238
The pressure field around a two-dimensional gas bubble
possible to substantiate some of the assumptions made in that work. The location of the bubble stagnation point (apex of the roof), the bubble floor and the sur(c) The bubble rise velocity face of the bed are plotted with respect to time in The position of the bubble stagnation point, Fig. 9 for a bubble whose stagnation point trajecobtained from the film strip frames, was plotted against the interpolated clock time and the rise tory passes over or close to the transducers. Also shown are the corresponding pressure-time velocity determined from the slope of that plot. Using the experimentally determined radius of curves recorded by two transducers, one located three inches vertically above the other. curvature, the value of Vd (ga) l/2 was calculated in each run. The data are given in Table 4. On injection, the bubble displaces its own The average rise velocity was O-45(ga) 1’2 volume of fluidized particles raising the surface of the bed. An increase in pressure is immediately which compares well with the theoretical value sensed by the transducers demonstrating that of 0.5(g~)~‘~[l], the experimental data of Pyle and Harrison [ 151 and previous measurements by point 0 on the pressure-time curve represents the the authors [ll]. Note in Table 4 that only 2 of beginning of bubble injection. The injection lasts about O-1 set (in agreement with Partridge and the 10 data points fall outside the limits of scatter Lyall [ 141) and in that period, the bubble roof and of Pyle and Harrison’s data. floor move in opposite directions relative to each (d) Distinguishable points on the pressure-time other. The upper surface of the bed rises to a curve along the vertical axis of symmetry stable height increasing the hydrostatic pressure In a previous paper [ 111, the authors identified seven distinguishable points on the pressuretime curve along the vertical axis of symmetry and used that information to measure bubble rise velocities. Synchronization of the pressure-time curves with the motion picture films makes it nation point and four radii from the vertical axis of symmetry. The wake is closed.
Table 4. The bubble rise velocity-4454p spherical glass particles- density 248 gm/c3
Run No. 1.5 16 RR 18 AAA RR 27 35 36 37 42 44 48
Bubble rise velocity (in./sec)
Bubble radius of curvature (in.)
r Injection
ud(gaY.5
9.2 13.2
1.8 1.9
0.35 o-49
:
13.4 8.9 12.6
2.0 2.0 1.7
0.48 O-32 0.49
e a
8.4 10.0 12.5 16.1 11.2
2.0 1.4 1.7 2.0 1.9
O-32 044 0.49 0.58 0.41
h .o E z 0”
Average this work 0.45. Collins [ 11-Theoretical 0.50. Pyle and Harrison [15]-Experimental upper limit of scatter 0.67, Lower limit of scatter 0.35. Littman and Homolka [ 111-Experimental 0.53.
, point.
I”
Ol
-6
,
nl/
1
A\
Pitran
’
I
‘A\/
,
I
I
2
t
1
i
I
0 Time
from
I
I
0.4
0.6
the start
I
I
I.2
I.6
of injectian,
1
SBC
Fig. 9. The relation of the bubble stagnation point, bubble floor and the surface of the bed with respect to the distinguishable points on the pressure-time curve.
2239
H. LITTMAN
and G. A. J. HOMOLKA
at all points in the bed. Although this change is felt both at the measuring and reference sides of the transducer equally the measuring side responds much faster causing positive dynamic pressures to be recorded between 0 and 1. The transducer output goes through a minimum at point 2t as the increase in hydrostatic pressure is registered on the reference side. Well before point 3, however, the injection transient has died out and the dynamic pressure increases monotonically to a maximum at point 3. There the bubble nose has reached the transducer. Since the pressure inside the bubble is constant, the dynamic pressure begins a linear decrease as the bubble rises past the transducer. This decrease continues until the bubble floor passes the transducer (point 5). When points 3 and 5 are recorded on the pressure-time curve, the bubble boundary is within a l/4 in. of the lower edge of the transducer. Point 4 records the time the center of pressure of the bubble reaches the transducer tap. The results found for points 3 through 5 agree well with those given by Reuter [ 161. Pressure recovery begins as soon as the bubble floor has passed the transducer but before point 6 is recorded, the upper surface of the bed begins to deform. The bubble is about three radii below the upper surface of the bed when the deformation begins. Note in Fig. 9 that at that point the bubble roof begins to accelerate but that the bubble floor decelerates causing the bubble to elongate. The acceleration of the roof continues until it bursts through the surface throwing the particles outward. The bubble floor is then about one radius of curvature below the surface of the bed and the bubble pressure is reduced to atmospheric. The floor then decelerates rapidly coming to rest at the upper surface of the bed. Point 6 occurs when the bubble stagnation point has reached the level the upper surface was at before it was disturbed by the approaching bubble. The pressure increase following point 6 is caused by the increased hydrostatic pressure tNegative dynamic pressures are not always recorded.
resulting from the particles above the bubble roof falling back onto the bed. The recovery is completed by the collapse of the bubble cavity returning the bed to its incipiently fluidized state (point 7). These results agree with previous work by the authors except for point 6 which they identified as occurring when the bubble begins to distort the upper surface of the bed (taken as occurring, when the stagnation point is about 3/4 of a bubble radius from the surface of the bed). This incorrect identification of point 6 does not affect their rise velocities significantly. DISCUSSION
The total pressure field above the center of pressure line is Laplacian and reasonably well described by Davidson’s theory. Jackson’s method of calculating the pressure field, on the other hand, predicts dynamic pressures that are much too low and his total pressure field is not Laplacian. The failure of Jackson’s method makes it doubtful that particle momentum and gas pressure changes are linked as Stewart assumes [ 171. Apparently it is the interphase drag force in the dense phase and not the inertial force caused by the flow of particles around the bubble which determines the local pressure gradient. The behavior of large air bubbles (14 and 2 in. radius of curvature) rising in water is quite different. In that case the inertial force is important and the frontal pressure field computed assuming irrotational flow around an oval shaped body (bubble and wake) agrees well with the experimental pressure field. Above the center of pressure line, Reuter’s data for 3D half bubbles also follow Davidson’s theory but Stewart[ 171 has noted that Reuter’s actual bubble rise velocity corresponds to (g~)“~. This is 50 percent faster than the 3D Davies and Taylor value [5]. Inserting Reuter’s experimental value into Jackson’s equation raises the pressure ahead of the bubble considerably improving the agreement between the theory and the data. But even with this change, Davidson’s equation provides the best fit to Reuter’s data (see Fig. 8 of Ref. [ 171).
2240
The pressure field around a two-dimensional gas bubble
Since the bubble velocity in this work is close to the 2D Davies and Taylor value and follows Davidson’s theory there is no possibility of justifying a relationship between particle momentum and the pressure field based on a fit of Jackson’s equation. The fact that Davidson’s theory fits both Reuter’s data and that reported in this paper despite considerable variation in the value of Ub2/ga relative to the Davies and Taylor value indicates that the pressure field is independent of the bubble velocity. It is not obvious why Davidson’s theory works but the reason may lie in the particle motion above the bubble roof. We have observed particle trajectories similar to those reported by Gabor [6], that is, vertical movement of the particles seems to be more extensive than horizontal movement and there are particles falling through the roof. There does not seem to be much horizontal relative motion between the particles and bubble. Perhaps the net effect of all this is to make Darcy’s law a good approximation of the local pressure gradient, Apfi Some caution in using the results are indicated as a result of the fact that these experiments were performed using spherical glass particles of a single size. However, the fit of all literature data by Davidson’s theory indicates the results are general. The major differences between Reuter’s data and that found in this work are in the wake. Those differences are: (1) Dynamic pressures are observed up to 12 radii below the stagnation point whereas Reuter’s wake ended 4 radii down, (2) a pressure minimum is observed in the wake (Fig. 8) whereas Reuter found no maxima or minima, (3) pressure magnitudes in the wake, 2- 10 bubble radii below the stagnation point, are larger than predicted by Davidson’s theory whereas Reuter’s pressures are lower than predicted. These results show that the wakes in the two cases are different. Apparently a three diniensional half bubble rising along a wall has a more restricted wake than that of a bubble rising normally. Davidson and Harrison in their book[3] de-
fined a ‘Bernoulli’ pressure, P, which is the same as that calculated by Jackson and their pressure, pr, is the one calculated using the Davidson theory. The relationship between these two quantities is defined as P = p,+pp where pp is the interparticle pressure. It is easy to show that if the bubble rises at the Davies and Taylor velocity then P is always less than pf. In order to equate P and pf at the stagnation point it is necessary to raise the bubble velocity to (2ga)‘12.t Since this is nearly three times the theoretical rise velocity, it is apparent that P is always less than pf making pp negative. Following Davidson and Harrison [33 this would imply tensile forces between the particles which must be interpreted as meaning that the particles separate increasing the local voidage [ 181. Experimental evidence for this is given by Lockett and Harrison [ 191. Perhaps p does tell us something about the interparticle pressure and the local voidage near a bubble but it is really a defined quantity unrelated, as far as the writers can tell, to any measured pressure. The rise velocities determined using high speed motion pictures agrees well with the authors’ previous method based on the use of the pressure-time curve along the vertical axis of symmetry. Since the bubble accelerates when its stagnation point is within three radii of curvature of the surface of the bed, the velocities reported do not apply in that region. The acceleration of the bubble roof which occurs as the bubble approaches the upper surface of the bed, can be explained by considering the pressure field which results from the interaction of a bubble with a free surface. The pressure is constant and above atmospheric throughout the bubble cavity; it is also decreasing as the bubble approaches the surface of the bed so that the pressure drop across the particulate layer, between the bubble roof and the upper surface of the bed is also decreasing as the bubble rises through the bed. As the bubble approaches the upper surface
2241 CES Vol. 28. No. I2 - I
IThe same value applies for a 3D bubble [3].
H. LITTMAN and G. A. J. HOMOLKA NOTATION
of the bed the average pressure gradient across the particulate layer between the stagnation point and the upper surface of the bed becomes increasingly greater than at incipient fluidization. The forces on the particulate layer above the roof force the layer upward until the roof bursts.
a 4 Pa Pb
CONCLUSIONS
Pd
1. The total pressure field above the center of pressure line of a single bubble in a two dimensional fluidized bed is Laplacian and follows Davidson’s theory reasonably well. Although the experiments were performed using spherical particles of a single size, the results are believed to be general. 2. The pressure wake is closed. It extends 12 bubble radii below the stagnation point and four radii on either side of the vertical axis of symmetry. 3. A pressure minimum is found in the wake two bubble radii below the stagnation point and half a bubble radius from the vertical axis of symmetry. This shows that the wake is not Laplacian. 4. Dynamic pressures in the field below the center of pressure line are greater in magnitude than predicted by Davidson’s theory for up to 8 bubble radii below the floor. 5. The rise velocities determined using high speed motion pictures agree with those based on the use of the pressure-time curve along the vertical axis of symmetry. 6. When the bubble is within 3 radii of the upper surface of the bed, it elongates because the roof accelerates while the floor decelerates.
Pf Ph
PP
P Pll PT
r R
u: umf X
Y
bubble radius of curvature gravitational acceleration distance of center of pressure line below top of bed total pressure above bed total pressure inside bubble dynamic pressure in field = pf - ph total fluid pressure in field hydrostatic pressure in field interparticle pressure defined by Davidson and Harrison [3] “Bernoulli” pressure defined by Davidson and Harrison [3] measured dimensionless interstitial pressure, pf/pb theoretical analogue to P,, calculated assuming Laplacian field distance from center of bubble to a point in the field distance from stagnation point to a point in the field r/a bubble velocity minimum fluidizing velocity horizontal distance coordinate vertical distance coordinate
Greek symbols pb bulk density of bed
Acknowledgement-The authors are grateful to the Atomic Energy Commission for their support of this work under Grant No. AT(30-l)-3639.
0 angle off vertical axis of symmetry (origin of coordinates - center of bubble) + angle off vertical axis of symmetry (origin of coordinates - stagnation point of bubble) cr standard deviation.
REFERENCES ill COLLINS R., Chem. Engng Sci. 1964 20788. I21 DAVIDSON J. F., Trans. Znstn Chem. Engrs 196139 320. I31 DAVIDSON J. F. and HARRISON D., Fluid&d Particles. Cambridge University Press, London 1963. [41 DAVIDSON J. R., PAUL R. C., SMITH M. J. S. and DUXBURY H. A., Trans. Insrn Chem. Engrs 1959 37 323. [51 DAVIES R. M. and TAYLOR G. I., Proc. R. Sot. 1949 A200 375. [6] GABOR J., Proceedings of the Znternafional Symposium 011Fluidization, Eindhouen, p. 230. Netherlands University Press, Amsterdam 1967. [71 HOMOLKA G. A. J., Ph.D. dissertation, Rensselaer Polytechnic Institute, Troy, N.Y. 1971. 181 ISAACSON E. and KELLER H. B., Analysis ofNumericalMethods. Wiley, N.Y. 1966.
2242
The pressure field around a two-dimensional
gas bubble
[9] JACKSON R., Trans. Instn Chem. Engrs 1963 4122. [lo] LAZAREKG. M., Ph.D. dissertation, Rensselaer Polythecnic Institute, Troy, N.Y. 1972. [l I] LITTMAN H. and HOMOLKA G. A. J., Chem. Engng Prug. Symp. Ser. No. 105 1970 66 37. 1121 MOTAMEDI-LANJANI M., Ph.D. dissertation, Imperial College, London 1970. [13] MURRAY J. D.,J. FluidMech. 1965 22 57. 1141 PARTRIDGE B. A. and LYALL E., J. Fluid Mech. 1967 28429. 1151 PYLE D. L. and HARRISON D., Chem. Enana Sci. 1967 22 531. [16] REUTER H., Chemie Inger. Tech. 1963 35 Sk[17] STEWART P. S. B., Trans. Insfn Chem. Engrs 1968 46T60. [181 STEWART P. S. B. and DAVIDSON J, F., Powder Tech. 1967 161. [19] LOCKETT M. J. and HARRISON D., Proceedings ofthe International Symposium on Fluidizatiun, Eindhooen, p. 257. Netherland University Press, Amsterdam 1967.
2243
Vol. 28, pp. 2231-2243.
Pergamon Press.
Printed in Great Britain
The pressure field around a two-dimensional fluidized bed
gas bubble in a
HOWARD LITTMAN and GEORGE A. J. HOMOLKAt Rensselaer Polytechnic Institute, Troy, N.Y., 12181, U.S.A.
(First received 16 August 1972; in revisedform
5 February
1973)
Abstract-Measurements of the entire pressure field around a two-dimensional bubble rising in an incipiently fluidized bed are reported. Above the center of pressure line, the field is Laplacian and reasonably well described by Davidson’s theory. The pressure wake is closed. It extends for 12 bubble radii below the stagnation point and four radii on either side of the vertical axis of symmetry. INTRODUCTION PRESSURE field measurements around an isolated bubble in an incipiently fluidized bed have been used previously for studying flow patterns near bubbles and for testing the theoretical models which describe those flows. ’ Reuter [ 161 measured the pressure field around a three dimensional half bubble rising along a plane wall and his data along the vertical axis of symmetry have been compared with theory by Stewart[ 171. Stewart’s analysis showed that in front of the bubble there was good agreement with Davidson’s theory [2, 33 and that the Murray [ 131 and Jackson [9] theories forecast dynamic pressures there which are much too low. Below the bubble Reuter’s dynamic pressures, although lower in magnitude than Davidson’s theory predicts, are in fair agreement with that theory. More recently, Littman and Homolka [ 111 measured the pressure field along the vertical axis of symmetry of a two dimensional bubble. Their results are in general agreement with the Davidson theory except below the bubble where the magnitude of the dynamic pressure exceeds that predicted. Those results were confirmed by Motamedi[12] who, in addition, found that the pressure field extended downward along the axis of symmetry for about 8 bubble radii below the floor. These two studies are in general agreement with Reuter except in the wake where apparently Reuters pressure field is modified by having the bubble rise along the wall. The magnitude of his
dynamic pressure and the extent of his bubble’s wake are both less than found by Littman and Homolka and by Motamedi. Measurements of the entire pressure field around a two dimensional bubble are reported for the first time in this paper. The data are compared with theory and discussed with regard to the nature of the pressure field and in relation to the forces which cause the local pressure gradient. The boundary of the entire detectable dynamic pressure field is shown. EXPERIMENTAL DETERMINATION OF THE NATURE OF THE PRESSURE FIELD ABOVE THE CENTER OF PRESSURE LINE
All experimental pressure field data above the center of pressure line of a rising bubble follow Davidson’s theory reasonably well. Since that theory predicts that the pressure field is Laplacian, studies were undertaken to test this prediction experimentally for a two dimensional bubble of arbitrary shape. Davidson’s pressure field is Laplacian because Darcy’s law is assumed to hold in the dense phase. The Jackson and Murray theories, on the other hand, are not Laplacian because the pressure gradient is related to the particle phase inertial force. Table 1 gives the pressure field equation for each theory. Davidson’s solution to Laplace’s equation is obtained analytically by considering an exterior problem whose boundary conditions are defined
tPresent address: Western Co. of North America, Forth Worth, Texas 76101, U.S.A.
223 1
H. LITTMAN
and G. A. J. HOMOLKA
Table 1. Comparison of expressions for the dynamic pressure due to the presence of a two-dimensional bubble in a fluidized bed
Theory
Field
Davidson
Laplacian
Pressure field equation
fi+,c,,
e=y
-&+scosp=~$
[
1-2sinzB-&
Field not Laplacian
Jackson
1 1
with iJb = i (ga) I’* Field not Laplacian
Murray
-&+scose=~
cos 28
Origin of the coordinates is the center of the circular bubble. 0 = 0 at the bubble stagnation point.
at the surface of the bubble and at infinity. The bubble is considered to be perfectly circular and the medium infinite in extent. Since experimental bubbles are irregularly shaped and the bed finite, testing the Laplacian nature of the pressure field
for arbitrarily shaped bubbles requires not only the measurement of the field but the formulation and numerical solution of a boundary value problem for Laplace’s equation. That boundary value problem is specified in Fig. 1.
_
Surface of bed
v2p,=o
Bubble
P‘
‘Pa
=P,t(l-_b)p*
Center
pressure 47 H*
of
llne
Fig. 1. Boundary value problem from which the theoretical pressure distribution around a bubble was calculated.
2232
The pressure field around a two-dimensional gas bubble Surface
At the upper surface of the bed, the pressure is atmospheric and motion pictures have shown that this surface remains flat as long as the bubble is more than three radii of curvature below the surface. The conditions along the vertical sides of the bed must be hydrostatic providing the bed is wide enough and this was easily verified experimentally by placing a transducer two inches from the outer edge of the bed. The pressure along the lower boundary of the region is constant if that boundary is composed of the center of pressure line and the bubble boundary. The distance of the center of pressure line below the surface of the bed is obtained from its definition. Thus
(1) The pressure inside the rising bubble, pb, required in Eq. (1) was measured using a single transducer positioned inside the bubble boundary at the instant the bubble was photographed (see Fig. 2a). This photograph gives the bubble size, shape and location in the bed and completes the specification of the boundary value problem. The pressure field can then be calculated and compared with the experimental one. The experimental field is obtained using four transducers set in a horizontal line one inch apart as shown in Fig. 2a. Their outputs together with
I--
-Time
Fig. 2a. Bubble and transducer locations for experimental measurement of the pressure field around a bubble.
that of the pressure transducer located inside the bubble were recorded continuously on a Honeywell 906C Visicorder from the time the bubble was injected. A sample of the transducer outputs appear in Fig. 2b. The photograph of the rising bubble and the Visicorder record must be synchronized in time and this is accomplished by recording on the chart the time interval over which the camera shutter is open (l/SO0 set). Experimentally this was done by placing two photo cells inside the
increasing
I
Carnero photocells
I
’ Time of bubble photograph
Fig. 2b. Sample of visicorder record-O gives the pressure at each transducer at the instant the bubble was photographed. Location of each transducer in field is shown in Fig. 2a.
2233
H. LI-ITMAN
and G. A. J. HOMOLKA
camera on either side of the photographic plate and recording the amplified signal from the photo cells on the Visicorder chart (top trace in Fig. 2b). The pressure at each location in the field at the time the photograph was taken is indicated by the circle on each pressure-time trace. The distance of the bubble from the various transducers at that time is easily obtained from the photograph. Repeating this procedure for many different bubbles varying the distance between the. bubble and the transducers gives the data required to test the field.
NUMERICAL
SOLUTION EQUATION
OF LAPLACE’S
The numerical solution of the boundary value problem is approximated by a relaxation method
PI. The method by which the relaxation is carried out was chosen on the basis of stability and the speed of convergence. It was found that an S.O.R. (Successive Over-Relaxation) iteration permitted the use of acceleration factors from l-6 to 1.9, depending on the size of the matrix, while maintaining its stability. This assured smooth and rapid convergence. The relaxation procedure gives only an approximate solution to the boundary value problem. An exact solution is obtained by means of Richardson’s extrapolation scheme which consists of solving the problem for several different grid spacings (h) and extrapolating to h = 0. pf (approximate)
= pr (exact) +f( h )
where f(h) = a,h2+a,h4+-
--
The advantage of this method is that, as the grid becomes finer, error due to the bubble shape approximation is diminished and extrapolation need only be done on those points which are of interest. Grid spacing of l/4 X l/4 in. and l/8 X l/8 in. were used in this study. The round off error was about 1.5 per lo6 per iteration. Details of the calculations are given by Homolka [7].
MAPPING
THE
ENTIRE
PRESSURE
FIELD
The experimental technique used for mapping the entire field is similar to that described for the pressure field above the center of pressure line except for the use of high speed motion pictures of the bubble instead of single photographs. Synchronization of the Visicorder traces and the film was achieved by placing a high speed clock in the field of view.
Equipment The two dimensional fluidized bed used was 57 in. high, 20 in. wide and l/2 in. thick. It was constructed from l/2 in. thick aluminum plates which were reinforced by steel structures to limit the maximum outward deflection of the plates due to hydrostatic pressure to O-002 in. In the front plate was a coated conductive glassPlexiglas window, 29 in. high and 9 in. wide, through which the bubbles were photographed. The air distributor was a sintered metal plate with a 20 p average pore size. The fluidizing air was humidified above 60 per cent relative humidity and this together with the grounded aluminum walls of the bed and conductive glass window prevented electrostatic charge buildup between the walls and the particles. The bubble injection system was similar to that of Davidson et al.[4]. The pressure transducers and the bubble injection tap were located in the rear plate. The field transducers were 12 in. above the injection tap, 3 in. above the bubble transducer and approximately 9 in. below the top of the bed. Two types of pressure transducers, an MKS Baratron pressure meter with a Type 77 head, and a Pitran (PT-3) transducer were used in this investigation. The Baratron is a secondary pressure standard and it was used to calibrate the Pitt-arts. The Baratron pressure head is a very sensitive capacitance transducer and the meter has a 5 place digital readout for 8 full scale pressure ranges from O-01 to 30 mm Hg. The instrument was calibrated against a deadweight tester to 0.01 percent of the actual pressure for any range used.
2234
The pressure field around a two-dimensional
The frequency response of the Baratron is greater than 20 Hz. The Pitran transducer consists of a planar silicon NPN transistor with a stress sensitive emitter-base junction which is connected to a diaphragm. It was mounted in a holder and placed in the wall of the bed. The positive pressure side of the transducer was covered with a 100 mesh screen to protect the diaphragm from direct particle contact because the transducer has a very high frequency response (150,000 Hz). The noise caused by the particle impact on the diaphragm in the absence of the screen would obscure the pressure signals. The Pitrans have a linear operating range from 0 to 26.5 mm Hg and a maximum output sensitivity of 143V/mm Hg using a 20 V power supply. Single photographs of the bubble were made with a 4 X 5 in. plate camera equipped with a Schneider Symmar lens and a Synchro-Compur shutter. The 4 X 5 in. negative was enlarged to make an 8 x 10 in. print for use in measuring the distances between the bubble nose and the transducers. Most of the motion picture data were obtained using a 16 mm Hycam high speed camera operating at a speed of 188.6 frameslsec. Two copies of the motion picture film were made. The original was projected and examined visually. The duplicate copy was cut up and examined frame by frame in a microfilm reader (enlargement factor = 25). The changes in the bubble’s size, shape, aed position which occurred during the run were obtained from every fifth frame of the mounted film strip. Details of the equipment are given by Homolka [7]. Bed and particle data are given in Table 2.
gas bubble Table 2. Bed and particle data Property
Shape Mesh size Particle diameter ( p) Density C,gm/c3 Experimental U,(ft/min) Voidage at U, Reat UN Bed height at U.&in.) Bed weight (lb)
RESULTS
The Laplacian character of the pressurefield above the center of pressure line
Eleven runs were made to test the nature of the pressure field and the results are given in Table 3. Distances from the stagnation point, R, were varied from O-79 to 3.91 in. for bubbles whose radius of curvature were of the order of two inches. The angle, * was varied from 0 to 77”.
round 30140 454 248 31.5 0.384 4.3 42.5 27.885
P,, the measured dimensionless interstitial fluid is defined as the total pressure pressure, (dynamic + hydrostatic) at a point in the field divided by the total pressure inside the bubble. PT is the analogous quantity to P, calculated assuming the pressure field is Laplacian. For the 44 data points listed in Table 3, the mean value of P,lP, is 0994 and the standard deviation, u, is 0.02 about the mean. The adequacy of this test of the field depends on the ratio of the dynamic to hydrostatic pressures since the dynamic pressure is not a large fraction of the total. The first and fourth points in Run 15 provide a representative test of the magnitude of the difference between a Laplacian and a Jacksonian field for the data presented. For those points, the dynamic pressures calculated from Jackson’s equation are O-270 and O-676 mm Hg respectively and this makes the corresponding values of P,/P, O-920 and O-880, well beyond the 3a limit or the maximum deviations found experimentally. Thus the pressure field above the center of pressure line is Laplacian.
(b) The entire pressure Davidson’s
(a)
Glass
field-
comparison
with
theory
The entire pressure field was examined for 5 radii of curvature vertically above and 15 radii below the center of the pressure line. It was also studied for 5 radii of curvature on either side of the vertical axis of symmetry of the bubble. The results of the investigation are presented in Figs. 3-7 as plots of the normalized dynamic pressure, pJp*ga, vs. distance from the stagnation point to a point along a line parallel to the
2235
H. LITTMAN
and G. A. J. HOMOLKA
Table 3. Comparison of the measured pressure above the bubble roof with a Laplacian pressure field- glass particles 454~
Run number 6
I
1OA
1lA
Superficial velocity Wmin) 32.5
Bubble height W 3.38
Bubble radius of curvature W 1.91
Transducer position rel. to bubble stag. point R ti (in) (de@
Dynamic pressure (mm. Hg)
(uu. H@
3.01-41.6 246-24.0 2.25 0.0 2.46 24.0 In bubble
22.544
2.338 2.530 2.598 2+&l 1404
24.882 25.074 25.142 25.204 31.067
0.801 0.807 0.809 0.811 l+OO
0.789 0.816 0.824 0.816 lJJO0
0996 1.011 1.019 lGO6
2 1.963
2.319 1.945 2~100 2.261 2.435
24.282 23908 24.063 24.224 31.718
0.766 0.754 0.759 0.764 lwl
0.742 0.753 0.759 0.758 lW0
0.969 0999 l+Oo 0.092
1.555 I.939 2.087 2.353 -0.119
23.555 23.939 24.087 24.353 29.214
0.806 0.819 0.834 0.834 lXlO0
0.812 0.832 0.846 0.844 1NNJ
1.007 1.016 1.014 1.012
1.737 2.781 3.551 3.412 - 2.598
23.764 24.808 25.578 25.439 26.573
0.894 0.937 0.963 0.957 lGO0
0.881 0.921 0.977 0995 IWO
0.985 0.983 I.015 lW_l
0.947 1Ml I.156 I.346 2.505
23.103 23.216 23.312 23.502 32.257
0.716 0.720 0.723 0.729 lM0
0.716 0.723 0.727 0.726 l+lOO
1xKm lGO4 1.006 0996
2.228 2.820 2.942 2.786 2.568
24.380 24.972 25.094 24.938 32.3 14
0.753 0.774 0.777 0.773 lJJO0
0.730 0.746 0.741 0.740 l+lOO
0.969 0964 0.954 0.957
1.385 I.609 1.542 1.419 3.054
23.534 23.758 23.691 23.568 32.804
0.717 0.724 0.722 0.718 l+OO
0.724 0.733 0.725 0.734 lGO0
I.010 I.012 I@4 1.022
2.738 3604 3.907 3.479 - 1.277
24.829 25.695 25.998 25.570 29.019
0.856 0.885 0.896 0.881 l+KlO
0.829 0.858 0.877 0.859 I+00
0.968 0.970 0.979 0.975
2.682 2.920 3.487 3.399 2.351
24.782 25.020 25.587 25.399 31.806
0.779 0.787 0.804 0.802 1GOCl
0.757 0.775 0.786 0.783 1.000
0.972 0.985 0.978 0.976
2.359 2.426 2605 2.495 2.655
24.447 24.514 24.693 24.538 32.115
0.761 0.763 0.769 0.765 lM0
0.754 0.768 0.773 0.768 I GO0
0991 1 Ml7 lMJ0 lNl4
2.335 3.022 3.445 4.083 0030
24.945 25.632 25.055 26.693 29.750
0.838 0.862 0.876 0.897 IWO
0,824 0.852 0.882 0.889 1NlO
0.983 0.988 1 xl07 0991
29663
32.9
2.38
1.67
3.72-42.3 3.13-28.6 2.80 - 10.3 2.80 10.3 In bubble
33 .o
3.08
2.33
3.40-54.0 266-41.2 2.14-20.6 2.02 7.1 In bubble
22J)OO
3.33-77.0 2.37-71.6 1.46-59.0 0.79 18.4 In bubble
22.027
32.1
3.67
2.50
29.283
29.333
29.171
13
33.4
3.22
2.08
3.91-39.8 3.35 - 26.6 3.049.5 3.04 9.5 In bubble
13A
33.4
2.86
1.72
3.75 -36.9 3.25 - 22.6 3.01-4.8 3.09 14.0 In bubble
22.152
3.35 - 26.6 30-9.5 3.04 9.5 3.35 26.6 In bubble
22.149
13AR
14
148
14BC
15
33.6
32.9
32.9
32.8
31.7
3.29
1.42
22.156
29.752
29.746
29.750
3.56
1.88
2.35 -58.0 160-38.7 1.25 0 1.60 38.7 In bubble
22.091
3.15
1.75
3.36-42.0 3.05-35.0 2.61- 16.7 5.7 2.51 In bubble
22.100
3.01-42.6 246 - 24.0 2.25 0 2.46 24.0 In bubble
22.088
3.35 -63.4 2.50-53.1 1.80-33.7 I.50 0 In bubble
22.610
3.21
3.79
1.88
2.09
Theoretical pressure PT
Hydrostatic pressure (mm. He)
30.296
29.455
29460
29.750
2236
Pfor
Pb
p.
pr p.
The pressure field around a two-dimensional gas bubble
0 Experimental
Davidson
Distance
data
PI/&
go
theory
from stagnation
point,radii
of curvature
Fig. 3. Normalized dynamic pressure along the vertical axis of symmetry (x/a = 0).
I
I
I
I,O-
$0.5’
0 Experimental
I
data P,/p,ga
I
I
I
I
I
1
A
0
0
OExperimental DavIdson
e -0
: 5
-0.4-
f n
-0.6-
P .I! 0 : c
-
thewy
I
0
a i
doto P&w
-0
2--
2-04-
-0.6
-10
-I.O-
L
0
-1.4 4
I 2 Distance
I 0
I -2
I
2
Distance
0
-1.2-
4
I
0 -4
I -6
I
0
I
-2
I
-4
from the stagnation pomt.mdli
I -6
-8
of curvature
Fig. 5. Normalized dynamic pressure along a line parallel to the vertical axis of symmetry (x/a = 1).
-6
fmm the stagnation polnt,radu of curvature
Fig. 4. Normalized dynamic pressure along a line parallel to the vertical axis of symmetry (da = 0.5).
bubble’s vertical axis of symmetry. These lines starting with the axis of symmetry (x = 0) are half a radius of curvature apart. On each plot, the data points are compared with the dynamic pressure (cos 0/s) predicted from Davidson’s theory. Pressures predicted from Jackson’s theory are not shown because they obviously do not fit the data.
Above the center of pressure line, the agreement between theory and experiment is good but the pressure rise in the nose is slightly less than predicted by theory. Below the center of pressure line, the agreement is much poorer becoming progressively worse as the distance from the axis of symmetry increases. The magnitude of the dynamic pressure below the center of pressure line is always greater than that predicted by the Davidson theory for about six radii below the stagnation point.
2237
H. LITTMAN I
Ei
I
I
and G. A. J. HOMOLKA
I
Vertical
I
$1.5
$
06-
9”
0.4 -
0 Experimental 0
of
0 4.0
pd/pbgu
data
axis
symmetry
I I
o"
E ‘C
-2.0
-
-4.0
-
0
4
I
I
2
0
I -2
I
I
-4
-6
I
: E z
Distance
from the stagnation
point, radii
of curvature
H v) Q) f
Fig. 6. Normalized dynamic pressure along a line parallel to the vertical axis of symmetry (x/a = 1.5).
g
$
I
I
I
E ~e
I
I +.2
04-
0
Q!
0 Experimental
data
pd/pbgo
J
8 -lO.O-
0
-0.4 -
0
6 j
-CO-
b := 0
E! 2 2 h .2 E e
-6.O-
a 2 S :
0 4
-06,-
2
0
of curvature from the vertical axis of symmetry
Radii
Distance
from
the stagnation
point, radii
of curvature
Fig. 7. Normalized dynamic pressure along a line parallel to the vertical axis of symmetry (x/a = 2).
It can be seen from the curves that the center of pressure for the bubble is less than one radius of curvature below the stagnation point in agreement with the findings of other investigators [ 11, 12,161. It can also be seen that the vertical distance between the bubble’s center of pressure and its stagnation point is approximately independent of the horizontal distance from its axis of symmetry. This is good evidence for the assumption that the center of pressure line is horizontal. In Fig. 8, the pressure field data are plotted in terms of radii of curvature from the stagnation point against radii from the axis of symmetry.
Fig. 8. Normalized dynamic pressure bubble -experimental.
field around
the
The pressure field above the center of pressure line shows dynamic pressures which are positive and the pressure gradient is largest near the bubble boundary. Below the center of pressure line, dynamic pressures are negative and there is a pressure minimum two bubble radii below the stagnation point and half a bubble radius off the vertical axis of symmetry. The presence of this minimum shows as expected that the pressure field in the wake cannot be Laplacian. It is interesting to note that the isobars follow the shape of the bubble floor for about three bubble radii of curvature below the floor. The pressure wake is much larger than the particle wake extending 12 radii below the stag-
2238
The pressure field around a two-dimensional gas bubble
possible to substantiate some of the assumptions made in that work. The location of the bubble stagnation point (apex of the roof), the bubble floor and the sur(c) The bubble rise velocity face of the bed are plotted with respect to time in The position of the bubble stagnation point, Fig. 9 for a bubble whose stagnation point trajecobtained from the film strip frames, was plotted against the interpolated clock time and the rise tory passes over or close to the transducers. Also shown are the corresponding pressure-time velocity determined from the slope of that plot. Using the experimentally determined radius of curves recorded by two transducers, one located three inches vertically above the other. curvature, the value of Vd (ga) l/2 was calculated in each run. The data are given in Table 4. On injection, the bubble displaces its own The average rise velocity was O-45(ga) 1’2 volume of fluidized particles raising the surface of the bed. An increase in pressure is immediately which compares well with the theoretical value sensed by the transducers demonstrating that of 0.5(g~)~‘~[l], the experimental data of Pyle and Harrison [ 151 and previous measurements by point 0 on the pressure-time curve represents the the authors [ll]. Note in Table 4 that only 2 of beginning of bubble injection. The injection lasts about O-1 set (in agreement with Partridge and the 10 data points fall outside the limits of scatter Lyall [ 141) and in that period, the bubble roof and of Pyle and Harrison’s data. floor move in opposite directions relative to each (d) Distinguishable points on the pressure-time other. The upper surface of the bed rises to a curve along the vertical axis of symmetry stable height increasing the hydrostatic pressure In a previous paper [ 111, the authors identified seven distinguishable points on the pressuretime curve along the vertical axis of symmetry and used that information to measure bubble rise velocities. Synchronization of the pressure-time curves with the motion picture films makes it nation point and four radii from the vertical axis of symmetry. The wake is closed.
Table 4. The bubble rise velocity-4454p spherical glass particles- density 248 gm/c3
Run No. 1.5 16 RR 18 AAA RR 27 35 36 37 42 44 48
Bubble rise velocity (in./sec)
Bubble radius of curvature (in.)
r Injection
ud(gaY.5
9.2 13.2
1.8 1.9
0.35 o-49
:
13.4 8.9 12.6
2.0 2.0 1.7
0.48 O-32 0.49
e a
8.4 10.0 12.5 16.1 11.2
2.0 1.4 1.7 2.0 1.9
O-32 044 0.49 0.58 0.41
h .o E z 0”
Average this work 0.45. Collins [ 11-Theoretical 0.50. Pyle and Harrison [15]-Experimental upper limit of scatter 0.67, Lower limit of scatter 0.35. Littman and Homolka [ 111-Experimental 0.53.
, point.
I”
Ol
-6
,
nl/
1
A\
Pitran
’
I
‘A\/
,
I
I
2
t
1
i
I
0 Time
from
I
I
0.4
0.6
the start
I
I
I.2
I.6
of injectian,
1
SBC
Fig. 9. The relation of the bubble stagnation point, bubble floor and the surface of the bed with respect to the distinguishable points on the pressure-time curve.
2239
H. LITTMAN
and G. A. J. HOMOLKA
at all points in the bed. Although this change is felt both at the measuring and reference sides of the transducer equally the measuring side responds much faster causing positive dynamic pressures to be recorded between 0 and 1. The transducer output goes through a minimum at point 2t as the increase in hydrostatic pressure is registered on the reference side. Well before point 3, however, the injection transient has died out and the dynamic pressure increases monotonically to a maximum at point 3. There the bubble nose has reached the transducer. Since the pressure inside the bubble is constant, the dynamic pressure begins a linear decrease as the bubble rises past the transducer. This decrease continues until the bubble floor passes the transducer (point 5). When points 3 and 5 are recorded on the pressure-time curve, the bubble boundary is within a l/4 in. of the lower edge of the transducer. Point 4 records the time the center of pressure of the bubble reaches the transducer tap. The results found for points 3 through 5 agree well with those given by Reuter [ 161. Pressure recovery begins as soon as the bubble floor has passed the transducer but before point 6 is recorded, the upper surface of the bed begins to deform. The bubble is about three radii below the upper surface of the bed when the deformation begins. Note in Fig. 9 that at that point the bubble roof begins to accelerate but that the bubble floor decelerates causing the bubble to elongate. The acceleration of the roof continues until it bursts through the surface throwing the particles outward. The bubble floor is then about one radius of curvature below the surface of the bed and the bubble pressure is reduced to atmospheric. The floor then decelerates rapidly coming to rest at the upper surface of the bed. Point 6 occurs when the bubble stagnation point has reached the level the upper surface was at before it was disturbed by the approaching bubble. The pressure increase following point 6 is caused by the increased hydrostatic pressure tNegative dynamic pressures are not always recorded.
resulting from the particles above the bubble roof falling back onto the bed. The recovery is completed by the collapse of the bubble cavity returning the bed to its incipiently fluidized state (point 7). These results agree with previous work by the authors except for point 6 which they identified as occurring when the bubble begins to distort the upper surface of the bed (taken as occurring, when the stagnation point is about 3/4 of a bubble radius from the surface of the bed). This incorrect identification of point 6 does not affect their rise velocities significantly. DISCUSSION
The total pressure field above the center of pressure line is Laplacian and reasonably well described by Davidson’s theory. Jackson’s method of calculating the pressure field, on the other hand, predicts dynamic pressures that are much too low and his total pressure field is not Laplacian. The failure of Jackson’s method makes it doubtful that particle momentum and gas pressure changes are linked as Stewart assumes [ 171. Apparently it is the interphase drag force in the dense phase and not the inertial force caused by the flow of particles around the bubble which determines the local pressure gradient. The behavior of large air bubbles (14 and 2 in. radius of curvature) rising in water is quite different. In that case the inertial force is important and the frontal pressure field computed assuming irrotational flow around an oval shaped body (bubble and wake) agrees well with the experimental pressure field. Above the center of pressure line, Reuter’s data for 3D half bubbles also follow Davidson’s theory but Stewart[ 171 has noted that Reuter’s actual bubble rise velocity corresponds to (g~)“~. This is 50 percent faster than the 3D Davies and Taylor value [5]. Inserting Reuter’s experimental value into Jackson’s equation raises the pressure ahead of the bubble considerably improving the agreement between the theory and the data. But even with this change, Davidson’s equation provides the best fit to Reuter’s data (see Fig. 8 of Ref. [ 171).
2240
The pressure field around a two-dimensional gas bubble
Since the bubble velocity in this work is close to the 2D Davies and Taylor value and follows Davidson’s theory there is no possibility of justifying a relationship between particle momentum and the pressure field based on a fit of Jackson’s equation. The fact that Davidson’s theory fits both Reuter’s data and that reported in this paper despite considerable variation in the value of Ub2/ga relative to the Davies and Taylor value indicates that the pressure field is independent of the bubble velocity. It is not obvious why Davidson’s theory works but the reason may lie in the particle motion above the bubble roof. We have observed particle trajectories similar to those reported by Gabor [6], that is, vertical movement of the particles seems to be more extensive than horizontal movement and there are particles falling through the roof. There does not seem to be much horizontal relative motion between the particles and bubble. Perhaps the net effect of all this is to make Darcy’s law a good approximation of the local pressure gradient, Apfi Some caution in using the results are indicated as a result of the fact that these experiments were performed using spherical glass particles of a single size. However, the fit of all literature data by Davidson’s theory indicates the results are general. The major differences between Reuter’s data and that found in this work are in the wake. Those differences are: (1) Dynamic pressures are observed up to 12 radii below the stagnation point whereas Reuter’s wake ended 4 radii down, (2) a pressure minimum is observed in the wake (Fig. 8) whereas Reuter found no maxima or minima, (3) pressure magnitudes in the wake, 2- 10 bubble radii below the stagnation point, are larger than predicted by Davidson’s theory whereas Reuter’s pressures are lower than predicted. These results show that the wakes in the two cases are different. Apparently a three diniensional half bubble rising along a wall has a more restricted wake than that of a bubble rising normally. Davidson and Harrison in their book[3] de-
fined a ‘Bernoulli’ pressure, P, which is the same as that calculated by Jackson and their pressure, pr, is the one calculated using the Davidson theory. The relationship between these two quantities is defined as P = p,+pp where pp is the interparticle pressure. It is easy to show that if the bubble rises at the Davies and Taylor velocity then P is always less than pf. In order to equate P and pf at the stagnation point it is necessary to raise the bubble velocity to (2ga)‘12.t Since this is nearly three times the theoretical rise velocity, it is apparent that P is always less than pf making pp negative. Following Davidson and Harrison [33 this would imply tensile forces between the particles which must be interpreted as meaning that the particles separate increasing the local voidage [ 181. Experimental evidence for this is given by Lockett and Harrison [ 191. Perhaps p does tell us something about the interparticle pressure and the local voidage near a bubble but it is really a defined quantity unrelated, as far as the writers can tell, to any measured pressure. The rise velocities determined using high speed motion pictures agrees well with the authors’ previous method based on the use of the pressure-time curve along the vertical axis of symmetry. Since the bubble accelerates when its stagnation point is within three radii of curvature of the surface of the bed, the velocities reported do not apply in that region. The acceleration of the bubble roof which occurs as the bubble approaches the upper surface of the bed, can be explained by considering the pressure field which results from the interaction of a bubble with a free surface. The pressure is constant and above atmospheric throughout the bubble cavity; it is also decreasing as the bubble approaches the surface of the bed so that the pressure drop across the particulate layer, between the bubble roof and the upper surface of the bed is also decreasing as the bubble rises through the bed. As the bubble approaches the upper surface
2241 CES Vol. 28. No. I2 - I
IThe same value applies for a 3D bubble [3].
H. LITTMAN and G. A. J. HOMOLKA NOTATION
of the bed the average pressure gradient across the particulate layer between the stagnation point and the upper surface of the bed becomes increasingly greater than at incipient fluidization. The forces on the particulate layer above the roof force the layer upward until the roof bursts.
a 4 Pa Pb
CONCLUSIONS
Pd
1. The total pressure field above the center of pressure line of a single bubble in a two dimensional fluidized bed is Laplacian and follows Davidson’s theory reasonably well. Although the experiments were performed using spherical particles of a single size, the results are believed to be general. 2. The pressure wake is closed. It extends 12 bubble radii below the stagnation point and four radii on either side of the vertical axis of symmetry. 3. A pressure minimum is found in the wake two bubble radii below the stagnation point and half a bubble radius from the vertical axis of symmetry. This shows that the wake is not Laplacian. 4. Dynamic pressures in the field below the center of pressure line are greater in magnitude than predicted by Davidson’s theory for up to 8 bubble radii below the floor. 5. The rise velocities determined using high speed motion pictures agree with those based on the use of the pressure-time curve along the vertical axis of symmetry. 6. When the bubble is within 3 radii of the upper surface of the bed, it elongates because the roof accelerates while the floor decelerates.
Pf Ph
PP
P Pll PT
r R
u: umf X
Y
bubble radius of curvature gravitational acceleration distance of center of pressure line below top of bed total pressure above bed total pressure inside bubble dynamic pressure in field = pf - ph total fluid pressure in field hydrostatic pressure in field interparticle pressure defined by Davidson and Harrison [3] “Bernoulli” pressure defined by Davidson and Harrison [3] measured dimensionless interstitial pressure, pf/pb theoretical analogue to P,, calculated assuming Laplacian field distance from center of bubble to a point in the field distance from stagnation point to a point in the field r/a bubble velocity minimum fluidizing velocity horizontal distance coordinate vertical distance coordinate
Greek symbols pb bulk density of bed
Acknowledgement-The authors are grateful to the Atomic Energy Commission for their support of this work under Grant No. AT(30-l)-3639.
0 angle off vertical axis of symmetry (origin of coordinates - center of bubble) + angle off vertical axis of symmetry (origin of coordinates - stagnation point of bubble) cr standard deviation.
REFERENCES ill COLLINS R., Chem. Engng Sci. 1964 20788. I21 DAVIDSON J. F., Trans. Znstn Chem. Engrs 196139 320. I31 DAVIDSON J. F. and HARRISON D., Fluid&d Particles. Cambridge University Press, London 1963. [41 DAVIDSON J. R., PAUL R. C., SMITH M. J. S. and DUXBURY H. A., Trans. Insrn Chem. Engrs 1959 37 323. [51 DAVIES R. M. and TAYLOR G. I., Proc. R. Sot. 1949 A200 375. [6] GABOR J., Proceedings of the Znternafional Symposium 011Fluidization, Eindhouen, p. 230. Netherlands University Press, Amsterdam 1967. [71 HOMOLKA G. A. J., Ph.D. dissertation, Rensselaer Polytechnic Institute, Troy, N.Y. 1971. 181 ISAACSON E. and KELLER H. B., Analysis ofNumericalMethods. Wiley, N.Y. 1966.
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The pressure field around a two-dimensional
gas bubble
[9] JACKSON R., Trans. Instn Chem. Engrs 1963 4122. [lo] LAZAREKG. M., Ph.D. dissertation, Rensselaer Polythecnic Institute, Troy, N.Y. 1972. [l I] LITTMAN H. and HOMOLKA G. A. J., Chem. Engng Prug. Symp. Ser. No. 105 1970 66 37. 1121 MOTAMEDI-LANJANI M., Ph.D. dissertation, Imperial College, London 1970. [13] MURRAY J. D.,J. FluidMech. 1965 22 57. 1141 PARTRIDGE B. A. and LYALL E., J. Fluid Mech. 1967 28429. 1151 PYLE D. L. and HARRISON D., Chem. Enana Sci. 1967 22 531. [16] REUTER H., Chemie Inger. Tech. 1963 35 Sk[17] STEWART P. S. B., Trans. Insfn Chem. Engrs 1968 46T60. [181 STEWART P. S. B. and DAVIDSON J, F., Powder Tech. 1967 161. [19] LOCKETT M. J. and HARRISON D., Proceedings ofthe International Symposium on Fluidizatiun, Eindhooen, p. 257. Netherland University Press, Amsterdam 1967.
2243