AIR VELIOCITY PROFILES IN THE PRESENCE
OF COCIJRRENTLY TRANSPORTED PARTICLES IAN D. D O l G AND GEOFFREY
H. ROPER
Unioersity of ,Yew South Wales, Kensington, Aust,alia
Air velocity profiles have been measured and correlated for a 43-mm. diameter glass riser when transporting 300- and 750-micron glass spheres. The measurements were made with a calibrated transversely orientated cylindrical Pitot tube and the results compared with solid-phase drag force density profiles derived from measur’ements of the velocity and density distribution of the 750-micron particles. At low average ratios of particle to air concentration the dispersed phase causes the air velocities relative to the single phase profiles to increase in the core and decrease at the wall. At high average concentration ratios there are a decrease, a tendency toward plug flow in the core, and an increase in the air velocities toward the wall. HE velocity profile of a fluid flowing through a conduit is T d e f i n e d as a plot of its local time-averaged axial velocity a t a point LS. the displacement of that point from the conduit axis or wall. \.\-hen the shape of this plot is independent of the axial distance from the entrance to the conduit, the fluid flow structure is called ‘.fully developed f l o ~ . ” Measurement of the velocity profile is one of the simplest means of gaining an insight into the flow structure of a fluid flowing through a conduit: An approximately constant eddy diffusivity in the core corresponds to the velocity defect law; a linear increase of the eddy diffusivity with increasing distance from the wall corresponds to a logarithmic law, those of Prandtl (77) and Deissler ( 2 ) being examples. For single-phase fluids the transport of energy and mass between the fluid and the wall is important. T h e velocity profile in the core region is of lesser interest and the “universal velocity profile” (23) is frequently presented as a correlation of all fully developed turbulent velocity profiles. The universal velocity profile is a plot of a velocity ratio, u+, z’s. a Reynolds number, u+, incorporating the distance from the wall. I n two-phase flow, transport processes between a dispersed and continuous phase may be more important for study than transport behieen the continuous phase and the wall. Logarithmic laws dvscribe the velocities near the conduit wall but not the velocities at the core of a duct. The core region is better described by a velocity defect law (77) which gives little indication of events near the conduit wall. The empirical po\vei- law (75, 77) possesses several advantages for studies in the presence of a dispersed solids phase. I t describes the velocity profile in the core better than a logarithmic law and represents much of the region described by logarithmic laws with little error (8, 7 7). Also, in single-phase flow its index, l,’n, is simply related to the resistance coefficient for rough- and smooth-walled conduits ( 7 4 , thereby providing a measure of the velocity gradient at the wall (75). I n this investigation, uniformly graded glass spheres were transported up a vertical glass pipe by an air stream and the air velocity profiles measured in the presence of the dispersed phase using a Pitot tube. Txvo particle sizes, 304 and 756 microns, \$ere used. A robust cylindrical transversely orientated Pitot tube (Figure 1) was developed and calibrated ( 3 ) .
Dimensional Consideraitions
If the axial pressure gradient is considered dependent, the folloiving set of dimensionless ratios may be obtained on analysis ( 3 ) .
p
I\’\
I
I
\I
A
Sectional Elevation On A r r o w s A-A
Figure 1 . General arrangement and details of cylindrical Pitot tube and piezometer tappings
I n the system studied the variables p P , p , D, g, and p were kept approximately constant, and the experimental work could be described by
5 =f
[i, 2
Re, Fr,
.u Introducing $ as a distribution parameter describing a general relationship between the ratios a/u and 20, D
#
=
f [Re, Fr, d / D , MI
(2)
The conduit diameter was fixed. The Reynolds and Froude numbers were both described by the mean velocity and their separate contributions could not be determined. Equipment
The apparatus consisted of 40 feet of vertical glass pipe of 1.7-inch inside diameter, in which particles could be concurrently transported upward by an air stream, separated from the air stream a t the top, and returned to the bottom to be continuously recirculated through the system a t a constant mass flow rate. The glass pipe was selected for the regularity and circularity of its bore. All joints were matched for circularity and diameter, ground square and butted together so that no step occurred between any two mating pieces on VOL. 6
NO. 2
M A Y 1967
247
assembly. Diameters and circularity varied by 10.012 inch for the lower and 1 0 . 0 0 5 inch for the upper sections of the conduit. The arrang-ement of the cylindrical Pitot tube and the piezometer wall tappings is shown in Figure 1. The tube was located 35 feet up the vertical glass conduit. The cylindrical Pitot tube slides in holes drilled and lapped into the conduit with square edges on the inner wall. Any leakage between the Pitot cylinder and conduit wall is prevented by the external O-ring seals. The arrangement permits the Pitot tube to be traversed, rotated, or completely removed and replaced without the seals being damaged. Compressed air a t a pressure of 90 p.s.i.g. was fed fron main supply, reduced to 30 p.s.i.g., passed throug-ha 7-cubic f surge tank, a manually controlled valve, and a Venturi fli meter, and entered at the base of the vertical glass cond A small electric boiler whose output was controlled by a varinhle transformer suodied Steam to maintain the relative humidity of the incomidiair above 75% R H . This increased the surface conductivity of the glass by several orders, allowing electrostatic charges to be rapidly dissipated, minimizing their effect in the system (5,7). Pitot-static nressure differentials were measured using the micromanometer shown in Figure 2. T h e measured ‘time constant for the manometer leads, cylindrical Pitot tube, a n d 0.010-inch piezometer tapping (assuming its response to be approximately first order) was 7.5 seconds. ~~~
~~
~
~~~
At the start of a run, the desired air and particles feed rates were set and maintained. T h e Pitot tube was inserted with the manometer leads clamped, and the total head hole set a t the desired point using the micrometer. The plumb bob and protractor arrangement shown in Figures 1 and 2 ensured that the total head hole faced exactly downstream. The manometer leads were unclamped and a stopwatch was simultaneously started, so that a t least nine “time constants” elapsed before a reading was taken. The manometer leads were again clamped and the differential head in the micromanometer was carefully measured. (Except for positions close to the axis, and some runs a t high mass flow ratios, readings could he reproduced within 1 0 . 0 0 1 inch.) A check was then made that neither the total head nor piezometer hole had hecome blocked: The Pitot cylinder was rotated about 50“ and the manometer leads were unclamped. If the holes were free, the manometer levels would change rapidly; if they did not, the blockage was cleared and the reading repeated. Otherwise the pressure differential reading was logged as correct and the total head hole moved to the next position. Total head hole positions were chosen randomly to avoid systematic errors. I n this way a series of measured Pitotstatic differentials, I’ (inches standard water gage), were obtained for a series of relative distances from the wall (2a/D) for each run. T h e Venturi pressure differentials, wet- and dry-bulb temperatures, air pressure in the conduit at the Pitot position, and the particles mass flow rate were also logged several times during a run. Runs were conducted for a series of air flow rates and ratios of particles to air mass flow for the two sizes of spherical glass particles. Correction of Measured Pitot-Static Pressure Differentials
Calibration (3) of the cylindrical Pitot required that the measured manometer differentials, I’ (inches swg), he corrected according to Equations 3 , 4 , and 5: - a) (1 f b)-l (inches swg)
where log- [ a (inches swg)] = 1.7 log R e
(3)
- 9.20
= 1.7 log [(zi)(ft./sec.)]
- 4.24 (4)
and -log 6 = 0.05 f
=
248
2a ~
D
((4.11 X lo-? Re
2a
4-6.41
0.05’4- ~. (0.337 (zi)(ft./sec.) f 6.41
D
l&EC FUNDAMENTALS
(5)
ins
The
14
Experimental Method
I = (z’
Figure 2. Cylindrical Pitot tube, protractor, ond micromanometer arranged for experimental measurements
6
zi S 54 (ft./sec.l: (11 X lo3)
< Re
6
(z)
’ + D
JC
0
(u)
. (1
-
gy
(7)
Plots of the kind shown in Figure 5 were used to conduct the second part of this integration graphically, to reveal whether a velocity defect law (8, 7 7) uC - u
-~
(u*)
(22)
(9)
and from Equations 8 and 9
1 I V 0.04 0 0 6 0080.1 0.2 0.4 RELATIVE DISTANCE FROM THE WALL
ur
(9)
0 W n
(+Te5
- Du* -4E
-
/.
2~\+
, ,
/A\
/DG\
Dzi
,.*,
Measured values of the slopes, m, and the eddy diffusivities, estimated from them are presented in Figures 6 and 10. The reciprocal of the power law index, n, and the slope, m, of the above equation can both be used as distribution parameters defining portions of the velocity profile of the continuous phase in accordance with Equation 2. For convenience in reproducing velocity profiles, two further distribution parameters have been reported in Figures 9 and 8: the relative distance, (2u/D)zi, at which the local mean velocity, u, equals the superficial velocity, zi, and the ratio, uc/zi, of the mean velocity a t the duct axis, u,, to the superficial velocity, zi. Values of these four distribution parameters, n, (2u/D)zi, uc/zi, and m, derived from velocity profiles obtained for the particle free fluid are shown in Figure 6 as a function of the Reynolds number (Re). Attempts to present the distribution parameters derived from measurements made in the presence of the dispersed particulate phase directly proved untidy, and the ratios of these distribution parameters to those obtained from the single-phase studies (Figures 7 to 10) have been presented instead. These distribution parameter ratios, X, Y , W , and S, are defined as: E,
x=[
reciprocal of power law index, n, for continuous phase in presence of dispersed particulate phase reciprocal of power law index, n, for continuous phzse in absence of dispersed particulate phase
relative distance where u = zi for continuous phase in presence of dispersed particulate phase Y = [ relative distance where u = zi for continuous phase in absence of dispersed particulate phase
W =
1 1
velocity ratio, uc/zi, for continuous phase in presence of dispersed particulate phase velocity ratio, u,/zi, for continuous phase in absence of dispersed particulate phase
-core region slope, m,for continuous phase in presence of dispersed particulate phase S = core region slope, m,for continuous phase in absence of dispersed particulate phase (14)
These distribution parameter ratios are plotted against the ratio of particles to air mass flow, M , with the Reynolds and Froude numbers, Re and Fr, and the particle to duct diameter ( d / D ) as parameters in Figures 7 to 10. VOL. 6
NO. 2
MAY 1967
249
Gradients Of These A r e The SLopes ( m )
d = 756u
M=O 2810 (Run NOS 23&19)
18 NOTE: N o n - Z e r o
0
0:l
012
Ordinate
013
OJ4
015
Oi6 017
018
Ol9
1.0
SQUARE OF RELATIVE DISTANCE FROM AXIS
(h)2 D
Figure 5. Set of velocity profiles showing change in velocity defect law gradients, m, in core region and decrease in air velocities near wall as particles to air mass flow ratio, M, increases from 0 to 4.4
Use of ratios X,Y , W , and S permits a ready interpretation of the trends with the mass flow ratio and the Reynolds and Froude numbers. The separate contribution of the Reynolds and Froude numbers could not be presented since the conduit diameter was not varied during these experiments. Reliability
The reliability of the results may be judged from the comparison made between the superficial air velocities obtained from the Venturi measurements, 5,,,and those obtained from an integration of the Pitot tube-derived velocity profiles, zi. T h e differences between these two mass flow rates were calculated as the ratio
{(a)
- ( 4 v ) /(4
These differences are tabulated by Doig ( 3 ) . For the single-phase studies these differences were small, but in the case of measurements made in the presence of a dispersed particulate phase, they range in general u p to 5.5% and in one case of a low air rate and high mass flow ratio up to 8.6%. Although differences of this order were not unusual for integrated Pitot measurements, they appeared to exhibit clear trends and their source should be examined. At the lower Reynolds numbers, many of these differences may be attributed to experimental errors (at a Reynolds number of 12,000 to 13,000, Pitot-static differentials ranged from 0.080 to 0.029 inch swg), but the differences obtained a t the high Reynolds numbers cannot be entirely due to experimental error. There appear to be distinct trends with regard to these differences. Almost all the differences obtained using the 756250
l&EC FUNDAMENTALS
micron particles were positive, while almost all those obtained using 304-micron particles were negative. To some extent the differences appear to increase with the ratio of particles to air mass flow. Negative differences may be caused by the differential resistance offered to the air flow approaching the Pitot cylinder from upstream by the cloud of particles which are reflected from and accumulate in front of the transversely placed Pitot cylinder. Positive differences may be due to a partial contribution from the kinetic head of the dispersed particles. Van Zoonen (22) in his studies a t high mass flow ratios approximately equated his Pitot-static differentials to the kinetic head of the dispersed catalyst particles he was using and derived their velocities from these measurements. For a particle to contribute the whole of itskinetic energy to the Pitot-static differential head, it must be entirely decelerated by the stagnant air within the Pitot tube. Considering the dimensions of the totalhead hole (shown in Figure 12), contributions from the dispersed phase may be neglected. A possible explanation for the positive differences is that the wall correction factor, b (derived from single-phase profiles), introduces an error when the dispersed phase is present. Integration of the velocity profile is most sensitive to the velocities near the wall and an error in the power law index could introduce a considerable error in the integration to obtain the mean velocity, 2. While no direct check has been made of the applicability of the wall correction factor, b , to the two-phase situation, indirect evidence of its applicability has been obtained from the coincidence of the straight line through the points above and below a relative distance of 0.2 in Figure 4, which has already been commented upon. I t appears,
0.271
I 4.0
"261
I
I
I
I
4.2 4.4 4.6 LOG. [REYNOLDS NUMBER (Re)]
I 4.8
0 W
r0 . 4 2 -
\
0.34 LL W
n
4.0
4.2 44 4.6 LOG. [REYNOLDS NUMBER (Re)]
4.8 T7
r-0.70: W
n
-1.5-
; +
/BLasius
-.
Prandtl-Nikuradse Resistance Law
--0.75 Z
Resistance Law
+' -+-
--
3
_
5
--0.80
-\-
W
--0.85
-1.7-
0
a
Y
4.0
4.2 4.4 4.6 LOG. [REYNOLDS NUMBER (Re)]
I
-0.90 4.8
;
Figure 6. Distribution parameters for air velocity profiles obtained in absence of a dispersed phase (9)
therefore, that serious errors have not been introduced from this source except perhaps a t high particle to air concentration ratios. Discussion
Peskin et al. (75) have measured air velocity profiles in a horizontal 3-inch, square-section duct when transporting glass ballotini beads of 80- and 110-micron diameter a t mass flow ratios up to 10 and F.eynolds numbers, Re, ranging from 100,000 to 150,000. They were able to use small conventional Pitot tubes to establish that, in the presence of the dispersed solids phase, air velocities in the wall region could be represented by a power law ( 7 4 ) . When plotted on the coordinates of Figure 7 , their results differ from those of the present study in that all their power law index ratios, X,are greater than unity, whereas all those of the present study are less than 1.
There are important differences between their system and that of the present study. The system reported in this paper was vertical, not horizontal. The conduit was cylindrical, not of square section. The particle diameters were 756 and 304 microns, not 110 and 80 microns. The mean air velocities were much lower and close to the choking velocity for the larger particles (20.6 feet per second a t a mass flow of 1.6). T h e distribution of the 756-micron particle velocities and spatial concentrations at various air and particle mass flow rates in this system have been measured and reported (3, 72). Particle velocities were measured by photographing particles in flight using a high speed movie camera and analyzing the film record. The particles were illuminated in a narrow vertical slit of light projected across a conduit diameter at right angles to the camera axis (20). A similar technique using a VOL. 6
NO. 2 M A Y 1 9 6 7
251
Symbol
+ H X
0
+
0.40
I
0
I
I
1
R e No. 44,200 35,000 29,900 73,000 70,400
Fr No. 25.1 19.9 17.0 13.0 11.6 I
I
1 2 3 4 5 PARTICLES TO AIR MASS FLOW RATIO ( M )
6
Re =37 000 8 28,200 & 160 Fr.21.0
-LEGEND
O'*OI 0
+
0.70I
I
I
I
20,000 12,400 I
11.4 7.05 I
Figure 7. Graph of power law index ratio, X, for air velocity profiles obtained in presence of a dispersed particulate phase Plotted as a function of mass Aow ratio with Reynolds-Froude numbers and particle to conduit size ratios as parameters
still camera and a single short (approximately 3 microseconds) flash was used to obtain film records from which the mean particle concentrations at various radii could be determined. Mean values of the local axial particle velocities and spatial concentrations corresponding to a superficial air velocity of 36 feet per second and mass flow ratios of 0.5 and 5.0 have been abstracted from this work and are presented in Figures 11 and 13. The corresponding air velocity profiles for mass flow ratios of 0.0, 0.5, and 5.0 have been added to Figure 11. A similar set of air velocity profiles is shown in Figure 5. Figure 13 shows the distribution of the calculated drag force per unit volume, Fd, acting upon the dispersed solids phase derived from the velocity and density data and Equation 15. Fd
=
(plpp)(pd/d)
CD (U -
(1 5)
Drag coefficients, CD,obtained from studies employing uniform laminar velocity fields (70) were assumed for this purpose. Particle Reynolds numbers [ ( u - v ) d p / p ] ranged from 208 to 382, and the corresponding drag coefficients ranged from 2.46 to 1.75. Unless the trend of the drag coefficient with the particle Reynolds number is very different for the bounded flow, dispersed solids phase situation, the conclusions drawn below from Figures 11 and 13 will be unaltered by this assumption. 252
l&EC FUNDAMENTALS
If the resistance of the dispersed solids phase to air flow follows the same distribution as the drag force per unit volume, the deviations of the air velocity profiles a t mass flow ratios of 0.5 and 5.0 from the profile obtained for particle free flow are explained by Figure 13. Where the drag force per unit volume is greatest toward the walls and smallest a t the center (at a mass flow ratio of 0.5), the air velocities are smaller toward the walls and greater in the center. At a mass flow ratio of 5.0, the reverse holds for 90% of the conduit radius. I n the remaining 10% of the radial distance to the wall, the air velocities are shown to be smaller than those for the corresponding particle-free situation, contrary to the result expected from the plot of the drag force per unit volume presented in Figure 13. Air velocities reported for positions close to the wall depend upon the validity of the wall correction factor, 6 , which must be applied to pressure differentials measured with the cylindrical Pitot tube used. At particle to air concentration ratios, Mzi/v, below 3 to 1, this validity could be partly confirmed by a plot of the kind shown in Figure 4, where a straight line projected through the points above a relative distance from the wall of 0.2 (for which the wall correction factor was insignificant) passed through the corrected pressure differential points below it. No confirmation of this kind was obtained
" "-1
d
R e = 35,000 Fr = 19 9 \
(W)
=756p
Re = 23,000 'r = ' 3 0 -I
-13
0.60
!
0
r
4 5 6 PARTICLES TO AIR MASS FLOW RATIO ( M )
1
2
3
r =16G
Re = 17,400 Fr = 7 35
-
0
i
\
2
\ +Fr 3
=11 4
4
5
PARTICLES TO AIR MASS FLOW RATIO
6 (M)
Figure 9. Graph of mean velocity position ratio, W , for air velocity profiles obtained in presence of a dispersed particulate phase Plotted as a function of mass flow ratio with Reynolds-Froude numbers and particle to conduit size ratios as parameters
for runs a t higher concentration ratios. Conventional cantilevered Pitot tubes were rapidly damaged and could not be used to resolve this, problem. During the investigation, measurements were confined to positions within 9575 of the radius to the conduit wall. I n view of the above discussion, velocity profiles derived from the correlations presented in Figures 7 to 9 for particles to air concentration ratios greater
than 3 should be considered reliable only for points within 90% of the radius from the conduit axis.. I t should not be assumed, however, that a flattening of the velocity profile a t the core results in a higher velocity gradient at the conduit wall. The slip between the air and the 756-micron particles near the wall is about 1 4 feet per second, and if the postparticle recovery of fluid energy is substantially less when a particle is VOL. 6
NO. 2
MAY 1967
253
R e = 44,200 Fr = 25.1 /
/
A
x
R e = 35,000 Fr = 19.9 Re 29,900 Fr = 1 7 . 0
I
0
I
I
I
I
I
i
1
2
3
4
5
6
PARTICLES
TO
AIR MASS FLOW RATIO (M)
1 37,000 = 21.0
XY
0.8
= 28,200 = 16.0 See Fig.7 For Symbols Legend
0.4
1
0
Fr = 1 1 . 4 1
I
\ i\
I
1 2 3 4 5 PARTICLES TO AIR MASS FLOW RATIO ( M )
t
6.
Figure 10. Graph of velocity defect law gradient ratio, SI for air velocity profiles obtained in presence of a dispersed particulate phase Plotted as a functon of mass flow ratio with Reynolds-Froude numbers and particle to conduit size ratios a5 parameters
at, or close to, the wall, the air velocities close to the wall may well be less than those of the corresponding particle-free air and the real trend represented by the data presented in Figure 7. The assumption that the power law index yields a measure of the air velocity gradient a t the wall (74) in the presence of dispersed solids phase (75) cannot be adopted here, considering the particle sizes and low air velocities employed. Interest in the fluid velocity gradient a t the wall stems from investigations of heat transfer ( 7 , 6, 27) in the presence of a dispersed solids phase and attempts to evaluate the components of the axial pressure gradient along the conduit ( 4 ) . Results of investigations show a n initial decrease in the heat transfer coefficient from a mass flow ratio of zero before it begins to increase with the mass flow ratio. The effect is most pronounced for the larger particles ( 7 , 6) and the major resistance to heat transfer is considered to lie in the gas film a t the conduit wall (78). The rapid initial decrease of the power law index ratio with increase of the mass flow ratio (Figure 7) indicates a decrease in the air velocities near the conduit wall and the present results for low mass flow ratios appear to agree with the trends shown by the initial decrease of the heat transfer coefficient ( 7 , 78) with the mass flow ratio. The decrease in the velocity gradient predicted by the results shown in Figure 7 for the 304-micron ballotini is less than that predicted for the 7 56-micron ballotini under corresponding 254
I&EC FUNDAMENTA1.S
conditions. I t is considered that wakes would form downstream of the particles used in this investigation, which would provide local increases in heat and momentum transfer. At low mass flow ratios, the density of these local increases a t the conduit wall would be small and probably insignificant, particularly for the larger particles. At higher mass flow ratios they may be the cause of the observed increase in the heat transfer rate ( 7 , 6, 78) and the gas velocity gradient a t the walls may be less important to mass and energy transfer to the conduit walls than the eddies produced by particle wakes. Insufficient data were available to calculate similar distributions of the drag force per unit volume for the 304-micron glass ballotini. The average slip velocities for these particles ranged from 6.8 to 14.5 feet per second for air velocities up to 35 feet per second and the particle velocity distributions tended to be closer to the air velocity distributions than those obtained for the 756-micron ballotini under corresponding conditions (72). It is evident from Figures 7: 8, and 10 that the air velocity profiles will show similar trends with the mass flow ratio as those shown in Figures 5 and 11 for the 756-micron particles. Figure 7 shows that the corresponding values of the power law index ratio, X,are less than half of those for the 756-micron particles, and air velocities near the wall will be closer to those of the corresponding particle-free air. The occurrence of a maximum drag force density at the
z W
-
(TO SCALE-
n
0 RADIUS RELATIVE TO CONDUIT AXIS (2r/D)
0 4
0 6
0 8
2n
RADIUS RELATIVE TO CONDUIT AXIS ( 2 r j D )
Figure 1 l., Local mean air and particle velocities at mass flow ratios of 0.0, 0.5, and 5.0
Figure 13. Local mean spatial densities, P,, and drag force on particles per unit volume, F,, at mass flow ratios of 0.5 and 5.0
Superficial air velocities, 0, 36.0 feet per second for oll moss flow rotios 1 4 . 5 0 and Superficial particles velocities, 14.56 feet per second for mas, flow ratios of 0.5 and 5.0, respectively
Average values of ( p d ) . 0.091 5 and 0.897 pound per cubic foot Average values of ( F d ) . 0.31 2 and 3.1 3 pounds force per cubic foot for mass flow ratios of 0.5 and 5.0, respectively
v,
walls shown for a mass flow ratio of 0.5 in Figure 13 supports the finding that air velocities near the wall are reduced a t low mass flow ratios. The main factor producing this result is the higher particle concentration a t the w,alls occurring a t low mass flow ratios. This could be observed for both the 756- and 304-micron particles. At the end of a run when the particles feed was stopped and the air flow maintained, the particle inventory in the system diminished to very low concentrations. When this occurred, particles could be seen to be traveling up the walls of the conduit and few could be seen in the core. Although these particles possessed random tangential velocity components, there was no perceptible net swirl suggesting centrifugal force as the cause. Measurements of the distribution of 756-micron ballotini ( 3 ) show the particles to be more concentrated at the wall where the particle to air concentration ratio, Mi&/D,is less than 2.5. Higher particle concentrations at the wall at low mass flow ratios have been reported by So0 (79) for the pneumatic transportation of 115- and 230-micron ballotini through a horizontal square-sectioned duct. Higher concentrations a t the walls have also been reported by Van Zoonen (22) for the pneumatic
I
0.2
r 10
Pitot Cylinder 0.128"diarna
0.008"v Backing Piece
I
P i t o t Hole (0.016")
Figure 12. Details of total head topping in transverse Pitot tube
transportation of a crack:ng catalyst ranging in size from 20 to 150 microns u p a 5-cm. diameter vertical riser, but in this investigation the mass flow ratios employed generally exceeded 20. At the lower particle mass flow rates the concentration distribution profile tended to flatten in the upper half of the 10-meter high riser (22). (By comparison, all measurements reported during the present study were taken 35 feet above the air and solids inlet at the base of the vertical riser.) These observations ( 7 , 6, 78, 79, 22) and those of the present study suggest that particle density distributions for the 304micron particles follow a similar trend with the mass flow ratio to that observed (3) for the 756-micron particles. The power law index ratio in Figure 7 increases with the Reynolds number. If Nunner's correlation (75) relating the power law index to the resistance coefficient, A, were assumed, this would agree with the trend of the total axial pressure gradient for a particular mass flow ratio reported by Rose and Barnacle (76). These investigators also show the total axial pressure gradient to be almost independent of particle size where the particles are relatively large, which suggests the net resistance to air flow may be independent of the particle size. The 304- and 756-micron diameter glass ballotini fall within the range of Rose and Barnacle's investigations and the plot of the axial velocity ratios shown in Figure 8 indicates a similar degree of flattening of the velocity profile a t the higher mass flow ratios for both particle sizes. Rose and Barnacle's findings suggest, therefore, that a similar degree of core flattening would occur with larger particles. Peskin et al. (75) have reported a considerable flattening of the center of the velocity profile when transporting 1IO-micron diameter ballotini horizontally at a mass flow ratio of 3 and a Reynolds number of 100,000. The relative population density of two particle sizes having the same over-all concentration ratio, Mii/B, will be proportional to the cube of the size ratio. Drag coefficients would increase with a decrease in the particle size (70) and, unless the corresponding slip velocities decrease very rapidly with particle size, there would be a tendency for the net resistance of the dispersed solids VOL. 6
NO. 2
MAY 1 9 6 7
255
phase to increase with a reduction of the particle size; Rose and Barnacle’s results show that the axial pressure gradient increases with a reduction in the ratio of particle size to conduit diameter below 0.02 for glass spheres. Air velocities in the core region have been correlated by a velocity defect law and values of the eddy momentum diffusivity, e, may be calculated using Equation 10 after abstracting values of the velocity defect law gradient and the power law index from Figures 6 , 7 , and 10. Equation 10 is not applicable to this two-phase situation and values of the eddy momentum diffusivity obtained in this way are of doubtful significance since they represent a diffusion of momentum to both the dispersed solids phase and the continuous gas phase. Diffusion studies indicate that the eddy mass diffusivity of the continuous gas phase is reduced by the presence of the dispersed phase (75, 22), whereas a flattening of the velocity profile in the core region (75) suggests an increase in the eddy momentum diffusivity. This indicates that momentum transfer, independent of mass transfer through the gas phase, is occurring and probably signifies a decrease of the eddy scale of turbulence approaching the particle dimensions and an increase in the wave number in the core region. Figures 7 to 10 show both the Reynolds and Froude numbers as parameters. The results of an earlier review (4) suggest the Froude number will be the more significant of these, the effect of the Reynolds number being already accommodated in the plots shown in Figure 6 for the particle-free air. Attempts to produce single line correlations of the distribution parameters shown in Figures 7 to 8 in terms of the concentration ratio, Malo, proved unsuccessful.
UO ti (ti) u U* U+ U
o
W
X Y Z’ 2
proportionality constant of power law for distribution of velocities, It-1 superficial or mean fluid velocity obtained by integration of velocity profile, It-’ superficial or mean fluid velocity derived from Venturi flowmeter measurements, It-‘ wall friction velocity, It-1 dimensionless velocity term of the universal velocity profile, u/u* time averaged axial velocity of particles of dispersed phase at a particular radius, It-’ integrated mean particle velocity, It-’ ratio of mean velocity positions defined by Equation 13 ratio of power law index reciprocals defined by Equation 11 ratio of velocities at conduit axis defined by Equation 12 measured Pitot-static pressure differentials, ml-9-2 corrected Pitot-static pressure differentials, ml-lt-*
SPECIAL GROUP relative distance from conduit wall at which local velocity equals mean velocity eddy diffusivity in core region, l2 t - I Blasius wall resistance coefficient fluid viscosity, m1-lt-I density of continuous fluid phase, ml+ density of solid material of which particles are made, ml-3 density of dispersed solids phase, ml-3 distance from conduit wall, 1 dimensionless distance from conduit wall used in universal velocity profile (u . u* , p / p ) generalized distribution parameter defined by Equation 2 mass; 1 = length; t = time
Conclusions
The distribution parameters presented in Figures 6 to 10 correlate the radial distribution of the time-averaged air velocities in the presence of a dispersed solids phase except a t points within 10% of the conduit radius from the wall. At particleto-air concentration ratios, Malo, below 2 the power law indices derived from Figures 6 and 7 provide a measure of air velocities closer to the wall relative to those obtained in the absence of the dispersed solids phase. Nomenclature a
b
D d
f Fd
Fr g
GP Kd
m
M n I.
Re
S U U,
256
Pitot-static pressure difference correction defined by Equation 4, ml-1t-2 Pitot-static wall correction factor defined by Equation 5 diameter of conduit, 1 geometric mean particle diameter for particles of dispersed phase, 1 “is a function of” drag force per unit volume acting on dispersed solids phase, ml3tP2 Froude No. = ( a ) / ( g .D)O.5 gravitational acceleration constant, it mass flow rate of dispersed particulate phase, ml -l Pitot hole coefficient gradient of velocity defect law defined by Equation 10 particles to air mass flow ratio = G p (pa) reciprocal of power law index of continuous fluid phase radial distance from conduit axis, 1 Reynolds number, a .d .p / p ratio of velocity defect law gradients defined by Equation 14 time-averaged axial velocity at a particular radius in conduit for continuous fluid phase, It-’ velocity, u, at conduit center, It-1 l&EC FUNDAMENTALS
(1) Broetz, W., Hilby, J. IV., Muller, K. G., Chem. Zng.-Tech. 30. No. 3. 138 (1958). ( 2 ) Deissler; R. G., Nitl. Advisory Committee for Aeronautics, NACA Rept. 1210 (1955).
(3) Doig, I. D., Ph.D. thesis, University of New South Wales, Australia, 1965. (4) Doi I. D., Roper, G. H., Australian Chem. Eng. 4, No. 1, 9 (19637: (5) Ibid.‘, No. 4, p. 9. ( 6 ) Farbar, L., Morley, M. J., Znd. Eng. Chem. 49,1143 (1957). ( 7 ) Fordham Cooper, W., Brit. J . Appl. Phys., Suppl. 2, S71 (1953). (8) Hinze, J. O., “Turbulence,” pp. 514--36, McGraw-Hill, New York, 1959. (9) Knudsen, J. G., Katz, D. L., “Fluid Dynamics and Heat Transfer,” McGraw-Hill, Nevi York, 1958. (10) Lapple, C. E., “Chemical Engineering Handbook,” 3rd ed., p. 1018, McGraw-Hill, New -fork, 1950. (11) Laufer, J., “Turbulence,” pp. 517, 525, 536, McGraw-Hill, New York, 1959. (12) Mathews, C. T., M.Sc. thesis, University of New South Wales. Australia. 1966. (13) Nikuradse, J., “Boundary Layer Theory,” pp. 402, 403, McGraw-Hill, Xew York, 1955. (14) Nunner, W., “Turbulence,” p. 520, McGraw-Hill, New York, 1959. (15) Peskin, R. L., et al., U. S. Atomic Energy Commission Quarterly Reports 63-1, 63-2, Contract AT( 30-1) 2930 (1963). 116) Rose. H. E.. Barnacle. H. E.. Eneineer 203. 898. 939 11957). (17) Schlichting,’ H., “Boundary Llyer Theory,;’ Chap. XX, McGraw-Hill, New York, 1955. (18) Smith, K. L., M.Sc. thesis, University of New South Wales, Australia, 1963. (19) Soo, S. L., IND.ENG.CHEM.FUNDAMENTALS 1, 33 (1962). (20) Soo, S. L., Ihrig, H. K., El Kouk, A. F., Annual Meeting, ASME, Paper 59-A-59 (1959). (21) Soo, S. L., Regalbuto, J. A., Can. J . Chem. Eng. 38, 160 11960). (22) Van Zoonen, D., Proceedings of Symposium on Interaction between Fluids and Particles, Inst. Chem. Engrs., London, 1962. (23) Von Karman, T., Trans. A.S.M.E. 61,705 (1939). RECEIVED for review May 13, 1966 ACCEPTEDJanuary 6, 1967
