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The pressure drop created by cones settling on the axis of a pipe
Pergamon Press.
Chemical Engineering Science, 197 1, Vol. 26, pp. 693-696.
Printed in Great Britain.
The pressure drop created by cones settling on the axis of a pipe J. LANGINSP, M. E. WEBER and I. PLISKINS Department of Chemical Engineering, McGill University, Montreal I IO, Canada. (First received
4 February
1970; accepted
15 July 1970)
Abstract-Pressure drops have been measured for cones with apex angles between 30” and 120”, settling axially in a vertical pipe. The available data have been correlated on a plot of dimensionless additional pressure drop vs. conduit Reynolds number. Distinct low and high conduit Reynolds number regimes have been distinguished with the transition between them occurring over the range from 600 to 3000. Brenner’s theoretical predictions for low Reynolds numbers are verified.
INTRODUCTION
(3) Energy dissipation is Oseen-type linearization. Feldman [6] confirmed Eq. for sphere Reynolds numbers behaviour he found for higher is shown in Fig. 1.
pressure drop due to the presence of a particle in a conduit has been studied theoretically by Brenner [ 1,2] and Happel[3], and experimentally by Fayon[4], Pliskin[S] and Feldman [6] for spheres. Feldman [6] investigated the variation of the additional pressure drop with Reynolds number and proposed an explanation of the behaviour for spheres. The purpose of our work was to study this same effect for cones of various apex angles, and to compare the results with those for spheres. Brenner[2] has shown theoretically that the product of the additional pressure drop and the conduit cross section area is given by THE ADDITIONAL
AP+A -=D
voo %I ’
2
D
(1)
(2)
for any particle shape or orientation. These results were derived under the following assumptions: (1) Particle Reynolds number is small. (2) Flow in the conduit is laminar in the absence of the particle.
by an
(2) experimentally less than 120. The Reynolds numbers
EXPERIMENTAL
For a particle on the axis of a cylindrical pipe,
AP+A=
described
The experimental-set up was similar to that used previously [5,6]. Particles settled in one leg of a U-tube filled with a solution of known viscosity. The particles were released by an electromagnet beneath the liquid surface and fell along the axis of the vertical 4.0 in. i.d. by 4ft long glass pipe which formed one leg of the U-tube. Each leg was sealed by a flanged plate provided with a pressure tap. The pressure difference between the taps was measured with a Baratron pressure transducer (MKS lnstruments of Burlington, Mass.). The apparatus was maintained at 23°C % O*Y’C. The particles were spheres A-1.5 in. dia. and cones with apex angles 30”, 45”, 60”, 90” and 120” and base diameters from O-5 to l-5 in. The particle densities were 7.79 g/cm3 (steel) and l-45 g/cm3 (polyethylene-steel composites). The composites were made by inserting a steel cylinder in a piece of polyethylene and machining to the desired shape. The five solutions used were mixtures of water
tGulf Research Centre, Sheridan Park, Ontario. SUniRoyal Research Center, Wayne, New Jersey.
693
J. LANGINS, I
M. E. WEBER and I. PLISKIN I
I
I
2.2>
z $
1.8-
24p
1.6-
4 3 2 2 2 P
O/R0 l
2,0_
.
da
.
;* i.
l.A
FULL
SYMBOLS
- AUTHOR
1.L
OPEN
SYMBOLS
- FElOMAN
1. o-
I
FLAGS
INDICATE
40
b 0.5
q
COMPOSITES
0.8-
0.1
1.b PARTIGLE
Fig. I. Dimensionless
lb0
til REYNOLDS
NUMBER
pressure drop parameter spheres.
and Ucon 50-HB-5100 (Union Carbide), and provided viscosities from 0.94 CP (pure water) to 2860 CP (pure Ucon). The average particle velocity was measured by timing the fall over a measured distance. Terminal velocities were obtained by correcting analytically for particle acceleration. In no case was the terminal velocity more than 6 per cent greater than the measured average velocity. Particle drag at the terminal velocity was found by weighing the particles when suspended in the fluid. The average deviation for the measured pressure difference between replicate runs was 4 per cent. There was no significant difference between homogeneous and composite particles. Additional description of apparatus and procedure is given by Langins [7]. RESULTS
AND
DISCUSSION
Figure 1 shows the present results and those of Feldman[6] for spheres. The ordinate is the dimensionless pressure drop, 6, defined by
&$yl_pJ].
(3)
The abscissa is the Reynolds number, Re, based .
1,doo
to.boo
100,000
-v
versus particle Reynolds number
for
on the sphere diameter and terminal settling velocity. The term in brackets in Eq. (3) corrects for wall effects and, following Feldman, was omitted for Reynolds numbers greater than 60. There is qualitative agreement between these data. However, our spheres, which had smaller a/R,, than Feldman’s, show a drop off in 6 around a Reynolds number of 30 while his show it around 120. Figure 2 presents the results for the cones. The Reynolds number here is formed with the base diameter and the terminal velocity. There is no trend with particle orientation during fall or with apex angle as predicted by Brenner[2]. Since Brenner’s analysis is the same for a particle settling in a motionless fluid and for a stationary particle in a moving fluid, the second and third assumptions in his analysis can be tested by plotting the data against the conduit Reynolds number based on terminal velocity and pipe diameter, Fig. 3. On this plot all of the data are drawn together on one curve. At low Reynolds number values 6 = 2 while at high values 6 = 1, and the transition occurs between Re, - 600 and Re, - 3000. The data of Fayon [4] and Pliskin[S], not shown, would fall on the 6 = 2 line since their largest conduit Reynolds
694
The pressure drop created by cones settling on the axis of a pipe
3
2.0
3 k T 3 F s
1.8
B 2 2
1.2
%
s
1.8
1.4
FULL
1.0
OPEN
SYMBOLS- RELEASED SYMBOLS
APEX
- RELEASED
APEX
DOWN UP
0.8
0.1
1.0
10 PARTICLE
Fig. 2. Dimensionless
REYNOLDS
100 NUMBER
1,000
10.000
100,000
-y
pressure drop parameter versus particle Reynolds cones of Apex Angle 30”-120”.
number for
2. ‘0 z* k *3
2. 1.
s
1.
% 5
1.
s 2
1.
F
1.
o
AUTHOR’S
l
FELDMAN’S
o
SELECTED
SPHERES SPHERES CONES
0
CDNDUIT
Fig. 3. Dimensionless
pressure
REYNOLDS
NUMBER
- F
drop parameter versus conduit Reynolds spheres and cones.
number was 120. The high bound of the transition, Re,, - 3000, is in fair agreement with the well-known laminar-turbulent transition in pipe flow at Re, = 2300. It is probable that Brenner’s third assumption about linearized energy dissipation breaks down somewhat earlier.
number for
The transition Reynolds numbers shown in Fig. 3 appear reasonable for a particle fixed in a flowing fluid; however, an anomaly occurs for a particle settling in a quiescent liquid. In this case, the conduit Reynolds number can be made arbitrarily large by increasing the tube radius. The 695
J. LANGINS.
M. E. WEBER
plot shows that a dimensionless pressure drop of unity will be obtained in a sufficiently large tube, i.e., the small flows induced by settling in a large tube are not linear. An experimental test of this prediction can only be obtained in tubes very much larger than those used here. Acknowledgment-The National Canada provided financial assistance
Research Council for this work.
a
1. PLISKIN
D
drag on particle dynamic pressure difference induced by falling particle RO radius of conduit Re particle Reynolds number, 2avlu Reo conduit Reynolds number, 2Rov/v V terminal velocity of particle VOOfluid velocity in conduit at the particle location in the absence of the particle average fluid velocity in conduit V, pressure drop defined by 6 dimensionless Eq. (3)
AP+
of
NOTATION
A
and
radius of particle cross sectional area of conduit
REFERENCES
[II BRENNER
PI [31 [41 [51 [61 I71
H. and HAPPELJ.,J. FluidMech. 19584 195. BRENNER H., Chem. Engng. Sci. 1962 17 435. HAPPELJ.andBYRNEB.J.,fnd.Engng.Chem. 1954461181. FAYON A. M. and HAPPEL J.,A. 1. Ch. E. .I119606 S5. PLlSKlN I. and BRENNER H.,J. FluidMech. 1963 17 89. FELDMAN G. A. and BRENNER H.,J. Fluid Mech. 1968 32 705. LANGINS J., M. Engng. Thesis, McGill University, Montreal 1969. RhsumC- Les chutes de pression ont ttC mesurCes pour des cBnes avec des angles au sommet entre 30 et l20”, s’Ctablissant dans I’axe d’une canalisation verticale. Les donnCes disponibles ont ttt correlCes sur un relevC de chute de pression additionelle sans dimension contre le nombre de Reynold du conduit. Des rkgimes distincts hauts et bas du nombre de Reynold du conduit ont it& distinguCs avec une transition se produisant entre eux dans la gamme de 600 $ 3000. Les prkdictions thCoriques de Brenner pour des nombres de Reynold faibles sont vCrifiCes. Zusammenfassung-Es wurden Druckverluste gemessen fiir Kegel mit Spitzenwinkeln zwischen 30” und 120”. die sich axial in einem vertikalen Rohr absetzen. Die verfiigbaren Daten wurden mit einander in Beziehung gebracht, indem der dimensionslose zusPtzliche Druckverlust graphisch gegen die Reynoldssche Zahl des Rohres aufgetragen wyrde. Es wurden deutliche niedrige und hohe Bereiche von Reynoldsschen Zahlen festgestellt, mit Ubergangswerten im Bereich von 600 bis 3000. Die theoretischen Voraussagen von Brenner fiir niedrige Reynoldssche Zahlen werden beststigt.
696
Chemical Engineering Science, 197 1, Vol. 26, pp. 693-696.
Printed in Great Britain.
The pressure drop created by cones settling on the axis of a pipe J. LANGINSP, M. E. WEBER and I. PLISKINS Department of Chemical Engineering, McGill University, Montreal I IO, Canada. (First received
4 February
1970; accepted
15 July 1970)
Abstract-Pressure drops have been measured for cones with apex angles between 30” and 120”, settling axially in a vertical pipe. The available data have been correlated on a plot of dimensionless additional pressure drop vs. conduit Reynolds number. Distinct low and high conduit Reynolds number regimes have been distinguished with the transition between them occurring over the range from 600 to 3000. Brenner’s theoretical predictions for low Reynolds numbers are verified.
INTRODUCTION
(3) Energy dissipation is Oseen-type linearization. Feldman [6] confirmed Eq. for sphere Reynolds numbers behaviour he found for higher is shown in Fig. 1.
pressure drop due to the presence of a particle in a conduit has been studied theoretically by Brenner [ 1,2] and Happel[3], and experimentally by Fayon[4], Pliskin[S] and Feldman [6] for spheres. Feldman [6] investigated the variation of the additional pressure drop with Reynolds number and proposed an explanation of the behaviour for spheres. The purpose of our work was to study this same effect for cones of various apex angles, and to compare the results with those for spheres. Brenner[2] has shown theoretically that the product of the additional pressure drop and the conduit cross section area is given by THE ADDITIONAL
AP+A -=D
voo %I ’
2
D
(1)
(2)
for any particle shape or orientation. These results were derived under the following assumptions: (1) Particle Reynolds number is small. (2) Flow in the conduit is laminar in the absence of the particle.
by an
(2) experimentally less than 120. The Reynolds numbers
EXPERIMENTAL
For a particle on the axis of a cylindrical pipe,
AP+A=
described
The experimental-set up was similar to that used previously [5,6]. Particles settled in one leg of a U-tube filled with a solution of known viscosity. The particles were released by an electromagnet beneath the liquid surface and fell along the axis of the vertical 4.0 in. i.d. by 4ft long glass pipe which formed one leg of the U-tube. Each leg was sealed by a flanged plate provided with a pressure tap. The pressure difference between the taps was measured with a Baratron pressure transducer (MKS lnstruments of Burlington, Mass.). The apparatus was maintained at 23°C % O*Y’C. The particles were spheres A-1.5 in. dia. and cones with apex angles 30”, 45”, 60”, 90” and 120” and base diameters from O-5 to l-5 in. The particle densities were 7.79 g/cm3 (steel) and l-45 g/cm3 (polyethylene-steel composites). The composites were made by inserting a steel cylinder in a piece of polyethylene and machining to the desired shape. The five solutions used were mixtures of water
tGulf Research Centre, Sheridan Park, Ontario. SUniRoyal Research Center, Wayne, New Jersey.
693
J. LANGINS, I
M. E. WEBER and I. PLISKIN I
I
I
2.2>
z $
1.8-
24p
1.6-
4 3 2 2 2 P
O/R0 l
2,0_
.
da
.
;* i.
l.A
FULL
SYMBOLS
- AUTHOR
1.L
OPEN
SYMBOLS
- FElOMAN
1. o-
I
FLAGS
INDICATE
40
b 0.5
q
COMPOSITES
0.8-
0.1
1.b PARTIGLE
Fig. I. Dimensionless
lb0
til REYNOLDS
NUMBER
pressure drop parameter spheres.
and Ucon 50-HB-5100 (Union Carbide), and provided viscosities from 0.94 CP (pure water) to 2860 CP (pure Ucon). The average particle velocity was measured by timing the fall over a measured distance. Terminal velocities were obtained by correcting analytically for particle acceleration. In no case was the terminal velocity more than 6 per cent greater than the measured average velocity. Particle drag at the terminal velocity was found by weighing the particles when suspended in the fluid. The average deviation for the measured pressure difference between replicate runs was 4 per cent. There was no significant difference between homogeneous and composite particles. Additional description of apparatus and procedure is given by Langins [7]. RESULTS
AND
DISCUSSION
Figure 1 shows the present results and those of Feldman[6] for spheres. The ordinate is the dimensionless pressure drop, 6, defined by
&$yl_pJ].
(3)
The abscissa is the Reynolds number, Re, based .
1,doo
to.boo
100,000
-v
versus particle Reynolds number
for
on the sphere diameter and terminal settling velocity. The term in brackets in Eq. (3) corrects for wall effects and, following Feldman, was omitted for Reynolds numbers greater than 60. There is qualitative agreement between these data. However, our spheres, which had smaller a/R,, than Feldman’s, show a drop off in 6 around a Reynolds number of 30 while his show it around 120. Figure 2 presents the results for the cones. The Reynolds number here is formed with the base diameter and the terminal velocity. There is no trend with particle orientation during fall or with apex angle as predicted by Brenner[2]. Since Brenner’s analysis is the same for a particle settling in a motionless fluid and for a stationary particle in a moving fluid, the second and third assumptions in his analysis can be tested by plotting the data against the conduit Reynolds number based on terminal velocity and pipe diameter, Fig. 3. On this plot all of the data are drawn together on one curve. At low Reynolds number values 6 = 2 while at high values 6 = 1, and the transition occurs between Re, - 600 and Re, - 3000. The data of Fayon [4] and Pliskin[S], not shown, would fall on the 6 = 2 line since their largest conduit Reynolds
694
The pressure drop created by cones settling on the axis of a pipe
3
2.0
3 k T 3 F s
1.8
B 2 2
1.2
%
s
1.8
1.4
FULL
1.0
OPEN
SYMBOLS- RELEASED SYMBOLS
APEX
- RELEASED
APEX
DOWN UP
0.8
0.1
1.0
10 PARTICLE
Fig. 2. Dimensionless
REYNOLDS
100 NUMBER
1,000
10.000
100,000
-y
pressure drop parameter versus particle Reynolds cones of Apex Angle 30”-120”.
number for
2. ‘0 z* k *3
2. 1.
s
1.
% 5
1.
s 2
1.
F
1.
o
AUTHOR’S
l
FELDMAN’S
o
SELECTED
SPHERES SPHERES CONES
0
CDNDUIT
Fig. 3. Dimensionless
pressure
REYNOLDS
NUMBER
- F
drop parameter versus conduit Reynolds spheres and cones.
number was 120. The high bound of the transition, Re,, - 3000, is in fair agreement with the well-known laminar-turbulent transition in pipe flow at Re, = 2300. It is probable that Brenner’s third assumption about linearized energy dissipation breaks down somewhat earlier.
number for
The transition Reynolds numbers shown in Fig. 3 appear reasonable for a particle fixed in a flowing fluid; however, an anomaly occurs for a particle settling in a quiescent liquid. In this case, the conduit Reynolds number can be made arbitrarily large by increasing the tube radius. The 695
J. LANGINS.
M. E. WEBER
plot shows that a dimensionless pressure drop of unity will be obtained in a sufficiently large tube, i.e., the small flows induced by settling in a large tube are not linear. An experimental test of this prediction can only be obtained in tubes very much larger than those used here. Acknowledgment-The National Canada provided financial assistance
Research Council for this work.
a
1. PLISKIN
D
drag on particle dynamic pressure difference induced by falling particle RO radius of conduit Re particle Reynolds number, 2avlu Reo conduit Reynolds number, 2Rov/v V terminal velocity of particle VOOfluid velocity in conduit at the particle location in the absence of the particle average fluid velocity in conduit V, pressure drop defined by 6 dimensionless Eq. (3)
AP+
of
NOTATION
A
and
radius of particle cross sectional area of conduit
REFERENCES
[II BRENNER
PI [31 [41 [51 [61 I71
H. and HAPPELJ.,J. FluidMech. 19584 195. BRENNER H., Chem. Engng. Sci. 1962 17 435. HAPPELJ.andBYRNEB.J.,fnd.Engng.Chem. 1954461181. FAYON A. M. and HAPPEL J.,A. 1. Ch. E. .I119606 S5. PLlSKlN I. and BRENNER H.,J. FluidMech. 1963 17 89. FELDMAN G. A. and BRENNER H.,J. Fluid Mech. 1968 32 705. LANGINS J., M. Engng. Thesis, McGill University, Montreal 1969. RhsumC- Les chutes de pression ont ttC mesurCes pour des cBnes avec des angles au sommet entre 30 et l20”, s’Ctablissant dans I’axe d’une canalisation verticale. Les donnCes disponibles ont ttt correlCes sur un relevC de chute de pression additionelle sans dimension contre le nombre de Reynold du conduit. Des rkgimes distincts hauts et bas du nombre de Reynold du conduit ont it& distinguCs avec une transition se produisant entre eux dans la gamme de 600 $ 3000. Les prkdictions thCoriques de Brenner pour des nombres de Reynold faibles sont vCrifiCes. Zusammenfassung-Es wurden Druckverluste gemessen fiir Kegel mit Spitzenwinkeln zwischen 30” und 120”. die sich axial in einem vertikalen Rohr absetzen. Die verfiigbaren Daten wurden mit einander in Beziehung gebracht, indem der dimensionslose zusPtzliche Druckverlust graphisch gegen die Reynoldssche Zahl des Rohres aufgetragen wyrde. Es wurden deutliche niedrige und hohe Bereiche von Reynoldsschen Zahlen festgestellt, mit Ubergangswerten im Bereich von 600 bis 3000. Die theoretischen Voraussagen von Brenner fiir niedrige Reynoldssche Zahlen werden beststigt.
696