Gas Flow and Pressure Drop through Moving Beds Sok Moon Yoon and Daizo Kunii Department of Chemical Engineering, University of Tokyo, Tokyo, Japan
Characteristics of air flow through moving beds were observed using two kinds of tubes, 4.1 and 7.0 cm in
i.d.
-
and four kinds of glass beads,
d,
=
133, 261, 430,
and 1 130 microns. Longitudinal pressure gradient was determined by slip velocity of gas relative to the descending solids; gas flowing into the moving
bed
from the feed
hopper played an important role in determining the mass balance at the gas outlet.
MOVING
BEDS of solid particles have been increasingly applied in physical and chemical processes of gas-solid systems in the metallurgical, chemical, and petroleum industries. However, only a few investigations have been carried out on the pattern of gas flow through moving beds, notwithstanding extensive studies in the field of fixed beds. Happel (1949) carried out an extensive study of the factors influencing pressure drop due to gas flow through moving beds, and devoted most of his study to comparing the results with closely related work. Shanahan and Schwarz (1954) studied the downflow of glass beads of uniform size in a 1-inch-diameter tube with a superimposed air flow in either direction. Muchi et al. (1962) measured the radial and longitudinal diffusion coefficients of fluids in moving beds. Terada et al. (1963) investigated the pressure resistance through shaft kilns for calcination of limestone. All investigations previously reported were made on the assumption that if the rate of gas flow measured a t the outlet of the bed was the real gas flow through the test section, experimental results obtained in fixed beds could be applied to moving beds as far as the pattern of gas flow was concerned. Happel (1949) obtained experimental results as follows: Even a t a high rate of catalyst flow-for instance, 1000 pounds per minute (descending velocity of solids about 12 cm per second) with gas flowing upward-there was little change in the rate of air flow measured a t a given pressure difference between two levels in the moving bed. At first glance, these results might suggest that the downward movement of solids did not appreciably affect the real rate of gas flow through the moving bed. Since the pressure gradient in the longitudinal direction originates in the slip or relative velocity between the solids and gas stream, this conclusion cannot be accepted, especially when the slip velocity is comparable t o the descending velocity of solids. This paper presents evidence of the above seemingly contradictory phenomena both theoretically and experi-
mentally, employing fine solid particles, focusing attention on the slip or relative velocity between gas and solids. Basic Considerations
Nearly all workers doing research on moving beds seem to have assumed that the amount of gas flow measured a t the outlet of the bed was the same as the gas flow through the test section (Figure 1, left). If this assumption is not correct, this model may result in a wrong analysis of the moving bed system, accompanied by chemical reaction, adsorption, or even heat and mass transfer. For a moving bed of fine solids which descend a t a high velocity, say 1 2 cm per second, there is no gas flow across any section marked in the moving bed, especially when the upward relative velocity of the gas stream to the descending solids is just 12 cm per second. If solids descend faster than this, one should conclude that the gas flows downward against the pressure gradient, which originated in the slip velocity between solids and gas flowing through the void space in the moving bed.
Figure 1. Schematic diagram of movements of gas and solid in moving bed leff. Right.
Conventional model Present model
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4, 1970 5 5 9
A conventional flow model cannot explain the above phenomena. Here let us take into account both the solids and gas streams coming into the moving bed from the feed hopper (Figure 1, right). Mass balance gives
G,
-+ G = G ,
(1)
Consider a bed of cross-sectional area A,, containing N uniform spheres. The pressure drop due to fluid flowing through the bed will be the sum of the frictional resistance on all the spheres. Then
AP-At = R / * N (2) where J P is the pressure difference through the bed and Ri presents the frictional resistance on one sphere. When the solids move downward, the frictional resistance, R,, on one sphere should be determined by the slip velocity of fluid relative to the moving solid. Assuming both the descending velocity of solids and the superficial velocity of gas flowing upward based on the empty tube to be u, and u,,, respectively, the linear slip velocity of gas relative to the descending solids, A L L ,can be given by All = u,, f t u, (3)
,-Test
section
-+
According to Equation 2,
Hence we can presume that the constant slip velocity determines the constant pressure drop in moving beds. Therefore, if the rate of downflow of solids increases under the constant pressure drop, the amount of upflow of gas may decrease. When the linear slip velocity of gas ALL relative to the descending solids becomes less than the velocity of the descending solids, gas flows downward a t any fixed section of the tube against the pressure gradient with higher pressure a t the bottom of the bed. Experiments for Gas Flow
The experimental unit is illustrated in Figure 2. Two kinds of test sections were employed-a 7.0-cm-i.d., 90-cm long acrylic pipe, and a 4.1-cm-i.d., 45-cm long iron pipe. Four kinds of glass beads were used in the experiments (Table I ) . The average particle diameters were obtained by reciprocal mean diameter,
where
refers to the weight percentages, W , , of particle diameters d,,,. Referring to Figure 2, upper, solids supplied to the feed hopper, A , by a pneumatic conveying system or a bucket moved down through a test section, B , according to the gravitational force, and were discharged. The amount of solids was controlled by a regulating table feeder and a gate valve. For each run weight rates of solids were determined intermittently by measurement of the discharged solids taken over a definite short time. The air was admitted to pressure drum C from a blower, passed through the test section, and discharged at outlet 560
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4,1970
Injection tube 2 mm 0.0. Figure 2 . Schematic diagram of experimental moving bed and cross-sectional view of tracer injection tubes (X-Y) 1. 2. 3.
Supply of solids to hopper
6. Solids to table feeder 7. Water manometer A . Hopper B. Test section C. Pressure drum
Air out to flowmeter Sample to thermal conductivity cell by aspirator
4. Position of tracer injection 5 . Air from comoressor
Table I. Physical Properties of Glass Beads
Type of Glass Beod
Av. Particle Diameter, Microns
Particle Density, G/Cm3
Bulk Density (in Flowing Condition), G/Cm3
Void Fraction
GB704K GB706K GB708K GB503M
133 261 430 1130
2.47 2.48 2.48 2.48
1.51 1.57 1.55 1.56
0.390 0.367 0.375 0.371
2 (Figure 2 ) . At the outlet of the bed, the rate of air discharge was measured with a soap film flowmeter, by which a relatively small amount of gas could be measured. The pressure drop in the test section was measured by a water manometer. The pressure taps for the manometer consisted of three holes evenly spaced around the pipes. With the steady-state flow of the air through the moving bed, helium tracer was injected into the middle part of the test section-position 4 in Figure 2 . A plane source
of tracer might be expected with a radial injection (Figure 2, lower). Sampling tubes of 1.2-mm i.d. were positioned a t two levels 20 cm apart from the place of tracer injection, X-Y , upward and downward, respectively, for 7.0-cmi.d. pipe. For 4.1-cm-i.d. pipe the distance between the two levels and the place of tracer injection was taken to be 15 cm. The gas sample was aspirated from the center of the test section a t a constant rate of 30 cm per second, and the tracer concentration was measured for each run, by detecting the output concentration of gas sample flowing through the thermal conductivity cell. At steady-state operation, the rate of gas flow was determined by the measured concentration of tracer, knowing the injection rate of tracer gas. Keeping the pressure drop constant, flow rates of gas were determined for various descending velocities of solids, and similar experiments were made for different pressure drops between the test section, employing two kinds of pipe diameter and four kinds of glass beads. Experimental results for 261-micron glass beads are shown in Figure 3, where the linear velocity of the air, ugm/c, was correlated with the linear velocity of descending solids, u,. Figure 3 indicates that the linear velocity of gas flowing through the bed is determined by the descending velocity of solids a t a given constant pressure 3
drop, and the direction of gas flow changes from upward to downward a t a high descending velocity of solids, against the pressure gradient through moving beds, with higher pressure a t the bottom. Graphs similar t o Figure 3 were obtained for other sizes of solids-Le., 133, 430, and 1130 microns. Gas Flow
Void fraction is one of the important variables influencing pressure drop through a bed of solids, and it may be affected by the gas velocity or the descending velocity of solids in moving beds. Inserting slide valves into the test section of a bed of aluminum granules, Sissom and Jackson (1967) measured the variation of the void fraction with air velocity. Ernst (1960) measured the effect of the descending velocity of grains on the void fraction and obtained a slight increase with increasing solid velocity. Brown and Richards (1960), employing a high speed camera, observed the fluctuation of the void fraction with time and obtained the mean value of the fluctuation corresponding to a loose-packed condition. Happel (1949) indicated that the flowing bulk density of a catalyst in moving beds agreed within 1%with an apparent density for a loosely packed bed of the same material. The descending bulk densities of glass beads referred to in Table I are determined under flowing conditions in a moving bed. The bulk density, PO, can be defined by
per cm3)= descending rate of solids, glsec rate of air displaced by descending solids, cc/sec
ph(g
2
I U
%
\
E,
\
0
-I
-2
I
1
I
I
I
I
. .
I
2
I
I
4
3
u,, cm/sec Figure 3. Rates of gas flow through moving beds a t various pressure gradients and linear velocity of descending solids Upper.
Glass beads
dp
=
8
f
=
0 0.142
261 microns 0.367 Dr = 4.1 cm AP/L. g./cm’ 0 0.366 0.311 0 0.254 0.220 0 0.180
0 0.126 0 0.109 Lower. D,
e
a
0.171
=
where the rate of solids is obtained by weighing the representative sample descending for a given period of time, and the rate of air displaced by descending solids is measured by a soap film flowmeter connected to the hopper inlet. While the solids are moving down, the air flows into the displaced area in the hopper through the flowmeter. The mean air rates displaced by descending solids are obtained over a long measuring time, while displacing rates fluctuate over a short measuring time. The bulk densities measured directly under flowing conditions coincide with the densities measured a t a normal packed condition for the glass beads employed in this work. Void fractions referred to in Table I are computed from both flowing bulk and solids densities. From the experimental data shown in Figure 3, relations between the pressure gradient, l P / L , and the slip velocity, A U , were plotted in Figures 4 and 5 , including experimental data for zp = 133, 430, and 1130 microns. Even though data points are somewhat scattered, experimental data of longitudinal pressure gradient AP L in moving beds can be correlated well by applying the slip velocity of the air relative to the descending solids, including the experimental data observed in fixed beds. For predicting the pressure drop in fixed beds, KozenyCarman’s equation (Carman, 1937, 1938, 1939; Kozeny, 1927) can be applied.
7.0 cm
A
AP/L< g./cm’ 0.066
A A
0.032
0.050
Ind. Eng. Chern. Process Des. Develop., Vol. 9, No. 4,1970 561
04
Ergun's Equation
d U , cm/sec figure 4. Relation between slip velocity and pressure gradient a t various velocities of descending solids
A
A
Dr. Cm Fixed bed, 7.0 Moving bed, 7.0 Fixed bed, 4.1 bed, 4.1
Upper. d,
=
L
=
Lower. d,
=
c
=
0 Moving
133 microns 0.390 261 microns 0.367
0.2
/ /
0.2
6
E, Y
E,
Ergun's Equation
2-
I
0,
--. 2-
LL
a 0.1
0
Om
I
2 3 tAU, cm/sec
4
5
Figure 5 . Relation between slip velocity and pressure gradient a t various velocities of descending solids D,, 4.1 cm Upper. d, = 430 microns ~
0 Fixed bed 0 Moving bed
Ergun (1952) gave the following equation applicable for fixed beds a t a wider range of Reynolds numbers than Equation 5 .
=
t
lower.
fm
dp
=
e
=
0.375 1130 microns 0.371
Re, =
=
562
Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 4,1970
f m =
1000 __ Re, 207
fm
=
(1 -
!Jt
for Re, Observing the pressure drop through moving beds, Happel (1949) assumed that the amount of gas measured a t the outlet of the beds was the same as that of gas flowing through the test section, independent of solid velocity, and presented the following correlation.
4&Pi
200 + -Re:'?'
RT
200
200
(7)
However, Figure 3 clearly shows that the pressure gradient decreases linearly with increasing velocity of descending solids, u s , even a t a constant rate of gas flowi. e., u,,/c. This comparison suggests that Equation 7 could cause misunderstanding of the behavior of gas flowing through moving beds. Taking up to be AU in Equation 6, and using the void fractions of moving beds, the pressure gradient, AP L , was calculated from Equation 6, resulting in solid lines in Figures 4 and 5. The calculated lines fit fairly well the data of both fixed and moving beds. Comparing the calculated lines with the experimental data in Figures 4 and 5, Equation 6 can predict the flow characteristics in moving beds, if slip velocity is used instead of superficial gas velocity in fixed beds. Gas Flowing out of Moving Beds
Figure 6 shows the gas flowing into the moving bed accompanied by the solids from the hopper. Equation 1 can be rewritten as
A , u,, +
= Go (8) where A , is the cross-sectional area of the moving bed (square centimeters) and u,, is the superficial gas velocity through the moving bed (centimeters per second). I n previous investigations of moving beds, little attention was given to the amount of gas flowing in from the hopper, and it was believed that A,ua, = Gofor moving beds. Referring t o Figure 6, however, one can understand that the gas flowing into the bed from the hopper, Gh, plays an important role in the mass balance. When both the hopper and the outlet are open to the atmosphere, G can be given by Gh
Gh = A ,
t
U,
(9)
Equation 9 means that the amount of gas flowing from the hopper corresponds t o the amount of gas accompanied by the bulk flow of solids. Substitution of Equations 3 and 9 into Equation 8 gives
A,
t
AU = G ,
(10)
Theoretically, the volumetric rate of gas flow measured a t the outlet from the bed is exactly the same relative to the moving solids, as long as pressures a t both the top of the hopper and the outlet are identical. N o matter what value of descending velocity the solids have, the slip velocity, Au, remains constant a t a constant pressure gradient. Furthermore, the flow rate of gas measured should also be constant, independent of the descending velocity of solids, even though the proportion of gas coming from the hopper varies considerably, depending on the solids velocity. The relation between the longitudinal pressure gradient, AP L , and the apparent gas velocity, G,/A,, is shown in Figure 7, where Go is the rate of gas flow measured a t the outlet and A , is the cross-sectional area of the test section. If A U and G,/A, are compared a t the same pressure gradient in Figures 4, 5 , and 7 , the slip velocity is nearly the same as the apparent gas velocity, G,/A,. From Equation 10,
Go/A , = ~ A u
(11)
This comparison of two figures suggests that Equation 11 is adequate. T o prove the assumption which led to Equation 8, experiments were undertaken in the case where the direction of gas flow was downward through the test section against the longitudinal pressure gradient with higher pressure a t the bottom, and gas flowing through the outlet of moving beds was the difference between the gas flowing in from the hopper and the gas flowing downward. Experiments to Find Gas Coming from Hopper
Referring to Figure 6, unadsorbing tracer, He, was injected into the outlet of the hopper, 1, and gas samples were taken by aspiration from two positions-2 a t the outlet of gas flow and 3 a t the upper part of the test section. Tracer injection and measurement of gas concentration were carried out a t a steady state by the procedures previously mentioned. The following data illustrate that gas flowing from the hopper compensates for gas flowing through the outlet of countercurrent moving beds.
Experimental Conditions Bed diameter Bed area Gas Solids Solids flow rates Void fraction Pressure drop He injected
D, = 7.0 cm A, = 38.4 cm' Air Glass beads, d , = 261 microns u, = 0.676 cm second c = 0.367 AI' L = 0.05 gram/cm'. cm 0.1025 cm'/second
Results. From Figure 4, we obtain for AP = 0.05 gram/ cm'. cm t ~ =u 0.13 cm per second Figure 6. Flow pattern in countercurrent moving bed 1.
Tracer injection
2, 3. Sampling tubes for gas
Therefore, the rate of gas flow through the test section is Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4, 1970 563
1
i:'
0.5
'
'
Ergun's Equation
t // '
O ov
I
I
I
0.2
I
0.4 G,/A,,
1
I
I .o
0.8
0.6
1
I
I
I
cm/sec
Figure 7. Relation between apparent gas flow rate and pressure gradient a t various velocities of descending solids
A A
Dt, Cm Fixed bed, 7.0 Moving bed, 7.0 Fixed bed, 4.1
0 Moving bed, 4.1 Upper. lower.
XP
=
133 microns
2,
=
261 microns
n x .
I
I Ergun's Equation
/. t/
/
-
. --
A/ 0
O o
w
1
I
1
1
1
2
3
G,/A,,
I
I
4
1
5
cm/sec
Figure 8. Relation between apparent gas flow rate and pressure gradient a t various velocities of descending solids D,, 4.1 cm
0 Fixed bed OMoving bed
U,
= tAu
- €us= (0.13) - (0.367) (0.675) = -0.105 cm per second
G, = -(0.105)(38.4) = -4.04 cm3 per second (12) The negative sign in Calculation 1 2 indicates that gas flowing through the test section is downward against the longitudinal pressure gradient, with higher pressure a t the bottom. Measurement of tracer concentration a t position 3 in Figure 6 by the thermal conductivity cell gave 1.09% of He in the gas flowing through the test section. Therefore, the amount of He flowing through the test section with gas is 564
d, Jp
= =
430 microns 1130 microns
-(4.04) (0.0109) = -0.0440 cm3 per second
or
I
Upper. Lower.
and the remainder of He injected is the amount through the outlet of the bed: 0.1025 - 0.0440 = 0.0585 cm3 per second (13) On the other hand, the amount of gas leaving the bed was obtained by measuring gas flow at the Outlet,
Go = 5.3 cm3 per second At position measurement by the thermal conductivity cell gave l.loyc of H~ in G,. ~ h ~the ~amount ~ off He flowing through the outlet of the bed is
Y
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4, 1970
(5.3)(0.0110) = 0.0583 cm3 per second
(14)
~
~
Calculation 14 gives the same value as Calculation 13, suggesting the fundamental concept shown in Figure 6, as well as by Equation 8. The same conclusion was obtained from the other experimental data under different conditions. Especially with a high velocity of descending fine solids, the above findings become important when chemical reaction takes place in moving beds. High pressure a t the bottom might make one misunderstand the direction of gas flow, if the gas flowing out is taken to be the same gas which flowed through the moving bed. But the situation may sometimes be reversed. Consider a reactant gas injected steadily a t the middle of a moving fine catalyst, with the expectation that the product gas may flow out of a lateral outlet positioned a t the top of the moving bed and just below the inlet of solids from the hopper. I n this case, in practice, the product gas flowing out of the bottom of the moving bed may present a hazard, and only the inert gas may flow out of the outlet from the hopper. Conclusions
Flow characteristics of gas through moving beds were investigated experimentally and theoretically. Since pressure drop between two levels in a moving bed is determined by the slip velocity of gas relative t o the descending solids, correlations of the pressure drop for fixed beds can be applied to moving beds, using slip velocity, d u , in place of superficial gas velocity, u,, in fixed beds. Some portion of the outlet gas comes from the hopper, accompanied by the entering solids. With high descending velocity of solids, the gas flows downward to the bed against high pressure a t the bottom of the bed. Therefore, gas flowing into the moving bed from the hopper plays an important role in mass balance a t the outlet. I n the special case mentioned above, for instance, gas flowing out of the moving bed is only the part of inert gas which has just come from the hopper. Flow rates of gas measured a t the outlet of the moving beds are equal to A,du as long as pressures at both the inlet and outlet are identical. Nomenclature
A, A, d,, Dt fm
= = = = =
sectional area of moving bed, cm2 sectional area of test section, cm2 average particle diameter, cm diameter of test section, cm modified friction factor, defined by Equation 7
g , = gravitational constant, gcm/gram.sec‘ Gh, G,, Go = volumetric rate of gas through hopper, moving bed, and outlet, respectively, cm,’/sec L = length of test section, cm N = number of solid particles AP = pressure drop, gram/cm’ Re = Reynoldsnumber Re, = modified Reynolds number, defined by Equation 7 Ri = frictional resistance u/ = superficial fluid velocity, cm,’sec ue,* = superficial gas velocity through moving bed, cm/ sec u, = downward velocity of descending solids, cm/ sec ~u = linear slip velocity of gas relative to descending solids, cm/ sec W = weight percentages of screened fractions of material W1, W,, etc., having average diameters of d , , d,, , etc. t = void fraction pi = fluid viscosity, g/cm.sec ph = bulk density of solids, g/cm’ pi = fluid density, g/cm” p 3 = solid density, g/cm” C#I~ = sphericity Literature Cited
Brown, R . L., Richards, J. C., Trans. Inst. Chem. Engrs. 38, 243 (1960). Carman, P. C., Trans. Inst. Chem. Engrs. (London) 15, 150 (1937); J . SOC.Chem. Ind. (London) 57, 225 (1938); 58, 1 (1939). Ergun, S., Chem. Eng. Progr. 48, 89 (1952). Ernst, R., Chem. Ing. Tech. 32, 17 (1960). Happel, J., Ind. Eng. Chem. 41, 1161 (1949). Kozeny, J., Sitzber. Akad. Wiss. Wien, Math-naturu. K I . (Abt. l l a ) 136, 271 (1927). Muchi, I., Morita, N . , Mamuro, N., Kishibe, M., Matsushita, Y., preprint for Hamamatsu meeting of Chemical Engineering Society, Japan, 1962. Shanahan, C. E., Schwarz, J., M.S. thesis, Massachusetts Institute of Technology, Cambridge, Mass., 1954. Sissom, L. E., Jackson, T. W., J . Heat Transfer, Trans. A S M E 89, 1 (1967), Terada, K., Shimizu, S., Kamimoto, K., Nanba, J., Sekko to Sekkai, No. 63, 73 (1963).
RECEIVED for review March 24, 1969 ACCEPTED January 19, 1970
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4, 1970 565
