ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
Equipment Design Datu
.
olid-liquid Suspensions ertical Columns Correlations for Fischer-Tropsch and Relate A. BHAITACHARYA AND A.
Syntheses
N. ROY
lndion lnstitute of Technology, Khorogpor, lndio
F
LVID catalyst processes in industrial synthesis reactions have assumed considerable importance since the last world war. Finely powdered solid catalysts in the fluidized state are being successfully used in the large scale processing of petroleum. It has been found that the Fischer-Tropsch reaction for the production of synthetic liquid fuels can be efficiently carried out by suspending a finely pulverized catalyst in a coolant oil or in the synthesis gas stream, as in a fluidized reactor. The many advantages inherent in the slurry-phase Fischer-Tropsch synthesis have been well recognized ( 7 ) . Any attempt to design a slurry reactor necessarily requires a knowledge of the flow behavior of the slurries. In view of this, an investigation has been carried out on the flow characteristics of finely divided catalysts suspensions in water and oil in vertical tubes. A slurry may behave as a Xewtonian or non-Newtonian fluid. As it is composed of two components, the flow behavior will be influenced by a number of factors-e.g., size and shape of the particles, density, solid concentration and apparent viscosity of the suspension, and the surface properties of the two phases. A problem of special interest in such design work is the quantitative prediction of the pressure drop in different types of equipment from a knowledge of the flow behavior of suspensions. Previous investigations in this field have shoivn that viscosity or, more precisely, the coefficient of rigidity is the principal factor in determining the pressure drop in the viscous flow of suspensions. Babbitt and Caldwell(2) found an equation for the laminar flow of sewage sludge, clay slurries, drilling muds, etc., which exhibit the characteristics of a true plastic substance. They showed that the usual friction factor versus Reynolds number plots can be obtained in the turbulent region provided the viscosity of the fluid medium and density of the slurry are used in calculating the Reynolds number. Similar correlations of data on the turbulent flow of concentrated suspensions of cement rock and filtereel in water were obtained by Wilhelm, Wroughton, and Loeffel ( I I ) , using viscosity data corrected for yield value. Alves, Boucher, and Pigford ( 1 ) measured the flow of lime and titanium dioxide slurries in the turbulent region and found the turbulent viscosity from a plot off versus DVs:ps:. They have suggested a design procedure for sizing a pipeline or computing pressure drop from a knowledge of the viscosity data. I n all such correlations the materials studied were assumed to behave as single-phase fluids, and separate effects of the solid particles and the dispersion medium were not considered. In the present paper an analysis is presented in which the properties of both components have been taken into account t o correlate the pressure drop with other variables in the turbulent flow region. The concept of pneumatic conveying and the analysis presented by Vogt and White (IO) have been found useful in the interpretation and correlation of data of present investigation. 268
Observations of siurry tlow behavior are m a d e in model vertical glass column
Apparatus. The flow diagram of the experimental setup is shown in Figure 1. The flow studies were carried out in a specially designed apparatus consisting essentially of a vertical column, A , with a short horizontal section, B, a t the upper end terminating in a righbangle bend with arrangements, C, for either leading the fluid back to the tank, D, or collecting in a measuring cylinder, E. The horizontal section was illuminated by a beam of focussed light to determine the velocity a t which the solid particles would just begin to settle out of t,he slurry on emerging from the vertical section. Two borosilicate glass tubes of internal diameter 10 and 19 mm. and effective lengths of 50.5 and 100
r.
Figure 1.
Experimental apparatus
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
Vol. 41, No. 2
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT om., respectively, between the two open end manometers, F, were used. The slurry uniformly agitated in the tank by a paddle stirrer, G, was made t o flow through the column by means of a centrifugal pump, H. Two by-passes, one in the pump line, I , and the other in the inlet tube of the column, J, with regulators were used to control the flow. The tank, D,was provided with copper coils through which water could be circulated to keep the slurry a t the desired temperature. The manometer tubes were kept free of solids by flushing with oil, whenever i t was necessary, from an overhead vessel, IC.
Table 1.
Material Kieselguhr Iron-copperkieselguhr (No. 1) Iron-copperkieseliuhr (No. 2) Iron-copper
WATER KIESELGUHR SYSTEM
Physical Properties of Materials Disulacement Sp. GI!. 2.280
Mean Particle Diam., Ft. x 104 1.31
Diameter Ratio, 250.0
2.29
2.310
2.65
123.0
2.91
2.820 3.970
1.67 1.75
272.0 359.0
3.57 5.00
D/DP
Density Ratio, PS __
Pl
Materials. The solid materials used in this investigation were kieselguhr, alkalized iron-copper-calcium-kieselguhr, and alkalized iron-copper catalysts. Powdered kieselguhr was dried at 120" C. and sieved t o pass through 170-mesh screen. Three Fischer-Tropsch alkalized iron catalysts, prepared by the wellknown procedure (9),had the following compositions, in parts by weight: Iron Copper Calcium oxide Kieselguhr
h-0. 1
No. 2
100 3 10 30
100 8
5 15
Iron-Copper 9 1
....
The alkali content was 30 grams of potassium hydroxide per kilogram of catalyst. Fine powders of these catalysts were prepared by drying a t 200" C., grinding in a pebble mill, and sieving t o pass through 200-mesh screen. The properties of the materials are given in Table I. Density of solid materials was determined by water displacement for kieselguhr and by oil (kerosine) displacement for catalyst materials. For this experiment air bubbles were carefully removed from the surface of the solid particles in the liquid medium by alternate heating and cooling with shaking. Density of the slurries was determined by the standard density-bottle method. The average size of particles of the catalyst powders was determined from the data on hindered sedimentation of a suspension of known consistency in kerosine (4). This method, however, could not be used in the particle size determination of kieselguhr as the solid settling rate could not be observed because of highly turbid supernatant liquid layer. The size of the particles was therefore determined by the microscopic method (6). There were particles of different shapes and sizes. However, in calculating the average diameter, the particles were assumed spherical in shape. Viscosities of water and oil were determined in a standard Ostwald viscometer. For the determination of the viscosity of slurries a specially designed rotational viscometer, to be described in a separate publication, was used. Experimental Procedure. A run consisted of circulating a slurry of definite concentration and temperature through the apparatus; care was taken t h a t there were no air bubbles in the test section which would otherwise reduce the pressure drop. When a steady state was reached, the rate of flow was determined by measuring the volume of slurry flowing per unit time; the corresponding pressure drop across the test section was noted from the manometer readings. Several readings for each conFebruary 1955
Figure 2.
Pressure drop velocity for flow of suspensions
INDUSTRIAL AND ENGINEERING CHEMISTRY
269
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
Table II.
Substance
gave two different straight lines. The deviation from the usual f versus Re curve is shown in Figure 3. Logarithmic plots of against V,1 a t a particular concentration, c, and against c a t a constant velocity, V*Z, gave linear relations for all the systems investigated. For illustration, plots for the water-kieselguhr system are shown in Figure 4. The range of variation in the values of slopes of the straight lines was between 1.36 and 1.59 and 0.19 to 0.26, respectively, for the two types of plots. It was, however, found that if a group of the form V : ~ C O . ~ ~ / D was plotted against A p j / ~for a particular set of values of Val, C,D, and L,a linear relation was obtained (Figure 5 ) which can be expressed by an empirical dimensional equation of the form
Physical Properties of Suspensions
Concentration Grams Pounds visoosStatic Head, Ft. Watersolidi solidi Density ity Lb./ 100 co. cu ft. Lb./Cu: (Ft:)(Sec.) Small Large of slurry of slurry Ft. X 104 columna column5 Water-Kieselguhr System a t 32' C.
Water Slurry Slurry Slurry Slurry
1 2 3 4
4.3 6.9 9.9 12.9
2.68 4.31 6.18 8.06
62.11 63.68 64.50 65.70 66.83
5.16 5.64 6.07 6.74 7.53
.,. ... ... ...
0.0420
0.0680
0,0980 0.1280
Kerosine-Iron-Copper-Kieselguhr System No. 1 a t 30' C Kerosine Slurry5 Slurry6 Slurry7 Slurry8
3.2 6.7 9.1 12.4
2.00 3.56 5.69 7.75
49.61 50.90 52.20 53.70 55.20
8.04 8.52 8.91 9.52 10.00
0.0374 0.0682 0.1105 0.1520
. I .
... ...
(5)
,..
Kerosine-Iron-Copper-Kieselguhr System No. 2 a t 30' C. Kerosine Slurry 9 Slurry 10 Slurry 11
4.9 11.4 18.0
3.06 7.13 11.25
49.32 51.50 54.44 57.37
8.04 8.82 10.06 11.30
... ... ...
0.1150 0,2700 0,4250
Kerosine-Iron-Copper System a t 32' C. Kerosine Slurry 12 1.5 0.94 Blurry 13 3.4 2.13 Slurry 14 7.5 4.69 Slurry 15 9.7 6.06 Slurry 16 13.5 8.44 Tube diam. = 1.0 em.; tube b Tube diam. = 1.9 em.; tube Q
49.50 7.86 50.28 8.06 51.23 8.40 53.35 9.34 54.40 9.95 56.30 11.16 length = 50.5 om. length = 100 cm.
..
.
. .. .. .. ..
. ..
0.0388 0.0887 0.2010 0,2600 0.3600
centration were taken a t different velocities, and the flow rates were controlled by the by-pass valves. The concentration or the loading ratio was determined by collecting a 100-ml. sample during a run and determining the solid content by filtration and weighing in a Gooch crucible. Experiments were carried out in tubes of two different dimensions, using concentrations ranging from 3 t o 20 grams of solid per 100 ml. of slurry. Typical experimental data are presented in Table I1 and are graphically shown in Figure 2 . Additional experimental data are available (3).
This equation results in an excellent correlat>ionof experimental data. The term c in Equation 5 may be replaced by the loading ratio, R, and in that case the constant C will assume a different value. More data on various types of slurries are needed to determine the usefulness of this equation because there are two limitations -it is valid only when c is not zero and the constant, C, has a dimension that imposes restrictions in its use as a generalized empirical correlation. I n view of these difficulties the authors attempted a two-component analysis of the experimental data. It has been found that the concept of flow of solid-in-gas suspensions, commonly known as pneumatic conveyance, can be extended to develop an equation which gives a satisfactory correlation of the data of the present investigation. Two-component analysis, based on pneumatic conveyance, correlates pressure drop and properties of system
The pressure drop €or the isothermal turbulent flow of fluids in straight pipes is given by the familiar Fanning equation
where Partial correlation based on Newtonian flow is obtained
(7)
The pressure drop measured across a section, L, gave the sum of pressure drop due to the slurry flow and the static head due to the solids Apai = A p j f A b e
This relation has been derived by dimensional analysis of the variables describing the physical characteristics of the system.
(1)
A plot of A p j i ~ against linear velocity, Vel, gave a set of characteristic curves for each slurry system representing the trend of pressure drop as a function of concentration. Some typical plots are shown in Figure 2. Fanning friction factor, f =
221,TPdL
(4)
D V8ipal
0 001 IO0
and Reynolds number, Re = ___ were calcu-
1
2
4
6
8
1 1,000
2
4
6
8 1 10,000
2
Re
Pel
lated, and a plot o f f against Re for the turbulent flow of slurries in pipes of two different diameters
270
Figure 3.
Reynolds number correlation for Row of suspensions in glass tubes ( 7 2)
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 2
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT assuming t h a t the drag force on a particle obeys Stokes' law, the following equation was obtained: o
WATER-KIESELGUHR
SYS~EM
Vogt and White carried out studies on the flow of suspensions of sand, steel shots, clover seeds, and wheat in air through 0.5-
..
-1 I
n
9
0
0.1
0.2
0.4
0.6
vsI
1.0
inch commercial iron pipe and succeeded in correlating their data as well as the data of Gasterstadt and Segler on the pneumatic conveying of wheat for both horizontal and vertical flow by a generalized dimensionless equation of the form
1 1
2.0
I n Equation 13 the effects of particle shape and roughness have been ignored. It is, in fact, a n empirical modification of Equation 12 and the terms A and k are empirical functions of the dimensionless group
ftlscc
Figure 4. Relation between pressure drop and velocity and pressure drop and concentration for flow of suspensions
PI
I n functional form this may be expressed as
When suspended solids are introduced in a fluid stream flowing in turbulent motion through a pipe under a steady state the additional factors which will appear in the system are the effects of the solid phase and the local acceleration due to gravity. By taking into account several variables-e.g., the average effective diameter, D,, and density, p t , of the particles, the weight ratio of solids to fluid in flow, R, a roughness factor for the particles, 6, and a shape factor of the particles, E, a functional relation may be written that assumes constant pipe roughness and wetting of solid by liquid.
5?2- $2(D, VL,PZ,PL, Dm pa, R, 6 , L
6)
Attempts to correlate the data of the present investigation by assuming a n equation in the form of Equation 12 proved unsuccessful. It was, however, found that when a n empirical modification of Equation 12 was made in the following form a satisfactory correlation of the data could be obtained. a
- 1 = A (%) D 2
2 (RRe )k
(9)
Gasterstadt (6) studied the transport of wheat in horizontal pipes by means of air and obtained a linear relation between R and a, a dimensionless term defined as the ratio of pressure drop obtained in the flow of solid-in-fluid suspension to that obtained with fluid flowing a t the same velocity. Similar correlation was obtained by Seglar (8) for his data on the conveying of oats and wheats in vertical conduits. Vogt and White (10) extended this idea and showed t h a t a can be expressed as a function of the following seven dimensionless groups :
0
2 0
60
4 0
--~
100
80
1
120
: i ~ c ~ . ~ ~ D
Figure
5. Relation between pressure drop and Vs11.6C0.25
D
It was found, however, that itn empirical correlation of the many variables given in Equation 10 was very difficult and therefore a theoretical analysis of the problem was proposed. It was assumed that the drag force on a particle due to the fluid is a function of the relative velocity between the fluid and the particle and the properties of the fluid. Further, to keep the particles in motion the fluid expended some energy which was supposed t o balance the kinetic energy loss of particles caused by collision with the pipe walls. According t o Vogt and White, the ratio of average axial velocity of a particle t o the average fluid velocity is given by
By summing all the drag forces on the individual particles and
February 1955
for flow of suspensions
-
Figure 6 represents the relation between ( a 1) and Re/R for all the slurry systems investigated. This figure was constructed by selecting several sets of values of velocities and determining the corresponding ratio of pressure drops, a, for the different systems from Figure 2. The calculated data are presented in Table 111. Values of Re were calculated at the selected velocities, and the values of R were calculated from the reIation R = C For every system the plots could be represented by PI1 - c a straight line with a slope of minus 1. Equation 14 is, therefore, reduced t o
-.
a
- 1=A
(g)*(?)(&)
INDUSTRIAL AND ENGINEERING CHEMISTRY
271
ENGINEERING. DESIGN. AND PROCESS DEVELOPMENT
Table 111. Slurry NO.
R
Calculated Data
vegtF’ Sec.
A p f , Ft. Water Liquid Slurry
a
-
1
Re
Re/R
Water-Kieselguhr System 1 2 3 4
0.044 0.072 0.104 0.137
1.2
0.0455
0,0530 0.0580 0.0640 0.0885
0.165 0.280 0.410 0.505
4730
107400 65670 45460 34510
1 2 3 4
0.044 0.072 0.104 0.137
1.5
0.0685
0.0760 0.0820 0.0880 0.0935
0.110 0.200 0.285 0.365
5910
134300 82140 56830 43140
1 2 3 4
0.044 0.072 0.104 0.137
1.8
0.0940
0.1020 0.1090 0.1160 0.1225
0.085 0.160 0.235 0.305
7090
161200 98510 68210 51770
Keronine-Iron-Col,per-Kicselguhr System No I 5 6
1.2
0.0370
0.0480 0.0560 0.0640 0.0700
0.300 0.515 0.730 0.890
2425
8
0.041 0.073 0.119 0.163
59360 33170 20400 14870
5 6 7 8
0.041 0.073 0.119 0.163
1.5
0.0000
0,0700 0.0780 0.0870 0.0945
0.165 0.300 0.460 0.590
3030
74200 41470 25500 18580
5
0.041 0.073 0.119
1.8
0.0830
0.0940 0.1010 0.1110 0.1210
0.130 0.220 0.340 0.460
3640
89040 49760 30600 22300
7
; 8
0.163
The value of Vc,so derived, is less than Vatand is dependent on the concentration of the slurry. Hence, it has been defined a s the “reduced superficial liquid velocity.” If the A p f versus VBicurves are replotted and Apf versus 1.81 for each slurry system is plotted, curves similar t o those shown in Figure 2 are obtained. These curves are shown in Figure 7.
Table IV.
Effect of Solid and Liquid Properties on A and A’
System Water-kieselguhr Kerosine-iron-copperkieselguhr (No. 2) Kerosine-iron-copper Kerosine-iron-copperkieselguhr ( S o . 1)
Re
dC‘n
A .4 ‘ (Equation 15) (Equation 17)
0 , 673
0.575
4.00
0 705 0 965
0.670 1.225
3 73
1.200
2.140
14,fiO
ij,
ox
K h e n the values of o( were calculated from Figure 7 and the new values of ( a - 1) were plotted against Re/R for all the systems, a set of parallel lines similar to those in Figure 6 was obtained. The slope of these lines was slightly different-1.15, as shown in Figure 8. Equation 14 was thus modified to
Kerosine-Iron-Copper-Kieselguhr System No. 2 9 10 11
0.0632 0.1505 0.2440
1.00
0.0285
0.042 0.056 0.074
0.475 0.930 1.600
3820
O04GO 25380 15660
9 10 11
0.0632 0,1505 0.2440
1.50
0.05G5
0 070 0.097 0.123
0.346 0.715 1.175
5730
90670 38080 23480
9 10 11
0.0632 0.150; 0.2440
1.75
0.0770
0.096 0.122 0.149
0.245 0.585 0.935
6685
105800 44420 27390
9 10 11
0.0632 0.1506 0.2440
2.00
0.0990
0.120 0.148 0.176
0.215 0.495 0.780
7640
120900 50760 31320
12 13 14 15 16
0.019 0.043 0.0S6 0.125 0.176
1.00
0.030
0.037 0.043 0.056 0.062 0.071
0.230 0.430 0.830 1.060 1.360
3920
206700 91200 40830 31360 22240
12 13 14 15 16
0.019 0.043 0.096 0.125 0.176
1.40
0,056
0.062 0.069 0.085
5490
0.104
0.115 0.230 0.520 0.680 0.860
288900 127600 57190 43920 31200
12 13 14 15 16
0.019 0.043 0.096 0.125 0.176
1.80
0.091 0.099 0.118 0.128 0.140
0.085 0.180 0.405 0.525 0.665
7060
371500 164200 73600 56480 40120
Kerosine-Iron-Copper-System
O.OQ4
0.084
il comparison of the plots of Figures 6 and 8 shons that Equation 17 docs not give a batter fit of the data than that obtained with Equation 15. In fact, correlation was better with Equation 15. This was particularly true with the data obtained in the higher velocity region, a t higher slurry concentration, and in the larger diameter column. From the physical characteristics of the suspensions it was hoped that it might be possible t o express the values of A and A’ as functions of the dimensionless group, Re 46,as was done by Vogt and White. Accordingly, the values of A and A‘ were calculated from Equations 15 and 17, respectively, and plotd ’ l d ~ b- PdpzgD3,, as shown in Table ted against the group Pl
I V and Figure 9. These plots show that the relation bctween Re dcf;and the values of A and A’ is linear. Therefore, the data can be represented by Equation 1.5. However, present data are insufficient to determine the general applicability or Equations 15 and 17 to slurry systems outside the range actually investigated.
COLUMN
SYSTEM
Next, the value of the constant, A , for each system as calculated from Equation 15, and it was found that the vducs of A differed from system to systein. I n Equation 15, the Reynolds number has been calculated on the basis of the observed velocity, Vat. In two-component analysis it is, however, desirable t o consider the velocity of the dispersing medium (fluid) instead of the observed velocity of the slurry. I n the flow of solid-in-gas suspensions, a term such as “velocity of suspension” is difficult to define. However, in a solid-liquid system, the velocity of the slurry is a measurable quantity. I n the former case, the superficial gas velocity is obtained by dividing the total volume of gas flow by the area of cross section of the duct, neglecting the projected area of the solids. I n the case of solid-in-liquid suspensions, the velocity of the liquid can be calculated from the slurry velocity as the total volume of flow of liquid per unit time divided by the area of cross section of the conduit-that is,
272
DIA.(crn) K e r o s ne-~k-CuKresclguhrfil I 0
,A
u K C r m 1ne-FeCu-Kiesciguhr(2) 1.9 K C r o 5 1 ne-Fc-Cu 19 0 ‘#ate*- Kieselguhr I*Q
-0
I
2
4
6
8
1
104
Idr
Figure 6.
2
lo5
la
4
6
5 1 106
Effect of Re/R on relative pressure drop based on velacity of suspensions
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 41,No. 2
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
4 Figure 7. Pressure drop versus reduced superficial liquid velocity for flow of suspensions Conclusion
With a view t o designing slurry reactors for the Fischer-Tropsch and related syntheses, in which the reacting gases are contacted with a circulating fluid catalyst, studies have been carried out on the settling and flow behavior of solid-in-liquid suspensions. Flow experiments were performed in model vertical glass columns using slurries composed of very finely divided catalyst materials uniformly dispersed in liquid mediums such as oil and water. Observations, both qualitative and quantitative, were made in the region extending from critical t o turbulent flow. The present study was limited to slurry systems composed of finely divided solid particles (40 to 80 microns) suspended in liquids such as water and kerosine. The concentration of solid was relatively low-2 t o 11.5 pounds of solid per cubic foot of slurry. The average linear velocities of the slurries were varied from 0.5 t o 2.5 feet per second, which corresponded t o a variation in Reynolds Number from 2000 t o 9500. No data were taken below this velocity range as the solid particles were observed t o settle out in the horizontal section. The critical region for the commencement of settling was a t a Reynolds number between 1500 and 2000 for the concentration range investigated. The data of the present investigation indicate that the pressure differential required t o maintain the flow of solid-in-liquid suspensions in vertical columns can be represented in the turbulent region by the empirically derived equation 0 5
0
I O
KEROSINE IRONCOPPER KIESELGUHR(1) SYSTEM
I S
20
where A is the linear function of the dimensionless group, Re 6 . This*equation is based on the two component analysis of flow of slurries and was developed by analogy with the concept of pneumatic conveyance. A new term, reduced superficial liquid velocity, defined as
Vr =
(psz
-P I ') "',
was introduced to express separate effects
of solids and liquids, and Equation 15 was modified t o
Equations 15 and 17 gave essentially the same result. 0
SYSTEM Q
K e r o s t ne-Fc-Cu-Kieselguhr~t) I 0
o Water - Kle sc1 quhr
I
COLUMN D l A (cm
o4
I
o5
1.0
IO6
RC / R
Figure 8.
February 1955
Effect of Re/R on relative pressure drop based on superficial liquid velocity
INDUSTRIAL AND ENGINEERING CHEMISTRY
273
ENGINEERING. DESIGN. AND PROCESS DEVELOPMENT
L
A‘ 2
6
4
IO
20
Ap,
R
SYSTEM
Walcr-Kieselguh r v KerOSi ne-Fe-cu-Kierelquhrii) Q K e r o s i ne-Fe-Cu-Kiesclquhr 12) Kerosine-Fe-cu D
I I __
Re VL Vi
V? V,l
~
1
a, a’
I
0.1
0.2
0.6
0.4
1.0
2.0
A
Figure 9. Effect of solid and liquid properties on A (Equation 15) and A’ (Equation 17)
6 E
p1 pal
A partial correlation of the data was also obtained by assuming that the suspensions in turbulent region of flow behave as a single-component Newtonian fluid. However, this type of correlation was found t o be unsatisfactory. For efficient operation of the slurry process for the FischerTropsch and related syntheses, the catalyst requires very fine grinding to -200 mesh size in coolant oil. The solid concentration may be varied from 20 t o 25 grams per 100 ml. of slurry. Therefore, data from this investigation, obtained under similar conditions, can be applied in the reactor design. In order t o determine the applicability of Equations 15 and 17 to slurry systems in general, a much wider range of data is needed. It is therefore suggested that further studies should be carried out in the higher velocity region with suspensions of higher solid content and particles of known physical characteristics. Acknowledgment
The authors n-ish to espress their sincere thanks to S. R. Sen Gupta, director, Indian Institute of Technology, and to J. C. Ghosh, vice-chancellor, Calcutta University, for their kind interest and encouragement during the course of this investigation. Thanks are also due t o E. Wiengaertner and D. Venkateswarlu for helpful discussions. Nomenclature
A , A‘ = empirical constants = concentration of solids in vertical tube, lb. solid/cu. c ft. slurry C = dimensional constant C D = drag coefficient D = tube diameter, ft. D, = average effective diameter of solid particles, ft. = Fanning friction factor, dimensionless f = acceleration due to gravity, ft./(sec.)(sec.) g = conversion factor = 32.17 lb. mass-ft./(force-lb.)(sec.2) go k, k8 = empirical constants K = ratio of average axial particle velocity to average fluid velocity
2 74
ft. pressure drop due t o flow of pure fluid, lb./sq. ft. pressure drop due t o flow of suspensions, lb./sq. ft. pressure drop in flow of suspensions, ft. of water observed pressure drop in flow of euspensions in vertical column, ft. of water = static head due to solids, ft. of water = weight ratio of solid to fluid flowing = Reynolds number = velocity of liquid, ft./sec. = reduced superficial liquid velocity, ft./sec. = velocity of liquid relative to the particle, ft./sec. = velocity of slurrv, ft./sec. = relative pressure drops-ratio of pressure drop obtained in flow of slurry to that obtained with fluid flowing a t V L = V,Land V L = Vl, respectively, dimensionless = roughness factor for particles = shape factor for particles = viscosity of liquid, lb./(ft.)(sec.) = viscosity of slurry, lb./(ft.)(sec.) = density of liquid, lb./cu. ft. = density of solid, lb./cu. ft. = density of slurry in vertical tube, lb./cu. ft. = density of water, Ib./cu. ft. = length,
Ap = A ~ F= Apj = Ap*z =
pl
pa pal
pw
literature Cited (1) Alves, G. E., Boucher, D. F., and Pigford, R. L., Chem. Eng. Prop., 48, 385 (1952). (2) Babbitt, H. F.. and Caldwell, D. H., IND. Exa. CHEX.,33, 249
(1941). (3) Bhattacharya, -4.. and Roy, A. S . , Am. Doc. Inst., Library of Congress, Washington 25, D. C., Doc. 4415.
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(12) Zimmerman, 0. T., and Lavine, I., “Chemical Engineering Laboratory Equipment,” Industrial Research Service, Xew Hampshire, 1943. RECEIVED for review May 28. 1984. ACCEPTED October 6. 19.54. Sfaterial supplementary t o this article has been deposited as Document S o . 4415 with the AD1 Auxiliary Publications Project, Photoduplication Service, Library of Congress. ‘&’ashington 25, D. C. -4copy may be secured b y citing the document number and b y remitting $1.25 for photoprints or $1.25 for 3 6 m m . microfilm. Advance payment is required. Make checka or money orders payable t o Chief, Photoduplication Service. Library of Congress.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47,No. 2
