ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT plus power rectification investment 0.836 kw.-hr./liter X 3785 liters/hour X $85/kw.-hr. = $269,000 results in a total investment in equipment of $512,000. With an allowance of an estimated $100,000 for instrumentation, tanks, and piping, and another $50,000 for building, including lighting and heating, the total capital outlay for a continuous two-chambered cell installation would be $677,000, a figure which, together with the figures reported previously, may be resolved into a cost-per-gallon figure. Thus, an hourly output of 1000 gallons for a 350-day working year, amounting to 8,400,000 gallons, gives the following breakdown of costs:
Electrolytic cells Rectification equipment Building Tanks, piping, etc. Interest on investment
Depreciation/ Year, % 10 6 3 20 6
Less operating profit (0.045 cent/liter)
cost, Cents/Gallon 0.294 0,200 0 0179 0,238 0.491 1 ,2409 0.1704 1.071
Labor and maintenance would bring the cost to approximately 1.4 cents per gallon total. The figure of $677,000 is comparable to the capital investment of a lime neutralization installation as given in a recent private communication. This particular installation includes a sufficient number of holding tanks to permit satisfactory flexibility of operation for a contract treating organization and is reported to
Aid to Blast Furnace Studies
have cost $2,000,000. The plant produces 60,000 to 75,000 gallons of waste liquor per day. In addition to this investment the company pays 1.8 cents per gallon to the contract treating organization to haul and dispose of their waste liquor. Other plants are reported by the same authority to contract their disposal a t 2 cents per gallon. A figure of 1.4 cents per gallon, then, should be attractive to a company that is paying 2 cents per gallon for waste disposal plus the write-off on what investment it has in holding tanks. Electrolytic regeneration of waste liquors should also be considered where ground for expansion of productive capacity is limited, for expansion of facilities could be made over ground formerly reserved for lagooning. Significant reductions in operating power requirements are foreseen through modifications in cell design and methods of operation. These modifications, which would promote higher efficiency, would also mean lower capital investment for cells and rectification equipment. Even without these certain reductions i t may be said that electrolytic treatment of waste sulfate pickle liquor using Amberplex membranes holds promise, not only because it completely eliminates the need for disposal, but because it competes economically with present processes. Acknowledgment
The authors wish to acknowledge the assistance of J. Lirio of the Rohm & Haas Co. during the experimental program and the mechanical department of the Rohm & Haas Go., Research Laboratory, for their assistance in developing and fabricating the electrolytic cells. RECEIVED for review September 15, 1954.
A 4 c c E ~ TFebruary s~ 5, 1955.
.. .
Air Flow in Beds of Granular Solids J. B. WAGSTAFF
AND
E. A. NlRMAlER
Fundamental Research laboratory, United Stater Steel Corp., Kearny,
I T T L E is known about the flow pattern of the gas in the blast furnace except that it is extremely complicated. The air is blown in through a number of tuyeres around the base with sufficient velocity to form a turbulent recirculation zone (6, 14). Both the solid and gas are in motion and probably neither move uniformly past any horizon. This article is a report of a study of the flow of air through beds of granular solids, in order to determine the correct criteria for an investigation of the blast furnace process by means of models. Bennett and Brown (1) found that when there was relative movement in a bed of granular solids, a region of loose packing developed. It is also known that gas tends to flow,preferentially along a region of loose packing. Therefore, it seems highly probable that the gas and solid flows interact. As a result, resewch on flow in a blast furnace must be done either on an actual furnace, as by Kinney and coworkers (11), or on models in which both solid and gas are in motion. Work on an actual furnace is extremely difficult and the use of models would seem preferable, provided that models are so constructed that the flow pattern is the same as in the full scale unit, a t least in its principal features. It is probable that in an actual furnace the formation of liquids presents additional complications, but in the initial investigations reported in this article such effects are ignored. The literature on gas flow through granular beds is extensive. The greater part of the work reported has been done by measuring June 1955
N. 1.
the pressure drop through columns of presumably homogeneous packings of granular materials. I n most cases, the columns were less than 6 inches in diameter and the granular materials were stationary. I n spite of these restrictions, there are remarkable areas of disagreement. Therefore, the work of five different authors was compared with measurements made in the laboratory in order to determine which of the recommended formulas best indicated the scaling factors necessary in the model experiments. The comparison has been made from the point of view of both the ease and accuracy with which a pressure drop may be calculated if the properties of the packing are known, and, conversely, the accuracy with which the properties of the packing may be determined if the pressure drop is known. The second approach is important when rough and porous materials, such as coke, are used, because such simple properties as particle surface and particle density are not easy to determine. The investigations of Chilton and Colburn ( 5 ) , Carman ( 4 ) , Brown (6),Leva and coworkers ( l a ) , and Ergun (9) were considered in this study. Chilton and Colburn in 1931 correlated the results of earlier workers on the basis of a modified friction factor and a modified Reynolds number. I n this way they deduced equations for the viscous and turbulent regions, respectively. They obtained for the viscous region
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
1129
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT and for the turbulent region
The division between the viscous and turbulent regions was a t a modified Reynolds number of 40,based on the size of the particles. The packing was described by a walleffect factor, A f ,and by the nominal diameter of the particles. They believed that the accuracy of the formula, particularly the estimate of the wall effect which was based on the work of Furnas ( I O ) , did not justify any correction for variations in the void fraction. Carman (4)attempted to find an improved correlation by introducing the void fraction of the material and the surface area of the particles. He found difficulty in estimating the surface area and used a diameter and a shape factor (which was unity for spheres). He derived different formulas, depending on the value of the modified Reynolds number, (3)
For high Reynolds numbers or turbulent regions he obtained
where f c is a friction factor which is best represented graphically as a function of Nke. For the viscous region he sugpested
The determination of the shape factor, 4 , presents a problem. For any particular material it can be obtained, in theory at least, by measuring the pressure drop under otherwise known conditions. This is not very satisfactory and becomes even more difficult for a porous material such as coke. I n addition, with coke, particle density is not easy to define so that the void fraction becomes uncertain. I n 1950, Brown (%) proposed a method of calculating pressure drop through packed beds which appearf to I)e a correlation of the work of previous authors. He found that the bed must be described in terms of the void fraction, shape, and diameter of the particles. These properties can be simplified to two main ones-namely, a shape factor, or the ratio of the area of a particle to the area of a sphere of equal volume, and the bed porosity. Unfortunately, the method of calculation proposed depends on a number of charts rather than the appropriate formulas, which leads to considerable inaccuracy. It happens that sniall changes in some variables such as porosity result in disproportionately large changes in the dependent factors. Leva and coworkers ( I d ) made a very thorough study of previous work from u-hich they were able to deduce the form of an equation for pressure drop. This relationihip was a function of the properties of the fluid and three properties of the packing, the void fraction of the bed, the particle diameter, and a shape factor of the particles which mas the same as that used b y Brown -i.e., the ratio of the area of a particle to the area of a sphere of equal volume. These workers then made a lengthy study using a wide variety of particles, ranging from those with a very smooth surface, such as glass beads, porcelain balls, etc., to rough particles of Aloxite. They used shapes varying from spheres to cylinders and Raschig rings. With these materials and the data of Oman and Watson ( I S ) they investigated each of the variables in turn, studying both the turbulent and viscous ranges. They found the best relationship to be
became 5.25, and for very rough materials such as Aloxite or coke became about 8.0. While this formula appears to cover a very wide range of conditions, detailed study of the curves of Leva and coworkers reveals a considerable degree of scatter, and a conservative estimate of the accuracy seems to be =!=25%. Their work on the viscous range was not as detailed. They proposed the relationship AP =
200 GpLX’(1 - E ) * D?P~gct~
(7)
Ergun (7-9) has published a number of papers on various aspects of fluid flow in packed beds. His approach was basically theoretical, although he confirmed his calculations by careful experiment. He based his calculations on a consideration of the flow through capillary tubes and then modified the formulas when the tubes were broken up in a random manner. This analysis was in direct contrast to the theoretical approach of Burke and Plummer (S), who considered that the forces on an individual particle would be similar to those of the same particle in a packed bed. Ergun went further and considered that the resistance in any particular packing was made up of the sum of the resistances of viscous and turbulent forces. This resulted in an equation that is markedly different from those of the other investigators, since the same formula applies over both the turbulent and viscous ranges.
where (9)
The first term, a, approximately represents the effect of viscous forces while bG represents the effect of turbulent forces. The packing was described in terms of the void fraction, E, and the specific surface, 8,. T h e constants, K I and K z , should be universal constants b u t were found experimentally to vary somewhat. This approach has several advantages. If the pressure drop through a particular bed is plotted against the flow rate in the AP
form - versus G , a straight line will be obtained of slope b L and intercept a; therefore, if the void fraction is known, the surface area can be computed. However, with rough porous materials the void fraction usually cannot be obtained because particle density is not easy to measure. Ergun (8) has shown that in this case t h e particle density can be obtained in the following way:
u,,
whence
or
These equations can be transformed as followcvs : =
Pp
-
(K,pLS,2~p)~’~(Pa2ia)l’~
(14)
P5 =
Pp
-
(KzS~Pz)”~(Pbib)”~
(15)
Pb
and for the turbulent range. The constant, 3.50, was found to depend on the roughness of the particle surface. This value was satisfactory for smooth particles b u t for rough particles 1130
Then if a number of runs are made at different bulk densities, since S, and p p are properties of the particles and not the packing,
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 41, No. 6
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Table 1. Diameter, Ft. 0.00642 0.0108 0.0215 Width = 0.00925 Height 0.00716
Particle Wood blocks 1/10 inoh Wood blocks’ :/a inch Wood blocks: 1/4 inch Plastic
*
-
-
Properties of Particles Length, Ft. 0.0091 0.0103 0.0205 0.0114
Surface Area Sq. Ft. lo4 2.483 5.339 21.31 5.08
A
i t becomes possible to plot curves of pb against ( p 2 , 0 ) 1 1 3 and against (pb/g)1’3 and the intercepts and slopes can be used to determine p p and 8,. This method of calculation represents a considerable advance because it seems to measure the surface area of a granular particle which is of interest in fluid flow. I n the case of an extremely rough and porous material, such as coke, it is not easy to visualize or define exactly what is meant by the term “particle surface.” Coke is so porous and contains so many macro- and microcracks t h a t it is not certain how far an irregularity in the surface will behave as “roughness” in the true surface and how far as a pore. Therefore, a method that will measure effective surface is useful. One disadvantage of this method is that the experimental work and calculation are somewhat more tedious than those of the other methods; in addition, the calculations involve rather lengthy extrapolations so that the method is sensitive to experimental error, particularly of flow measurement. Air flow through beds of wooden cylinders i s studied experimentally
Most of the experimental work was carried out in a vertical glass tube, 28/, inches in diameter and 32 inches long. The bed, which was formed by pouring the particles into the tube, was supported on a wire gauze mounted on four legs. It was usually about 20 inches high. There was a Pinch length of empty tube beneath the bed to allow the air flow to even out before the air entered the bed. The air flow was measured with an orifice gage. Although it was not found practical to calibrate the orifice with the 0.25-inch-diameter plate used in this work, the apparatus was calibrated with a I-inch plate against a meter checked by the Public Utilities Commission of New York. The 1-inch plate gave a constant very close to that predicted by calculation, so that the appropriate caIculated constant was used for the smaller diameter orifice plate. The pressure under the bed was measured with a water gage connected to a copper tube closed a t the end, with a series of holes drilled through its sides. This was inserted into the empty
Table II.
Error in Estimation of Mean Surface Area per Particle Particle
Investigator Chilton and Colburn (6)
Type m700d blocks
Carman ( 4 )
Plastic Wood blocks
Brown (3)
Plastic Wood blocks
Leva (1%)
Plastic Wood blocks
Ergun (7-9)
Plastic Wood blocks Plastic
June 1955
Estimated Nominal Surface Area, size, inch Sa. Ft. X I O 4 l/io 1.11 l/8 1.64 9.54 ‘/4 4.44 1/10 1 .97 6.01 1/8 15.4 1/4 8.09 1/10 1.90 4.67 =/a 14.8 l/4 4.89 1/10 3.24 ‘/a 7.80 1/4 15.45 10.60 I/io 2.68 1/8 5.87 20.25 1/4 6.38
Difference from Actual Area,
-
%
55.3
- 87.3 - 60.1
+-+ -
12.6 20.6 12.6 27.7 59.2 23.4 12.6 30.5 - 3.74 30.6 46.0 - 27.0 108 8.06 9.94 - 4.93 25.6
++ + + + +
tube space below the bed. On blank runs with the tube empty, no pressure drop was found beSpecific Area Sphere of Surface Equal Volume, tween the point of measurement Sq, Ft./C;. Ft. Sq. Ft. X 104 and the atmosphere for the maxi844 1.391 562 3,036 mum flows used. Therefore no 282 12.11 end corrections were applied. 956 3.176 Further, on some runs, a hypodermic needle was inserted into the bed. The pressure drop per unit of bed height was found to be the same as that obtained from the normal pressure point. Toward the end of the work some readings were taken on a 4-inch plastic tube with similar pressure and flow connections. The wood blocks used as particles were cylinders cut from wood dowling. Their size was accurately measured with a micrometer; a sample of 50 particles was taken as typical of the batch. The properties of the particles are shown in Table I, together with the properties of the polystyrene particles which were also used. The polystyrene particles were very smooth and not quite so regular in shape as the wood blocks. However, measurements of their surface area were reasonably precise. The various packing densities were obtained by filling the tube a t different rates, by tapping the tube, by expanding the bed with a high air flow, and by varying the rate of reducing the flow. Calculation of shape factors and prediction of pressure drop are used to evaluate relationships
Various methods of measuring the accuracy of the different relationships were used. The first method is perhaps the most severe in that the measured pressure drop through the bed was used to calculate the shape factor or surface of the particle. This is a doubtful procedure for use with the formula devised by Chilton and Colburn because the required result cannot be obtained without the use of other information. I n this case, the charts of Brown were used. The results are presented in Table 11, with the estimated area of a single particle together with the percentage difference from the actual measurements. As might be expected, the formula of Chilton and Colburn gives results that are erratic. I n fact, the results from all the relationships, with one exception, are highly variable; this indicates that perhaps there is something not quite repeatable about flow through granular beds. By far the best estimate is obtained from the work of Ergun, which gives results within 10% for all particles except the very smooth plastic. Since the present work is mainly concerned with rougher particles, these results are encouraging. However, such a test of the formulas may be excessively rigid, so the same data were recalculated to show the comparison of estimated and numbered pressure drops. The results are shown in Figure 1, A , B, C, and D. The most disturbing factor is the lack of consistency among the different size particles, although the results for the l/lo-inch blocks (Figure 1, A ) seem to be much closer than for the other particles. I n Figure 1, B, Chilton and Colburn formula predicts a pressure drop that is much too high and the Brown equation one that is much too low for the I/a-inch blocks. In Figure 1, C, for ‘/r-inch wood blocks, Chilton and Colburn formula gives results that closely approach the measured value, while the relationship of Leva gives much too high a figure. Calculations based on Carman’s work produce very low values. Figure 1, D,shows a much wider scatter for the smooth particles and the calculation based on the work of Brown would seem to be highly unreliable. I n general, it appears that work of Leva and of Ergun seems to give rather more consistent results than the others. I n order to explore the possibility of improving the results, the formulas of Leva ( l a ) and Ergun (7-9) were re-examined in detail, I n Equation 6 Leva determined the constant, 3.50, empirically, and it was found to vary to 8.0 for very rough
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
1131
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
0 26
-
0 CARMAN
0 24
+. 020
a
z 0'
G.
0 16
_I
51
_I
012
a 0 3
4 0
.
008
0 04
0
0
004
0 08
0 12 OBSERVED
0 I6 0 20 LB,/SQ IN. P E R FT.
0 24
0 28
004 0 0 6 OBSERVED
0
A.
0 12
9 LBISOJN
'/m-lnch wood
6.
1
0 16
0 20
PER FT.
wood
'/&ch
1
044
i
LEGEND
0 - LEVA 0
- ERGUN
A-
BROWN
0 - CARMAN
A
A
C.
1/4-lnch wood 0
0
'
004
I
0.08
l
I
I
0 12
0.16
Accuracy of pressure drop calculations
Aloxite and magnesium oxide particles. This is a ratio of about 2.25 to 1, depending on the type of particle. There does not seem to be any reason to expect a sharp differentiation, and the experimenter is compelled to use his judgment. Therefore, the constant was investigated, and constants for the various particles are presented in Table 111. For convenience in calculation, the constant in Table I11 also includes the values of 1.0.' and gc and a conversion unit for D,,as proposed by Leva. On this basis, the values 3.50 and 8.0 become 4.27 X 1132
D.
l
0.20
'
0.24
1
~
0 28
~
032
I
LB./SQ, IN. PER FT.
OBSERVED
Figure 1 .
'
Plastic
10-11 and 9.76 x 10-11, respectively. Table I11 shows that the constants obtained for the l/lo-inch and 1/8-inch wood blocks are reasonable a t 5.65 and 6.38 X 10-11, respectively. However, the value obtained for the llrinch wood blocks was considerably lower, 3.0 x lo-". Observation indicates that there is no marked difference in roughness in these cases. I n a similar manner, it seemed worth while to investigate the constants used in the formula of Ergun. Unfortunately, this cannot be done completely because the formula involves two
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 6
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Table 111.
Estimated Constant in Leva Formula (72)
(Proposed constant = 4.27 X 10-11) Constant X 1011 ‘/ip-Inch l/s-Inch ‘/r-Inchwood particles wood particles wood particles 6.65 5.43 5.19 5.93 5.42 5.94 5.60 6.68 5.89 5.28 6.19 5.62 5 . 6 5 (Av.)
7.43 7.06 6.58 5.86 6.46 5.91 6.14 7.08 6.59 6.03 6.08 6.35 6 . 3 8 (Av.)
3.34 3.18 3.01 2.90 2.93 2.82 2.83 2.86 3.14 2.91 3.16 2.96 3 . 0 0 (Av.)
constants that appear as a sum. By combining Equations 8, 9, and 10,
Figure 2.
Effect of bulk density on KZ2 for K1
I/,-inch wood particles
where K1 and Kz are the two constants. However, i t is possible to obtain some information on the nature of the constants by equating Equations 12 and 13.
whence
By taking a number of measurements a t diRerent flow rates on the same packing it becomes possible to evaluate
K’ -.’ The
.
K1 results are shown in Figures 2 and 3. The value of the combined
Kz‘~has been plotted against bulk density and the 95% K1 confidence limit of the point is included for the case of the l/a-inch wood particles (Figure 2). The confidence limit is based on the confidence with which a and 6 can be predicted from the scatter of the original data, using the usual statistical techniques for obtaining the appropriate straight lines. There appears to be KZ some change in the value of -5 particularly a t the high bulk K1 density. This may be due to the fact that t h e particles were regular cylinders which tended to pack one on top of the other t o form “rods.” Under these conditions, i t could happen that the effective specific surface was reduced. This effect may also be present with the l/lo-inch particles, Figure 3, but does not show up with the l/&nch pieces. The experimental technique may have had two defects. The orifice was not calibrated and the tube ( z 3 / 8 inches in inside diameter) may not have been sufficiently large for the particles, particularly the 1/4-inch pieces. I n an attempt to check the first of these points i t seemed as though errors in the pressure calculations would be a function of the measured flow. Figure 4, A and B, have therefore been drawn to show the effect of air flow on the ratio of the calculated to measured pressure drop, The points fall into groups, and each group represents a particular set of particles or tube size. I n Figure 4, A , where the predictions of Leva have been applied to the 23/a-inch tube, there are four distinct groups that give, for the same air flow, different pressure drop ratios. It is true that within each group there is an effect of flow and this effect is always the same for all the sizes of particles, but it is small compared with the difference due to the use of different particles. I n Figure 4,B, where the results of the Ergun formulas are shown in the same way, very similar conditions are encountered, except that the scatter due to the different particles is much less. From this it is concluded constant,
lune 1955
o13p--y 1/4 PARTICLES
01 I
0 36
Figure 3.
038
0 40 0 42 BULK DENSITY
0 44
041
Values of Kzz - for wood particles
Ki that the differences observed between particles were not due to errors of flow measurements. Included in Figure 4, A and B, are some results obtained in a plastic tube 4 inches in inside diameter. It seemed reasonable that if the 23/8-inch tube were small enough to cause errors of prediction, then these errors should be less in the larger tube. Unfortunately, this does not prove to be the case and the errors seem to be just as large or larger. An attempt was made to correct the calculations by means of a correction to the specific surface in the formula of Ergun by including the internal surface of the tube. This calculation showed the correction to be minor. Range of Application. The authors were interested in a Reynolds number range of about 100 to 400 (calculated as suggested by Leva or Ergun). This range is perhaps a little extreme for the formulas of Leva which, because they are empirical, should be applied only in the ranges for whjch they were devised-Reynolds numbers of 200 and above. However, Figure 4, A , shows the error to the greatest for the l/4-inch particles, which had the highest Reynolds number, well within the permissible range.
INDUSTRIAL AND ENGINEERING CHEMISTRY
1133
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT 18
16
LEGENO DlA OF TUBE
I-
5
2% 4" - -
1.4
TYPE
k
o
$"WOO~
t
A
A
0
6
= I 2 W P
:P:I 0 LT
f
E VI
1.2-
I
-
9 5 1.04
-
20 3
4
9
0.6-
0.4
W
t
y 0.8 -
3
.
LO8
$0 6 a
a
+
wL
(
1
1
1
A.
1
1
1
1
I
I
1
I
k" WOOD
0 200
300
400
500 600 700 AIR FLOW LB./SP. FT, PER HR.
-
6.
leva formula
Figure
4.
0
0.2
800
900
Ergun formula
Effect of flow on error of calculation of pressure drop
The error was much less for the l/lo-inch particles, with lower Reynolds numbers (some even below 100). The relationship of Ergun is much less sensitive to Reynolds number, since it is theoretical. It is applicable in both the viscous and turbulent regions. Ergun formulas give most consistently reliable results
On the basis of this study of pressure drop through packed beds, it appears impossible to predict the pressure drop with great accuracy by any known method. The two best proposals are those by Leva and coworkers and by Ergun. Of these two, the work of Leva seems to have surprising irregularities and suffers from the fact that a constant is required whose value depends on an unmeasured property of the particles. This constant varies twofold so that the worker is required to exercise some judgment. Even so, the results on different size particles of the same roughness seem to be inconsistent to some extent. The work of Ergun seems to be the most consistently reliable. As a means of estimating particle surface it is good; and most certainly it is the best for rough porous materials where linear size is not known with precision. The use of the Ergun formulas for regular cylinders a t high packing density may well be questioned as the particles tend to arrange themselves in groups which behave as one large partide rather than as a number of small ones.
If the pressure drop, A P , and the velocity, U,, are taken in the same direction, it can be assumed t h a t
by writing p~ Urn for G in Equation 16. Unfortunately, thie equation was deduced for unidirectional flow and is a scaler, not a vector, description. However, the purpose of the model is to integrate suitably the various flow components. Equation 19 can be rewritten in the form
If the flow in the model corresponds to t h a t in the full scale unit (with subscript M for the model and F for the fill1 size unit),
and
Modified Reynolds number i s proposed for use in blast furnace studies by means of models
After it was established that the best equations for the calculation of flow in granular beds are those proposed by Ergun, attention was directed to the requirements for a suitable model. T h e essential feature is assumed to be that the flow a t various points in the model should correspond t o the flow at corresponding points in the full scale unit.
1134
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 6
-
- .
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Table IV.
Consistency of “Constants” in Ergun Formula (20runs) Constants, C.G.S. Units
then and whence or
or
%e((m)(-) 1
PG,F
UF
1 (1 -
=
E)F
(24)
Equation 24 is a modified Reynolds number multiplied by the reciprocal of 1 minus the void fraction. It follows then, that in order to model air flow in a bed of granular solids, the flow criterion of similarity is not a Reynolds number, as often supposed, b u t is P G m
MStJ
(2) 1E
which is a type of Reynolds number based on specific surface rather than particle diameter and further corrected for the void fraction of the bed. Unfortunately, this rigid criterion of modeling is not practical for the blast furnace. The model is a t uniform temperature so that the viscosity of the fluid is constant, whereas in the blast furnace, owing to the large temperature gradient, the viscosity changes considerably. In addition, there are several chemical reactions in the furnace stack that involve increases in the gas flow which are difficult to simulate. Further, the purpose of the model is to work on a smaller scale-that is, with smaller particles or a t larger values of S,. This, in turn, means that 7 7, must be increased since there is not much range possible with the other variables unless some fluid other than air is used. Use of another fluid might bring considerable complications. Equation 19 shows that, if both S , and U , are increased proAP
AP
As -L approaches (1 - E) ( p p - P O ) the bed starts to expand and becomes fluidized or blown out of the vessel. That is to say, any attempt a t modeling is impractical close to this limiting value of portionately,
increases as the cube.
y
=
(1
-
E ) (PP
- PO)
This might mean that further work along these lines is impractical, except for the common experience that considerable “crudities” are permissible. It has been found by a large number of workers on other furnace models that a wide range of Reynolds numbers gives sensibly the same flow pattern, provided the flow is turbulent in all cases. Acknowledgment
The authors wish to thank J. A. Sumpter and ht. McCleskey for assistance with the practical work. The advice of L. S. Darken is also gratefully acknowledged, as well as the constant encouragement of B. M. Larsen. Nomenclature
AP P
= pressure drop through bed, lb. force/sq. f t . = viscosity of gas, lb. mass/(ft.) (sec.)
Vo,U, = superficial gas velocity, based on empty tube, ft./sec. G = mass velocity of gas, lb. mass/(sq. f t . ) (sec.) June 1955
a
b
K:/K~
4.525 3.423 3.980 5.821 7.827
‘/lo-Inch Wood Particles 95.43 65.93 77.42 129.45 158.18
0.0204 0.0167 0.0161 0.0234 0.0202
1.2666 1.5026 1,5268 1.7204 1.9325 2,3594 2.8491 1.5450 1.7863 4.1914
I/s-Inch Wood Particles 40.08 40.26 44.09 50.05 53.54 61.02 74.50 43.16 53.21 114.03
0.0227 0,0185 0.0203 0.0214 0.0206 0.0205 0.0226 0.0203 0.0222 0.0263
0.7971 0.5018 0.5207 0.7014 1.1922
‘/c-Inch Wood Particles 37.42 22.18 25.03 29.92 48.69
0.0118 0.0132 0.0153 0.0135 0.0127
density of gas, lb. mass/cu. f t . height of packing, f t . c = void fraction, dimensionless = bulk density of bed, lb. mass/cu. f t . Pb = wall effect factor, dimensionless A/ = density of particle, lb. mass/cu. f t . PP 8, S, = specific surface, sq. ft./cu. f t . = shape factor (unity for spheres), sq. ft./sq. f t . 9 1 = shape factor = -, sq. ft./sq. ft. x 9 = diameter of sphere of volume equal to t h a t of particle, D, ft. N ’ R ~ = modified Reynolds number, dimensionless = friction factor, dimensionless fc = gravity constant = 32.2 ft./(sec.)2 9 sc = conversion factor = 32.2 (lb. mass) (ft.)/(lb. force) (sec.)2 K,, Kz = constants
2
=
=
Subscripts b refers to bed p refers to particle G refers to gas F refers to full scale unit M refers to model literature cited (1) Bennett, T. G., and Brown, R. L., J . Inst. Fuel, 13, 232-46
(1940). (2) Brown, G.G., and others, “Unit Operations,” W h y , New York, 1950. (3) Burke, S. P., and Plummer, W. B., IND.ENQ.CREM.,20,1196200 (1928). (4) Carman, P. C., Trans. Inst. Chem. Engrs. (London), 15, 150-66 (1937). (5) Chilton, T. H., and Colburn, A. P., Trans. Am. Inst. Chem. Engrs., 26, 178 (1931). (6) Elliott, J. F., Buchanan, R.A . , and Wagstaff, J. B., Trans. Am. Inst. Mining and Met. Engrs., 4,709-17 (1952). (7) Ergun, S., ANAL.CHEM.,23, 151-6 (1951). (8) Ibid.,24, 388-93 (1952). (9) Ergun, S., Chem. Eng. Progr., 48, 89-94 (1952). (10) Furnas, E.C., U. S. Bur. Mines, Bull. 307, 1929. (11) Kinney, S. P.,Royster, P. A . , and Joseph, T. L., U. 5. Bur. Mines, Tech. Publ. 391, 1-65, 1927. (12) Leva, M., Weintraub, N. M., Grummer, M., Pollchik, M., and Storch, H.H., U. S. Bur. Mines, Bull. 504, 1951. (13) Oman, A . O.,and Watson, K. M., Natl. Petroleum News, 36, R795-802 (1944). (14) Wagstaff, J. B.,Trans. Am. Inst. Mining and Met. Enps., 5, 895 (1953). RECEIVED for review July 31, 1954.
INDUSTRIAL AND ENGINEERING CHEMISTRY
ACCEPTED JEnUErY 3, 1956.
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