INDUSTRIAL AND ENGINEERING CHEMISTRY
September 1948
cussion of the potentialities of Vinyon N yarns would be out of place in this article, but it is pertinent to point to a few broad examples by way of illustration, Thus the industrial applicatibns of Vinyon yarns have been greatly expanded, and their inherent nonflammability can now be exploited in upholstery and drapery fabrics and in (26). Their shrinkage characteristic make possible new inexpensive processes for the manufacture of familiar seersuckers and crepes, and also novel fabrics which could be made from the older fibek only by more complicated processes. Similarly, the controllable thermoplasticity of certain grades of fiber makes possible interesting open netting which can be readily shaped into items of wearing apparel such as light-weight hats and shields. The whole program of research on vinyl fibers is based upon the fact that vinyl fibers, as a class, offer a combination of properties which is outstanding in many respects, and which cannot be approximated by any other fiber known today. A great number of chlorine-bearing resins have been studied intensively over a period of years, in the search for an alternative t o vinyl chlorideacetate copolymer which would have a substantially higher softening point than the older copolymer and yet be similar to it in all other essential properties. T h e Vinyon N family represents a substantial step forward along this line. LITERATURE CITED
(1) Anon., Rayon Textile Monthly, 28, 178-9 (1947); Chem. Inds., 61,589 (1947). (2) Anon., Silk & Rayon, 18, 36-40, 372-4, 438 (1944); Textile World, 97,No. 9,113-28 (September 1947). (3) Bacon, R. G. R., Trans. Paradau Soc.. 42.140 (1946). (4) Baxendale, J . H., Evans, M . G., and Park, G. S., Ibid., 42, 155 (1946). ( 5 ) Boyer, R. F., J . Phys. Colloid Chem., 51,80(1947). (6) Boyer, R.F.,and Spencer, R. S.,J . Applied Phys., 15,398(1944) (7) Chaney, N. K.,and Dexter, W. B. (to Carbide and Carbon Chemicals Corp.), U. S.Patent 2,060,035(Nov. 10,1936). (8) Charch, W. H., Finzel, T. G., Hansley, V. L., Houtz, R. C., Latham, G. H., Merner, R . R., and Rogers, A. 0. (to E. I.
1731
du Pont de Nemours & Co.), U. S. Patents 2,404,714-28 (July 23,1946). (9) Clash, R. F., Jr,, and Rynkiewicz, L, M., IND. ENG. CHEM,, 36, 279 (1944). (10) Crawford, J. W. C. (to Imperial Chemical Industries, Ltd.), U. S.Patent 2,054,740(Sept. 15, 1936). (11) D’AleliO, G. F. ($0General Electric CO.) I u. Patents 2,366,495 (Jan. 2,1945),2,412,034(Dec. 3,1946). (12) ~ ~c, c.,~and Blake, i ~ J , T,, “Chemistry , and Technology of Rubber,” pp. 71-2, New York, Reinhold Publishing- Cors., 1937. (13) Doolittle, A. K., IND. ENG.CREM.,30, 189 (1938). (14) Doolittle, A. K., IND. EKG.CHEM.,NEWSED.,18, 303 (1940). (15) Lawton, T. S., and Nason, N. K., Modern Plastics, 22,No. 2,145 (February 1944); IND.ENG:CHEM.,36,1128 (1944). (16) Mauersberger, H. R., “Matthews’ Textile Fibers,” 5th ed., p. 875,New York, John Wiley & Sons, 1947. (17) Morehead, F. F., Testile Research J., 17,96 (1947). (18) Powell, G.N., and Quarles, R. W., Oficial Digest, Federation Paint & Varnish Production Clubs, No. 263,696 (1946). (19) Quarles, R.W., IND. ENG.CHEM., 35,1033 (1943). (20) Quattlebaum, W. M., and Noffsinger, C. A. (to Carbide and Carbon ChemicalsCorp.) ,U.S.Patent2,307,157 (Jan. 5,1943). (21) Rugeley, E.W. (to Carbide and Carbon Chemicals Corp.), U. S. Patent 2,277,782(March 31,1942). (22) Rugeley, E. W., and Feild, T. A., Jr. (to Carbide and Carbon Chemicals Corp.), U. S.Patent 2,418,904(April 15,1947). (23) Rugeley, E.W., Feild, T. A., Jr., and Petrokubi, J. L. (to Carbide and Carbon Chemicals Corp.), U. s. Patent 2,420,565 (May 13,1947). (24) Shriver, L. C., and Fremon, G. H. (to Carbide and Carbon Chemicals Corp.), U. S. Patent 2,420,330 (May 13,1947). (25) Stoops, W. N., and Wilson, A. L. (to Carbide and Carbon Chemicals Corp.), U. S. Patent 2,403,960(May 16,1946). (26) Stowell, E., Papers Am. Assoc. Textile Technol., 3,30 (1947). (27) Wilkes, B. G.,and Denison, W. A. (to Carbide and Carbon Chemicals Corp.), U. S.Patent 2,381,020(Aug. 7, 1945). (28) Wirshing, R.J., IND. ENG.CHEW,33,234 (1941). (29) Yngve, V. (to Carbide and Carbon Chemicals Corp.), U. S.Patents 2,267,777, 2,267,779(Dec. 30, 1941),2,307,092(Jan. 5, 1943), RECEIVEDMarch 4, 1948. of
The word “Vinyon” is a registered trade-mark
Carbide and Carbon Chemicals Corporation.
Friction in the Flow of Sus ensions GRANULAR SOLIDS IN GASES THROUGH PIPE E. G. VOGT‘ AND R . R. WHITE, University of Michigan, A n n Arbor, M i c h . T h e pressure ditrerentials required to produce steady flow of suspensions of sand, steel shot, clQvcr seed, and wheat in air through 0.5-inch commercial iron pipe are presented. These values and data from the literature on the pneumatic conveying of wheat in pipe sizes ranging from 2 to 16 inches in diameter are correlated by the following type of equation for both horizontal and vertical flow:
(Zy (: x 41/3(w -
OL-l=A
.&)k
where A and k are given as empirical functions of the dimensions group
p)pgd’, and
OL
is the ratio of
pressure drop to pressure drop obtained for the flow of pure fluid at the same velocity. The development and limitations of the correlation are discussed.
* Present address, State College of Washington,
Pullman, Wash.
NEUMATIC conveyers have been used commercially in transporting granular solids for many years. Recently the development of fluid catalyst processes has brought further attention t o the need of information on the pressure differentials required to produce flow of suspensions of solids in gases through conduits. For design purposes i t is necessary to know the effect of such variables as pipe size, the amount of fluid and solid flowing, and the properties of the fluid and solid, on friction losses. Although i t is easy t o calculate friction losses in the flow of ordinary fluids through pipe, the relations ordinarily used for this purpose are not applicable generally t o the flow of suspensions of solids in gases because the meaning of such terms as the density, viscosity, and perhaps velocity of suspensions is rather obscure. PREVIOUS INVESTIGATIONS
Although the literature contains a large number of references on pneumatic conveying and fluid catalyst processes, most of the information is qualitative in nature. Cramp and Priestly (3, 4)
INDUSTRIAL AND ENGINEERING CHEMISTRY
1732 4.0
I
I
I / I
I
I l l
I
I
I
I l l
0 SLGLEA- WHFAT I H 420 HH PIPE
2 0
-0 A
1
h01
I I
x ID
S E G L E R - W M T IN 2 3 5 MY P I E S U E R - H M P i T 1N 113 MM. PIPE S U ; L L R - W Z A T I N 48 M N PIPE WTERSTADT-WHEAT IN 8 B 5 H M PIPE W E N T W O R K - W T IN 111 !W PIPE SEGLER OATS I N 420 M M PIPE X G L E R - CATS IN 295 MM. RPE
the weight ratio of the solids to fluid in flow, r; a roughness factor for the particles, d ; and a shape factor for the particles, E . An equation similar to Equation 2 may now be written as follows:
-
06 0 4
0 2
0 IO
OW
0 04
Vol. 40, No. 9
w,ooo ow
!o,m
Figure 1.
Horizontal Conveying of Wheat and Oats with Air
and Cramp (a)present information relating to the design of pneumatic conveyers for grain but the data on 1%-hichtheir conclusions are based are not adequately preeented. Apparently, they invest'igat,ed the vertical conveying of FT-heat with air, at pressures somewhat belon 1 atmosphere. Gasterstadt (6)determincd friction losses for the horizontal conveying of wheat in an apparatus similar to that used in the present investigation. Pressure differentials over a range of velocities and compositions m-ere measured for a pipe approximately 3 inches in diameter. Segler (8)invest,igatcd the effect of pipe diameter in the horizontal conveying of wheat by air and in addition obtained a few data for the conveying of oats and the vertical conveying of wheal. Kood and Bailey ( 1 0 ) measured the pressure differentials in a 3-inch smooth brass pipe with suspensions of sand and linseed in air but, unfort'unately,included in thc ir measurements the pressure differential across an injector used in moving the air; their measurements are apparently incompletely reported. The dat'a in the literature and the methods of design recommended (6) appear to be of dubious reliability for general use. The experimental ~vorliof the present, investigation consisted of measuring the pressure drop across horizontal and vertical sections of O.5-inch iron pipe through which suspensions of closely sized fractions of sand, si eel shot, clover seed, and wheat in varying proportions with air were Aoxing. THEORY
The mechmics of the turbulent flow of fluids are so complex that empirical methods have proved most effective in correlating the relat,ions involved. The turbulent flow of single phase fluids for long has been correlated by the familiar Fanning equation: 2puz
Some simplification might be gained by resoi ting immediately to dimensional analysis, but the choice of dimenaionless groups i s difficult as there are many combinations vi hich could be used. Gasterstadt ( 5 ) proposed the dimensionless term, relative pressure drop, represented by the symbol, cy. This term is defined as the ratio of the pressure drop obtained in the flow of a suspension t o that obtained with pure fluid flowing in the same pipe a t the same velocity. Both Gasterstadt (6) and Segler ( 8 ) shorn ~i t o be a linear function of r for the conveying of wheat in a given pipe n ith a given air velocity, but make no generalizations as to the effect of pipe size or air velocity on the proportionality factor. Although the linear relation between a and T did not hold for the data of the present investigation, the same terms were found t o be helpful in the correlation. Dimensional analysis indicates that o( can be expressed as a function of seven dimensionless groups. For reasons discussed belon. the functional equation was m i t t e n as:
The difficulty involved in ariiving at an enipiiical correlation with such a large number of variables prompted the authors t o attempt a theoretical analysis of the pioblem. Many assumptions are involved, and the data obtained do not fit the theory as developed. However, the theoretical treatment suggested certain dimensionless groupings vihich were useful in correlating the results. I t was assumed that \\hen a particle in motion strikes the pipe mall it loses a definite fraction of its kinetic energv. The pal ticle then nil1 be moving a t a velocity lover than that of the fluid. The drag force exerted bs the fluid on the pal tick n.ill bc: a function of the relative velocity bctn-ern the fluid and the particle and the properties of the fluid. The energy expended by the fluid to maintain the motion of the particle will balance the loss in kinetic energv caused by the collisions with the pipe wall. By further assuming that the number of collisions that each particle makes with thc pipc ~vallis proportional to the quotient of the particle velocity and the pipe diameter, it can be shown that the velocity ratio, K , may be represented by the functional relation:
By a summation of all the drag forces on the individual particles it may also be shoTn that
=
where
f = $
(?)
This relation may be obtained by the dimensional analysis of the following functional relation w-ritten in a general form based on experimental observation- in long pipes ( 7 ) :
Khen suspended solids are present in a fluid phase floning in turbulent motion under steady conditions through a pipe, the only additional variables required t o define the system, assuming the pipe roughness to be constant, are those describing the solid phase and the local acceleration due to gravity. The variables dewribing the solid phase are: thc average effective diameter of the particles; d : the density (displacemcnt) of the particles, us;
Since u7is proportional t o ua, Equation 7 may be modified to:
By further assuming that the drag force on a particle is given by Stokes' law, Equation 8 transforms to: (9)
Equation 9 is in excellent agreement with the data of Gasterstadt and Segler for wheai in several pipe sizes as shown by Figure 1. Honever, as mentioned above, this does not agree with the data of the present investigation. It was believed that the
INDUSTRIAL AND ENGINEERING
September 1948
30.TITAP
-
on the values of ( a l), several sizes of sand were determined over a range of Reynolds numbers. Steel shot and clover seed of about t h e same size as one of the sand fractions were run t o determine the effect of density ( p / w ) . The ratio of solid t o fluid was varied in all cases t o determine the effect of the term, T .
39.663 44.IO
{TAP
TAP 12
TAPII
1733
CHEMISTRY
9
EXP EkIMENTA L APPARATUS
GAGE
Oi
r TAP 8 STOR AG E HOPPER
ORIFICEL
Figure 2.
Flow Diagram of Experimental Apparatus
Stokes law assumption might be responsible for this, so Equation 8 was written in exponential form as follows:
It was believed t h a t the values A and k could be expressed as functions of t h e properties of t h e solid and fluid. Also, as the particles were i n a field of two opposing forces-namely, friction a n d gravity-the problem appeared t o be one similar in some respects t o a problem in free settling. Hence, the group,
4
1/3(w-
p)pgda,
(1
was assumed t o be a parameter for A and k.
This latter group is t h e product of the Regnolds number and t h e square root of t h e drag coefficient for a spherical particle under free settling conditions and involves no velocity term. Equation 10 is the final result of the theoretical analysis. However, i t did not fit the d a t a until a n empirical modification was made. A slight shifting of the grouping in Equation 10 was made which seemed t o fit all of the d a t a of this investigation a s well as the work of Gasterstadt and Segler. The final equation used was:
-
1=A
w Re
where A and k are functions of the dimensionless group
41/3(w p)pgdS,
This derivation has i g n o e d the effects of
particle shape and roughness. T o determine the effect of the dimensionless group ( d / D ) ,
The flow diagram of the experimental apparatus is shown in Figure 2. Air at about 7 pounds per square inch gage pressure enters the system through a needle valve and calibrated orifice p-here the rate is measured and controlled. It then passes under the feed hopper from which solid is introduced. T h e mixture of air and solid passes through the horizontal calming section, through the vertical test section, through t h e horizontal test section, and into the receiving hopper where the mixture separates; the air leaves at t h e top. The feed hopper is connected t o the air line by a large rubber hose, T h e rate at which the solid falls into the air stream is regulated by adjusting a screw clamp on the rubber hose. The storage and feed hoppers were 6 inches in diameter and 1 foot high. T h e receiver hopper was 4 inches in diameter and 4.5 feet high. T h e capacity of the hoppers was based on the assumption t h a t the maximum ratio of solid to air passing through t h e apparatus would be about 10. This proved to be unduly small, as it was found t h a t solid t o air ratios of 40 would flow smoothly through the 0.5-inch pipe without difficulty. The hoppers were constructed of sheet metal and fitted with external connections consisting of standard pipe nipples brazed into place. The bottom of the receiver hopper was connected t o the top of the storage hopper by a rubber hose. During a run, a screw clamp on the rubber hose was used to close this connection. The pressures at the various points in the test pipe shown in Figure 2 were measured through pressure tops consisting of pipe nipples 0.125 inch in diameter connected t o a water-fillcd manometer. The nipples were inserted SO t h a t the ends were flush with the inside of the pipe and were brazed in place. MATERIALS USED. The materials used in this investigatlon were sand, steel shot, clover seed, and wheat. The sand, was screened to give four fractions, each having a relatively uniform particle size. T h e properties of the materials are given in Table I. T h e particle size of wheat was determined from micrometer measurements on about 100 representative grains. Specific gravity was determined b water displacement for sand, clover seed, and wheat and by arcohol displacement for the steel shot Photographs of typical particles are shown in Figures 3 through 6: EXPERIMENTAL PROCEDURE. The initial operations of the equipment consisted of determining the friction factor of the test sections for air. Following this determination sand suspensions were circulated under steady conditions through the equipment until the friction factor for the test section when determined with air became constant with time. The friction factor decreased about 10% during the first nineteen runs, presumably because of the polishing action of the sand on the inside surface of the pipe. A run consisted of charging the food hopper with the material to be run, adjusting the air flow to the desired rate as measured by the control manometer. T h e rate of solid flow was adjusted by the screw clamp on the solid feed line. Slugging or unsteady flow was detected easily by whether or not the manometers connected t o the test sections gave steady readings. No particular difficulties in this respect were noted. T h e amount of solid transported during each run was determined by weighing the receiver hopper before and after each run.
TABLE I. MATERIALS USED & aterial 'I
Displacement specific gravity Screen analysis Mesh (Tyler standard) 10 14 20 24 28 32 35 48 65 80 100 Through 100 Mean particle diameter, inch Diameter ratio d / D Density ratio (s'olld to air), w / p
EXPERIMENTAL RESULTS
80-Mesh 48-Mesh 35-Mesh 24-Mesh 35-Mesh Clover Wheat Sand Sand Sand Sand Steel Shot Seed 2.66 2.63 2 . BO 2.56 7.21 1.23 1.28 Cumulative Weight % Retained on Screen 0.0 *.. ... 59.8 ... ... .... 22.2 99.7 0.0 70.0 .... 9Q.Q ... ... 0.0 1.0 .93.0 0.0 ... 0.8 21.0 99.6 6.0 ... ... ... 0.0 14.0 73.5 100.0 62.0 ... ... 1.0 79.0 99.5 ... 99.5 ... 19.5 98.5 100.0 100.0 85.5 ... ... 98.5 99:o *.. ... ... ...
- ...
... ...
... ..* .... ..
100.0
99.0
0 0080 0 01285 2300
0 013 0 029 2230
... *..
... .:
0 0172 0.0277 2200
...
.,. ...
...
0 0287
0.0462 2100
... ...
.. .. .. ..
... ... ...
..... ...
....
0 0165 0 0265
6000
...
...
...
046 0 074 1065
0 158 0 254 1070
(3
.
T h e pertinent experimental d a t a are presented in Table 11. Pressures were recorded at points 3 feet apart in the horizontal calming section behind t h e point of solid feed introduction; these data indicated t,hat the ' solid had not been accelerated t o the full stream velocity within 9 feet of the feed point a t all conditions encountered during the investigation. As shown in Figure 2, the calming section was 15.5 feet long. Only the pressures at points 7 and 9 of the vertical section are presented. It was dis-
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
1734
covered late in the investigation t h a t the pressure tap a t point 8 was improperly installed and gave inaccurate readings. Thus the vertical flow data may be subject t,o error because of the absence of a calming section between the bend and point 7. However, this calming section may no1 be essential as wide radius bends (2 f w t ) iwre used. CALCULATIOS O F R E S U L T S
The experimental results were calculated as friction losses frorii the first Iaw of thermodynamics statcd 11s the familiar flow equation:
This equation s t a h an energy balance where the units are those of energy per pound of material flo~ving and assumes t h a t there is no in. crease in other forms of energy-such as elect'rical, chemical, surface, or magnet,ic energy. Where t w o materials, a and p , fiow through the apparatus under steady conditions, a n equation similar t o Equation 11 ma?; be m i t t e n for each material.
(12)
Run so.
O
F.
Air Rate, Lb./Seo. 0.00735 0.00635 0.01116 0.00705 0,0047 0.00468 0.00454 0.00933 0,00845 0.00688 0.01110 0,00642 0.00471 0.00472 0,00470 0.00483 0.00840 0.00943 0,00897 0.00861 0.00840 0.01250 0.00545 0.00303 0.00503 0.00500 0.0197 0.00303
Atmos. Press.,
83 82 81 81 81 81 81 81 78 78 86 82 82 82 82 82 78 80 80 81 82 80
751 733 733 736 736 736 744 744 744 744 744 744 741 741 746 746 730 725 726 725 725 725 742 742 742 742 742 744
74 75 76 77 78 79
80 80 78 73 72 72
742 742 742 741 741 741
0.00859 0.00950 0.00930 0.01310 0.01488 0.01970
80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
82 82 82 82 82 82 86 86 85 84 84 74 73 73 72 72 72
739 739 739 739 739 739 737 737 737 737 737 741 741 741 741 741 741
0,00555 0.00570 0.00584 0,00595 0.00591 0.00585 0.00918 0.00890 0.00858 0,00860 0,00845 0.01295 0.01250 0.01320 0.01240 0,01830 0.01685
76 80 80 80 80 82
=
0.01 inch
Gage Presrure a t Tap, Inches of Water S o . 9 No. 11 2N .o
'Yo. 7
14,coo
0.00408
None None
12,560 22,100 18,900 9,320 9,250 8,790 17,450 16,700 13,600 21,900 12,700
0.00224 0,0096 0.0165 0.0035 0 00209 0.00367 0.00909 None None
0.0025 0,0051 0.0108 0.0201 0.0182 0 093 0.054 0,0223 Kone
0,320
9,350 9,430 9.570 16,700 18,700 17,750 17,100 18,600 24,700 10,900 9,930 9,990 9,900 38,800 9,9GO
Sone
0.195 0.0389
0.0800
0,0261 0.00855 0.1075
17.90 6.94 19.34 15.25 15.00 18.50 20.70 20.85 20.80 12 90 t.84 8.05 11.40 16.70 19.95 47.65 74.30 89.95 50.20 13 50 26.10 63.05 33.15 44.40 23.30 76.80 46.80
14.40 5.80 16.18 12.30 11.80 14.20 7.80 17.20 17.05 16.05 16.90 6.70 6.50 9.15 12.9B 1.5.25 37.90 56.70 47.45 40.20 11.34 21.87 40.35 25.75 33.30 19 73 63.95 35.33
7.20 2.71 7.75 6.25 5.45 6.40 3.95 8.90 8.50 7.80 8.80 3.63 3.00 4.55 6.00 6.60 17.05 23.15 21.70 19.30 6.23 11.77 18.40 11.00 13.60 8 GO 34.40 14.70
4.65 1.52 4.82 3.95 3.15 3.55 2.63 5.55 5.35 4.80 5.70 2.59 2.30 2.90 3.55 3.60 10,65 13.7.6 12.65 11.65 4.25 7.75 10.10 0.65 7.85 5.15 22.85 7.45
35.90 33.90 25.45 43.25 52.75 74.75
28.55 25.95 19.85 33.75 41.10 69.80
10.45 10.50 8.29 13.94 17.05 25.70
6.00 5.80 4.50 7.65 9.65 14.38
3 9 . 7 0 29.25 43.55 31.40 37.85 26.60 2 5 . 8 5 19.03 20.50 15.00 16.75 12.30 64.80 48.10 46.95 3 5 . 3 0 32.23 2 4 . 4 5 28.75 22.25 17.12 1 3 . 3 2 51.65 40.45 6 2 . 2 0 47.65 74.20 56.95 69.35 53.40 75.95 60.25 63.45 50.30
12.35 11.75 9.60 7.80 5.80 4.75 18.85 14.20 9 75 9.20 5.87 17.20 19.40 23.30 21.95 26.30 21.75
7.80 6.65 5.25 4.65 3.10 2.55 10.50 8.05 5.25
Y.60
Air and Wheat
35-R.Iesh Steel Shot Magnified Approximately 10 Diameters 1 scale division
Air Reynolds NO.
Air and 58-Alesh Sand 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46
I n the above equations, no subscripts are necessary on the AX terms, because the increase in the potential energy of position, for 1pound of mass, is a function of the increase in height only. I n Equation 12, material a may be considered as the system and material p as part of the surroundings. Under such conditions, weewould include any work done by material a on material p , as well as work done by material a on some external device such as a turbine. However, it must always apply t o the amount of work done by 1 pound of material a. $n analogous situation exists when considering Equation 13. Letting m arid n denote the pounds flowing per second of
Figure 3.
Solid Rate, Lb./Seo.
M m , Hg
Air
I'emp.,
Vol. 40, No. 9
0.084 0.0615 0.0263 0.0542 0.0553 0,0544
17,000 18,800 18,400 25,900 29,400 39,000
Air and Clover Seed 0.139 11,000 0.158 11,280 0 146 0.0708 0.0494 0,0375 0.169
11,560 11,750 11,700 11,550 18,150 17,600 16,900 17,000 16,700 25,600 24,700 26,200 24,500 36,200 33,300
0.101 0.058 0,0315 0,0109.5 0.0542 0,1052
0.109
0.114 0.0555 0.0440
5.00 2.75 9.60 10.90 13.35 12.50 14.85 12.20
materials a and p , respectively, the follon-ing equations may be written in which each term is exprrssed in units of energy per second: m
* fv,dp + m a 2Yc 2 + m AX
Figure 4.
26
=
mw,. -
rnj.
(14)
28-Mesh Sand Magnified Approximately 10 Diameters 1 scale division = O.0linch
INDUSTRIAL AND ENGINEERING CHEMISTRY
September 1948
I n applications of the type considered in this work, there is no way in which the materials in flow can do work on external devices, and the work done by material a on material p must be equal to minus the amount of work done by material p on material a. Therefore, mWsa 72wt~lp
DATAON FLOW OF SUSPENSIONS (Continued) TABLE 11. EXPERIMENTAL Run NO.
Air Temp., O F.
Atmos. Press., hIm. Hg
Air
Rate Lb./deo.
"
Solid Rate
Lb./Se'c.
Air Reynolds
No.
Gage Pressure a t Tap. Inches of Water No. 9 No. 11 No. 12
No. 7
Air and 35-Mesh Sand 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121
70 70 70 72 72 72 72 72
82 82 81 81 81 86 88 89 89 90 90 79 78 80 80 82 82
745 745 745 743 743 743 743 743 729 729 729 729 729 736 736 736 736 736 736 750 750 750 750 750 750
0.01730 0.01760 0.01730 0.01900 0.01275 0.01292 0.01250 0.01273 0.01010 0.01021 0.00910 0.00830 0.00780 0.00510 0.00522 0.00550 0.00844 0.00560 0.01720 0.00545 0.00550 0.00542 0.01240 0.01880 0.01885 0.00483 0,00494 0.00484 0.01820 0.00713 0.00488 0.00477 0.00489 0.00953 0.00870 0.00885 0.00865
'
0.0364 0.0281 0.0195 0.00723 0.0728 0.0458 0.0385 0.0213 0.115 0.1385 0.089 0.0767 0.0172 0.2080 0.150 0.1075 0.0533 None None 0.0134 0.0953 0.0440 0,1570 0.0700 0.0370
34,200 35,400 34,200 37,600 25,200 25,600 24,700 25,200 20,000 20,200 17,950 16,400 15,460 10,100 10,350 10,850 10,750 11,100 34,000 10,800 10,850 10,700 24,500 37,200 37,200
102.45 93.30 75.85 62.80 84.75 73.55 65.65 51.60 78.10 83.85 60.85 50.30 28.75 61.90 53.90 39.70 28.10 5.87 39.30 13.85 37.80 20.15 99.30 116.75 89.60
82.60 75.20 61.35 50.85 65.60 57.85 52.15 41.00 59.45 63.55 45.80 38.00 22.75 46.70 41.10 29.30 21.15 4.67 31.65 10.80 27.95 15.15 75.80 93.10 72.60
5.60 28.40 40,45 32.05
18.55 16.60 13.65 11.35 13.05 11.70 11.05 8.95 11.10 12.25 8.50 7.00 4.90 7.60 5.80 5.00 3.75 1.00 6.50 2.00 4.90 2.65 15.66 21.80 17.10
36.70 33.05 27.10 22.30 25.25 23.55 21.95 17.50 22.40 23.20 16.50 13,70 9.65 15.75 15.60 10.50 7.90 2.02 13.55 4.25 10,20
1735
+
=
0.
The two friction terms may be coinbined into one term which includes 'the total loss due t o friction per second: Wjt =
mWr.
+ nW1,
Equation 16 then reduces to:
Air and 80-Mesh Sand 47 48 49 50 51 52 53 54 55 56 57 58
83 84 84 84 84 80 80 80 82 82 82
739 739 739 739 739 739 739 739 728 728 728 728
59 60 61 62 63 04 65 66 67 68 69 70 71 72 73
78 78 76 76 76 76 76 78 80 81 81
741 741 736 736 736 736 736 739 744 '744 744 744 744 744 744
82
81
81 81
81
0.112 0.0743 0.0288 None None 0.0433 0,0144 0,00617 0.0813 0.0126 0.0215 0.00953
9,560 9,800 9,550 36,000 14,100 9.650 9,450 9,660 18,850 17,200 17,500 17,100
40 35 32.45 22.20 48.05 9.83 27.60 18.75 13.00 64.10 32.60 41.95 28.45
30.70 24.55 17.05 40.65 8.13 21.00 14.90 10.50 48.50 26.20 33.45 23.10
12.50 10.15 7.40 21.20 4.43 8.75 7.10 5.30 19.90 13.00 15.95 11.50
7.05 5.50 4.35 13.50 2.90 5.05 4.30 3.40 12.30 8.10 9.55 7.30
Air and 24-Mesh 0.1668 0,057 0.0396 0.1145 0.0507 0.0743 0.0238 0.0260 0.0459 0.107 0,0184 0.1083 0.034 None None
Sand 10,300 10,200 9,820 10,200 18,300 19,500 19,250 18,800 18,750 18,400 18,250 15,070 15,000 36,300 12,100
60.20 39.30 29.45 48.10 50.55 63.55 51.45 46.90 53.85 75.00 36.55 57,05 34.85 42.20 6.35
44.30 29.00 21.60 35.60 37.95 48.45 40.90 37.05 41.35 56.55 29.05 42.50 26.70 34.35 5.05
fos7 .:59":5
7.78 5.50 3.25 6.40 7.65 9.45 9.25 7.95
DISCUSSION OF RESULTS
All terms in the above equations are now expressed in units of energy per second, and the. two equations can be added to give : m I & p
+nSu,dp
4- m A UO Z 42gc
(m fn ) A X =
Figure 5.
nA@
4-
2ge
- m'CtT,,- n?v,, - mwf, - nWj,
(16)
Clover Seed Magnified Approximately 10 Diameters 1 scale division = 0.01 inch
Equation 17 was used in the present work t o evaluate the pressure differential because of friction from the measured pressure differential. The calculated data are presented in Table 111.
13.50 14.70 18.35 17.60 15.90 16.40 21.40 12.65 16,OO 10.45 14.70 2.20
The data have been correlated according to the theory described earlier. Figures 7 and 8 show the effect of the group
8.20
111.35
6.40 8.00 5.45 7.45 1.10
on the values A and k of Equation loa in both horizontal and vertical convering. This figure was constructed by selecting the best value for K for each set of data from a plot of (a
Re
- 1) against - and then plotting the data as shown in 1'
Figures 9 and 10. The values for A mere then taken from the intercepts of the latter curves, which were drawn as straight 45 O lines. Some of the points are seen to represent values of 01 considerably
Figure 6.
Wheat Magnified Approximately 10 Diameters ~1 scale divimion = 0.01 inch
INDUSTRIAL AND ENGINEERING CHEMISTRY
1736
Vol. 40, No. 9
foot; P A = (0.076) = 0,0000359 p 14.7 X 144 where p = pounds per square foot . Specific volume of air, cubic 27,900 1 =foot per pound; .0 = P
PA
Solid rate of flow, pounds per second; n = (5)(2000)/(3800) = 2.78 Air rate of flow, pounds per second; ~ f = l n/l5 = 0.185 Air mass velocity, pounds per second per square foot; G = (0.185) = 3.77 Effect of Solid and Fluid Properties on '4 i n the Equation
Figure 7 .
above the rest of the data of the same set. The values for wheat in the present investigation are higher than those obtained from the literature. I n all the runs using wheat in the 0.5-inch pipe and some of those with 35-mesh sand the flow was somewhat erratic. I n general, erratic flow results in higher apparent average frictional losses, so these points irere not given much weight in the final correlation. No attempt was made in this investigation to account for the effects of particle shape and roughness because the accuracy and quantity of data available did not justify any IT ork along these lines. It appears, however, that the particle shape is not of primary importance. The complete range of variables by no means has been covered in this and previous investigations. Therefore, caution should be exercised in any attempt t o use these data for design purposes oubide the range actually investigated, although the general method of correlation seems promising.
Linear air velocity, feet per second; tio = Gv, = (3.77) X 105,000 27,900 __ =P P Assume: Over reasonable small sections f vadp = (0, W ) (;t.p, ; up = ua (this is conservative); energy required t o lift the a1r IS negligible; loss due to bends in pipe is negligible; and the term n J v , d p is negligible Equation 17 then reduces to: u2
mvaAp -I- (rn -I- % ) A 2 4- n 4 X = - J ~ / x 290
or :
Substituting numerical values gives: 0.185 X o-
Five tons of wheat are to be conveyed per hour. The conveyer will consist of a 3-inch inside diameter pipe running 100 feet vertically and 200 feet horizontally. A blower is installed a t the discharge end of the pipe which is oDen to the atmosuhere. Calculate the pressure'drop through the line if the weight ratio of wheat to air flowing is 15. SOLUTIOX. Atmospheric pressure, pounds per square foot; 14.7 X 144 = 2120 Air density, pounds per cubic
pl
-
PI
+ 2.78 + 0.18564.3
[
(
7
)
2
-
P2
+ Pi + 5.08 X
108
[ (:%)'-
(+,)'I.
+ 2.784.X
(i)'
By trial and error: ps = 2072Ib./'sq. ft.
6
;' 2
IO
00 04
e)
mo
ai
Figure 9. w Re
= -Wit
Equat,ion 17a now will be applied to separate sections of the conveyer. INLET NOZZLE.Assume that in the nozzle, friction and lift are absent and that only inertia effects are responsible for pressure loss. I n this special case the term should not be included because the velocity is zero before entering the nozzle. Then:
QO
d
- W/t
(17a)
20
-1=
+
2.78 AX =
40
Figure 8. Effect of Solid and Fluid Properties on k i n the Equation
(-)2]
and simplifying: 10,300 X
SAMPLE PROBLEM
Pav
Relative Pressure Drops for Horizontal Conveying of Various Solids with Air
INDUSTRIAL AND ENGINEERING CHEMISTRY
September 1948
~
~~~~~~
~~~
~
1737 ~
TABLE 111. CALCULATED DATA Wt. Ratio Solid t o Air, r
Air Reynolds Number, Re
Run No.
Fractional Increase in Pressure Drop Horizontal Vertical (a 1 ) ~ (a 1 ) ~
Air and 48-Mesh Sand 14,600 0.655 12,560 22,100 .... 13,900 0.318 9,320 2.04 9,250 3.53 8,970 0.772 18,450 0.224 16,700 0.435 13,600 1.32 .... 21,900 12,700 ..,. 9,320 0.531 9,350 1.08 9 430 2.25 9:570 4.17 16,700 2.17 18,70 9.85 17 750 6.02 17:lOO 2.59 .... 16.600 24.700 10,900 35.5 9,950 7.73 9,990 15.9 9,900 5.22 38,800 0.282 9,960 21.4
....
-
0.655
...
-
..*. ....
...
...
8.78 7.1 4.87
..
8.9 3.02 1.26 8.53 1.45 2,43 1.10
Air and 24-Mesh Sand 32.1 11.0 7.98 22.2 5.48 7.54 , 2.45 2.74 4.85 11.5 1R,250 1.99 15,070 14.2 15,000 4.5
Air and Wheat
.... ....
36,300 12,100 17,000 18,800 18,400 25,900 29,400 39,000
9.78 6.48 2 83 4.13 3.72 2.76
(a
7
-
1 ) ~
(a
-
1)v
3.61 3.87 2.95 2.27 1.47 1.04 2.36 1.65 1.10 0.97 0.32 0.76 1.05 1.21 1.33 0.455 0.415
7.18 8.13 8.95 4.00 3.20 2.47 5.3 3.76 2.4 1.38 0.57 1.73 2.42 3.04 3.03 1.21 1.30
1.70 1.36 0.985 0.35 1.97 1.85 1.80 1.13 2.47 2.67 2.29 2.30 1.67 8.38 6.92 4.53 3.41
2.37 1.97 1.37 0.63 4.07 3.05 2.67 1.75 5.68 5.35 5.30 4.58 2.25 12.7 10.43 7.75 5.2
Air and 35-Mesh Sand
...
10,300 10,200 9 820 10:200 18,300 19,500 19,250 18,800 18 750 18:400
72 73 74 75 76 77 78 79
Fractional Increase in Pressure Drop Horizontal Vertical
Air and Clover Seed 25.0 27.7 25.0 11.9 8.35 6.41 18.4 11.35 6.75 4.36 1.265 4.18 8.42 8.25 9.20 3.03 2.63
1.2
Air and 80-Meah Sand 23.2 15.1 5.95
Air Reynolds Number, Re
Run No.
...
....
9,560 9,800 9,550 36,000 14,100 9,650 9,460 9,660 18,850 17,200 17,500 17,100
Wt. Ratio Solid to .4ir,
... ...
1.12 0.87 0.565 0.425 0.362 0.155
... ..*
3.05 1.99 1.21 1 20 1.21 0.85
97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
.
....
... ...
4.55 1.44 2.21 2.12 1.30 0.845 4.1 2.67 1.54 1 .9 0.75 0.594 0,442 0.75 7.13 ,
a - 1 = 20,000
3 0.068 15 (m)(m 7m0) 2
... ...
7.65 1.96 3.5 5.11 2.46 1.38, 6.95 4.9 2.45 3.4 1.37 1.01 0.78 1,25 14.7
... ...
0.86
a = 13.5
W,f, = 13.5 X 366 X 0.186 = 913 (ft.)(lb.)/sec. Substituting in 17a gives:
VERTICALSECTION. R e = DG -= p
0'25 3'77 = 78,000; 0.018 X 0.000672
10,300
" - 2072 + 5.08 X G-Tzm
108
[$
-
&I*
+ 2.78 X 100 =
-913
By'trial and error: p , = 1619 lb./sq. ft.
Fanning friction factor = 0.0048. Assume average p in this section = 1900 lb./sq. ft.; average P A = 0.068 lb./cu, ft. If only air were flowing, head loss due to friction would be: 2fGzL =.
u
z
2 X 0.0048(3.77)2100 32.2 X (0.068)' X 0.25
366 (ft.)(lb.)/lh. air
d'/aX A
=
62.4 X 1.28 X 0.068 X 32.2 (0.158/12)2 0.018 X 0.000672
950
20,000 (from Figure 7)
k = 0.85 (from Figure 8)
Figure 10. Relative Pressure Drops for Vertical Conveying of Various Solids with Air
INDUSTRIAL AND ENGINEERING CHEMISTRY
1738
HORIZONTAL SECTIOK.Assume average p in this section = 1500 lb./sq. ft.; average pA = 0.054 lb./cu. ft. If only air were flowing, head loss due to friction would be:
Vol. 40, No. 9
volving suspensions of fine particles in large pipe, as no experimental data are available for this range. NOMENCLATURE
dI/a(w
- p)
gda = d I / a
v
X 62.4 X 1.28 X 0.054 X 32.2 (0.158/122 0.018 X 0.000672
E
848
A
D f
= empirical constants = ipe diameter, feet
== Flocal riction factor acceleration due t o gravity
A = 26,000 (from Figure 7)
g g.
k = 0.93 (from Figure 8)
mass pounds-feet 32’17 force pounds-(seconds) L,k , , k , = empirical constants R = ratio of average axial particle velocity to the average fluid velocity L = length, feet m,n = mass ratio of flow of fluid and solid, respectively, pounds per second p = pressure, pounds per square foot r = weight ratio of solid t o fluid flowing ua, up = average velocity of fluid and solid, respectively, feet per second ur = velocity of fluid relative to that of particle (u,- u,l v = specific volume, cubic feet per pound W = energv, feet per pound w = solid density, pounds per cubic foot X = vertical distance above a given datum, feet a: = relative pressure drop, dimensionless p = viscosity of fluid, pounds per foot second p = fluid density, pounds per cubic foot
a:
-1
a:
-
=
A
(6 x g)k
2);(
1 = 26,000
(.3- )z 0 108
o!
(0l4 79.8
98,000
= 4.75
Wj.lirt = 4.75 X 1160 X 0.185 = 1020 ft. lb./sec. Substituting in 17a gives: 10,300 X
$: t:ii
+ 5.08 x
1 0 8 [pi -
&]
= -1020
By trial and error: p2 = 1309 lb./sq. ft. The ressure at the blower intake is then 1309 pounds per square root of 9.1 pounds per square inch absolute and the pressure drop through the line is 5.6 pounds per square inch.
= conversion factor in Newton’s second law =
BIBLIOGRAPHY
CONCLUSIONS
The data of this investigation together with published data on the pneumatic conveying of wheat indicate t h a t the pressure drop due to friction in flowing suspensions can best be represented by the folloying type of equation:
where A and k are functions of the dimensionless group 4 1 : 3 ( ~ p)pg@
IJ
The data are insufficient t o evaluate the effect of particle shape but indicate t h a t this factor is not of primary importance. The above equation should not be used for design work in cases in-
(1) Badger, W. L., and McCabe, W. L., ”Elements of Chemical En-
gineering,” New Yorlc, McGran~-HillBook Co., 1936. (2) Cramp, W., J.SOC.Chem.lnd.(London), 44.207-B (1925). (3) Cramp, W., and Priestly, J. F., Engineer, 137, 34 (1924). (4) Cramp, W., and Priestly, J. F., J. SOC.Brts., 69,253 (1924). (5) Gasterstadt, H., 8.D.I. Forschungsarbeiten, No. 265 (1924). (6) Hudson, W. G., “Conveyors and Related Equipment,” New York, John Wiley & Sons, 1944.
(7) Perry, J.,H., “Chemical Engineer’a Handbook,” New York, McGraw-Hill Book Co., 1934. (5) Segler, W.. “Untersuchungen an Kornergeblason und Grundlagen fur ihren Berechnung,” Mannheim, Wiebold Co., 1934. (9) Van Driest, E. R., J . A p p l i e d Mechanics, 12, A 3 4 (1946). (10) Wood, S . A , , and Bailey, A . , Proc. Inst. M e c h . Eng. (London),
142, 149 (1939).
RECEIVED J u n e 11, 1947.
Fugacities in Gas Mixtures JOSEPH JOFFE Nezcark College of Engineering, ivewark, N . J .
F
UGBCITIES of components of gas mixtures are of importance in the study of chemical equilibria in gaseous systems a t high prbssures (.5). In view of the difficulties inherent in the determination of fugacities of individual coinponents in mixtures, the Lewis-Randall fugacity rule, which is based on the law of additive volumes: has come into general use. Accordi:ig t o this rille the fugacity of the ifk component in the mixture, j ~is, given by j &= fpxi (1) wheref: is the fugacity of the pure ith component a t the temperature and total pressure of the mixture, and zi is the mole fraction of the ith component in the mixture. FUGACITIES OF CORIPOSENTS OF GAS MIXTURES
By employing the Levi-is-Randall fugacity rule in conjunction with generalized fugacity charts, n’ewton and Dodge have shown
that the effect of pressure on the equilibrium constant of a gaseous reaction may be predicted with the aid of critical data alone ( 1 7 ) . While the method of Kewton and Dodge gives good results for the ammonia equilibrium which they studied, its validitv rests on tho applicability of the Lev, is-Randall rule, which is known to fail in some gaseous systems at high pressures (18). In this paper a method is presented for the evaluation of fugacities of components with the aid of generalized charts, which is similar in principle to the mrthod developed by Gamson and Watson (3)in connection with high pressure vapor-liquid equilibria, but which is believed to be applicable over a different range of conditions. This method does not involve the assumption of the Leais-Randall fugacity rule and should prove useful in thc study of chemical equilibria at high pressures. It is assumed, 111 accordance with the suggestion macle by Kay (11), that the pseudocritical temperature, To, and the pseudocritical pressur c, P,, of the mixture are given by the relations:
