INDUSTRIAL AND ENGINEERING CHEMISTRY
MAY, 1939
However, many flow problems require the determination of the flow rate or of the diameter, the other quantities in either case being fixed. I n both types of problem it is impossible to calculate directly either Rel or fl,and many texts state that solution may be obtained only by trial and error. It is clear, however, that if the other two ratios K and S. F. are plotted against f 1 , then w or D, respectively, may be solved directly without trial and error, as well as p i n both cases. The plot of f1-ll2 us. log K (3, 7,9) accomplishes this result for w, with the added advantage that the line obtained in the turbulent region a t usual velocities is straight, as far as can be determined by the data available. The plots of f - l I 2 vs. log S. F. and log Re have also been considered (6). I n order to make it possible to compute directly whichever variable in a fluid flow problem is unknown, Figures 1 and 2 were plotted. A decrease to 2 in the number of curves necessary is obtained by plotting fi, Rel, K , and S. F. against each other in pairs. I n Figure 1the curves of 1 / f l u s . log K were obtained from the equations of Drew and Genereaux (2) by converting to the present f1. From them, all other curves in this paper were computed, point by point. Figure 1 has the advantage that all of the turbulent flow plots are sensibly straight. The equations for the four lines shown are as follows: For turbulent flow: smooth pipe:
45 - 2.22 bgi& - 0.56
log S. F. = 2.39 log Rel
- 0.36
which condenses to: sgp =
2
13
12
II
IO
0.1906
-&&wi.ia
3.0
0.22
=
0.1906 ?f
0 6
4.0
(x)-o'22 Dp 5.0
619
average for commercial pipe: log S. F.
=
1
z= -
1.777 log&
2.41 log Re1
+ 0.399
0.42
which condenses to : sgp =
[Genereaux sgp = 0.1252
0.1445
wl.az
0.18
=
0.1445
w" D6 (%)-'"* Dp
(4) obtains by another means the equivalent of ( w ~ / D ~ ) ( w / D ~ which ) - O ~includes ~~, a margin of
safety beyond average conditions.] For viscous flow: sgp =
128 w p - (Poiseuille) T D4
Each of these formulas may be used to calculate either of the unknowns for which its plot is used. The condensed formulas may also be used in calculating w, a t some sacrifice in accuracy. In the graphical determination of s, D, or p, Figure 2 is more accurate and for w equally accurate. The pairs of dimensionless ratios are picked in such a way that in each case the ratio omitting a given variable is plotted against the ratio in which this same variable occurs to the highest power with respect to the other variables present. Table I is included to facilitate replotting the curves. Logarithms on linear scales were plotted for ease in interpolation, but logarithmic scales may be found more convenient in replotting the curves. All of the principles and methods described here also apply when the commoner definitions of Re and f are used, but the scales for these ratios would have to be multiplied by the proper correction factor. If for Re the ratio D V p / p were to be used, the scale should be multiplied by 4 / ~ . If for f, sgD/(2pV2) were substituted, the ) multiplier is n2/32. -7 The scales are labeled with the dimensionless ratios, instead of sym-8 bols (such as f ) , and thus eliminate the need for an auxiliary defining equation (particularly the Fanning -9 equation, or the Darcy equation, with their additional constants of 2 and l/2, respectively). -10
9
-I I
8
-12
7
-13
6
-14
5
-15
4
-16
3.
FIQURE 1. STRAIGHT-LINE CHARTOF (FRICTION FACTOR)' 1 2 us. VON KLRMLN AND REYNOLDS NUMBER us. SIZEFACTOR NUMBER,
Direct Solution by Friction Factor-Reynolds Number Chart As previously stated s (or p) or p may be determined directly by the fl us. Rel chart (Figure 3). I n solving for w by trial and error, a value of w is assumed, j1and Rel are calculated, and the point is plotted on the fl-Rel chart. If this point does not fall on the desired curve, another is calculated with a new value of w. It is obvious that all such points in a given problem will lie on a straight line on the log-log plot. Furthermore, the lines obtained in different problems will always have the same slope (namely, -2) since Rel is directly proportional to w, and f i to w-2. Therefore, a direct solution may be obtained for any problem in which w is unknown by assuming
VOL. 31, YO. 5
INDUSTRIAL AND ENGINEERING CHEMISTRY
620
TABLEI. Smooth Pipe
Log
K
Rei
2.6 398 3.0 1,000 3.4 2,512 3.8 6,310 4.2 15,850 4.6 39,810 5.0 100,000 5.4 251,000 5.8 631,000 6.2 1,585,000
Rex 3.317 2,075 3.785 6,100 4.244 17,560 4,696 49,700 5.143 139,000 5.585 384,400 6.023 1,054,000 6.458 2,870,000 6,890 7,770,000 7.321 20,940,000
1 -
fl
fi
dr
-1.434 -1.571 -1.689 -1.793 -1.886 -1.970 -2.046 -2.116 -2,181 -2.242
0.0368 0.0289 0.02046 0.0161 0.01302 0.01072 0.009 0.00766 0.00659 0.00573
5.21 6.1 6.99 7.88 8.76 9.65 10.54 11.43 12.32 13.2
DATAFOR REPLOTTING CURVES
k?#. 7.6 8.68 9.77 10.85 11.92 12.98 14.03 15.09 16.14 17.18
S.F. 3.98 X 107 4.77 X 108 5.85 X 108 7.01 X 10'Q 8.23 X 10" 9.5 X 1012 1.08 X 1014 1.22 X 1016 1.37 X 1018 1.52 X 10"
a convenient value for it (usually 1, 10, etc.) that will give values of fl and Rel on the chart, plotting this point, and passing through it a line of slope -2 which intersects the appropriate fl-Rel curve, then reading off Rel and calculating back to obtain w. Direct solution for D may be obtained similarly, except that the lines of varying D have a slope of -5. Figure 3 is the standard plot of f1 us. Rel, with lines of varying w and of varying D drawn in. (A combination of versions of this plot and of Figure 4 is presented by Johnson, 6.)
Direct Solution for w , D , and s Inasmuch as the problem of determining p would seldom arise, simplification is obtained by omitting fl. Rel may be plotted against K , and lines of varying D (slope = -"z) superimposed. Inspection shows that a line of varying or unknown D corresponds to a definite value of S. F. Figure 4 is plotted on logarithmic scales for scale values of S. F. I n using this chart on a problem in which w,s, or D is unknown, the dimensionless ratio not containing this quantity is calculated, the corresponding point on the appropriate curve found, and one of the other two dimensionless ratios read off and used to calculate the unknown. TYPICAL PROBLEM. What water flow rate will be obtained through a 4-inch, nominal size steel pipe, 1000 feet long, if the water is a t 68" F. ( Z O O C.) and the pressure drop available is 93.73 pounds per square inch?
Re1
Average Commercial Pipe 1 Log
-
Re1
fl
3.299 1,999 -1,401 3.758 5,730 -1,517 4.209 16,180 -1.618 4.654 45,140 -1.709 5,096 124,700 -1,791 5.533 341,400 -1,867 5.968 928,000 -1.936 6.4 2,511,000 -2.0 6.83 6,755,000 -2.059 7.257 I.8,100,000 -2.117
fi
dz
k?8.
S. F.
0,0397 0.03045 0.0241 0.01955 0.01617 0,0136 0.0116 0.01001 0.00865 0,00763
Using Figure 1, we obtain from the commercial pipe plot of
K against (fi) -1 (.fi)-'/'
=
'2:
8.73 =
W
(13.5 X 32.2 X 62.3)1'2(0.3355)6/2 and finally w = 93.5 lb./sec. or 1.50 cu. ft./sec.
Using Figure 2, from K against Rel: W
log Re: = 5.62; Re1 = 415,000 = 0.3355 X 0.000672 w = 93.5 lb./sec.
Using Figure 3, let us assume a trial value of w of 50 pounds per second: Reltrlal fitrial
= 222,000; log Relt,,,l = 5.346
=
0.0460; log fitnal
-1.337
Plotting this point and going from it to the commercial pipe line along a varying w line, we obtain:
93.73 X 144-1000 13.5 lb./sq. ft./ft. 4.026 D = 12 = 0.3356 ft. p = 62.3 lb./cu. f t . p = 0.000672 X 1.00 = 0.000672 lb./f t./sec. S '
-
Since w is the unknown quantity, K is the dimensionless ratio that can be calculated, giving: K = (13.5 X 32.2 X 62.3)1/*(0.3355)a'a0.000672 47,600 log K = 4.677
CHART OF REYNOLDS NUMBER us. vox KARMANNUMBER, AND FIGURE 2. PRECISION FRICTION FACTOR vs. SIZEFACTOR
MAY, 1939
621
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
log,,
W
(Re ,)
FIGURE 3. FRICTION FACTOR us. REYNOLDS NUMBER, WITH LINES OF VARYING w D,FOR DIRECTDETERMINATION OF w OR D
FIGURE 4.
REYNOLDS NUMBERus.
VON
A N D OF
VARYING
KARMAN NUMBER,WITH VERTICALLINES OF CONSTANT SIZEFACTOR, FOR DIRECTDETERMINATION OF S,
D,OR W
log Beltrue = 5.62 Beltrue= 415,000 w = 93.5 as before
Using Figure 4,for K = 47,600, we read Rel = 415,000, and get w = 93.5 as before. We could also read S. F. from the curve and calculate w from it, but it is easier from Rel.
A = & = -a2 = n
4m2(n n
+ 1)2
+
Therefore, D should be replaced by2a/(n 1) and w b y n n w/ (n 1)2. The unknown a may be determined by the same plot used for D in circular-section pipes.
+
Literature Cited Noncircular Cross Sections of Flow Viscous flow in noncircular cross sections requires an equation derived for the particular case (IO). The charts here presented apply directly only to flow through a single circular section, and if a number of such pipes is connected in parallel, w must be the flow rate per pipe. These charts may be applied, however, to turbulent flow in noncircular sections by making the proper substitutions. I n most problems involving noncircular cross sections, the dimensions are fixed, and the hydraulic radius, m, and the cross section of flow, A , can be calculated. I n order to solve for s or w, D should be replaced in the dimensionless ratios by 4 m and w by ( 4 m 2 / A ) w . Figures 1 to 4 may then be used as before. For a constant cross-sectional shape, A will be proportional to m2, and it will be possible to solve also the problem of unknown dimensions directly. Thus, for a rectangular duct cross section, let the sides be a and b and a/ b = n. If n is fixed, we will have:
Buckineham. E.. Phvs. Rev..4. 345 (1914). Drew, 5.B.‘, and Genereaux,‘R. P., Trans. A m . Inst. Chem. Engrs., 32, 17, 18 (1936). Drew, T. B., Koo, E. C., and McAdams, W. H., Ibid., 28, 56 (1932). Genereaux, IND.ENG.CHEM.,29, 387 (1937). Johnson, S. P., papers presented before Aeronautics and Hydraulics Div., Am. Soc. Mech. Engrs., June, 1934, p. 105. Miller, B., Trans. Am. Inst. Chem. Engrs., 32,21 (1936); Chem. & Met. Eng., 44, 616 (1937). Nikuradse, J., Forsch. Gebiete Ingenieurw., Forschungsheft, 356 (1932). Walker; W. H., Lewis, W. K., McAdams, W. H., and Gilliland, E. R., “Principles of Chemical Engineering,” 3rd ed., p. 76, New York, McGraw-Hill Book Co., 1937. Tbid., p. 83. Tbid., p. 94.
Esters of Sodium Sulfosuccinic Acid-Correction An error occurs in our article which appeared under the above AND ENGINEERING title in the January, 1939, issue of INDUSTRIAL CHEMISTRY.I n Table I on page 47, compound 3 in group B should have been printed as R’-Ar-SOs Na.
C. R. CARYLAND W. P. ERICKS
able for handling slurries. A propeller stirrer, supplemented in some cases by air agitation, revented settling in the mixing tank. This tank was further fittefwith a cooling coil of 0.5-inch copper tubing in order to keep the suspensions essentially at room temperature. Without the coil, temperature increased rapidly as a result of the work done upon the fluid. The weight rate of flow was measured in duplicate in the weigh tank. The measuring section of each pipe was 27 feet long and was followed by a 2-foot calming section. The brass pipes were preceded by 6.5-foot calming sections, and the 3-inch iron pipe by an 8-foot section. All pipes were in two lengths joined by carefully fitted butt connections held together by sweated sleeves. The pressure taps for the brass pipes consisted of piezometer rings with an internal groove connecting four holes in the pipes. Two 0.25-inch Dole fittings served as pressure taps in the 3-inch pipe. In all cases the pressure was transmitted to the manometers through 0.25-inch copper tubing. Two vertical separating chambers, S, prevented solids from reaching the pressure lines. The entire pressure-measuring system was filled with water, and to keep it full, vents V were o ened at frequent intervals to allow an escape of air that might lave collected in the system. The piezometer connections were kept free of cloggage by the occasional injection of a small amount of water from the separators into the pipes. Pressures over a wide range were measured on a mercury-water and an air-water manometer (not shown), each of which was connected in turn through manifolds. The apparatus was fitted with air and water lines to aid in cleaning the pipes after slurries were pumped through them. The operation of all three pipes was tested with water. The data are given in Table I. The pressure drops in the 0.75- and 1.5-inch brass pipes agreed closely with accepted values for such pipe (17) (see Figure 6 also). The 3-inch pipe gave pressure drops about 40 per cent higher than would be predicted.
Flow.of (Y
0
3uspensions
through Pipes1 RICHARD H. WILHELM, DONALD M. WROUGHTON,2 AND WILLIS F. LOEFFELI Princeton University, Princeton, N. J.
F
INELY divided industrial solids are frequently transported by being pumped in the form of water suspensions. Comparatively little has been published, however, concerning the friction losses that occur in pipes with the flow of such heterogeneous fluids. Gregory (6) gave the results of pressure drop experiments conducted with clay suspensions in a 4-inch iron pipe. Shishchenko and Baklanov (16) worked with a similar material. Ambrose and Loomis (1) studied the flow of bentonite suspensions through 0.5- and 1-inch pipes, and data on the flow of paper pulp and sand suspensions (2, 4, 9, 11, 12, 14) have been reported. The viscometry (6, 8, 10, 13,16, 19) of suspensions as an approach to the theory of such materials has received more attention. However, the results of viscometer measurements have not been readily applicable to pipe design. The present paper deals with a comparison of the flow behavior of cement rock and Filter-Cel suspensions in a viscometer with that in three sizes of pipe. Data were taken simultaneously with the viscometer and the pipes to eliminate from consideration any changes in the properties of the suspensions due to working or aging. Gregory (6), Ambrose and Loomis ( I ) , and Merkel (11) previously compared suspension flow in pipe and viscometer. Herrick (7) developed an equation for the flow of drilling fluids through pipes by using plastometer measurements together with the Williams and Hazen formula for hydraulic flow. However, this equation was not verified for tubes larger than capillary size.
Materials and Method
TABLE I. PIPEFLOW DATAFOR WATER NO.
1
to
C.
25
26
27 2
3
4
5
,
The cement rock suspensions were made from a material ground for kiln feed, 92 per cent passing No. 200 sieve. The Filter-Cel suspension was prepared from standard Filter-Cel mixed with some Super-Cel. Figure 1 shows the pipe system used. The 0.75- and 1.5-inch brass pipes were connected parallel to a 3-inch iron pipe by means of end manifolds so fitted with valves that any one ipe could be placed in operation. Fluid circulation was provifed by a 45gallon-per-minute, 120-foot head, Gould centrifugal pump, suit-
100 Ft. AP
Run
26 27 28 30 31 32 24 26 27 28 29 30 31 32 33 33 34 36 33 35 36 37 37 25 26 28 29 30 31 32 32 33 34 34 35
Re 3-Inch Pipe (3.05 Inches I. D.) 3.00 105 73,100 3.09 104 75,000 2.60 78 70,000 2.15 52 58,000 1.23 20 33,600 3.15 95 88,600 3.15 97 88,600 2.84 84 80,000 2.33 60 65,500 1.5-Inah Pipe (1.59 Inches I. D.) 1450 10.40 148,000 1450 10.40 148,000 8.56 940 126,000 5.12 383 76,000 2.72 126 43,000 22,200 1.38 37 8.92 1070 119,000 957 119,000 8.51 920 119,000 8.23 805 113,000 7.70 108,000 7.16 712 102,000 6.58 606 455 5.76 90,000 4.61 73,000 322 167 3.26 53,000 57 1.75 28,800 1.13 27 19,100 16 0.81 14,000 0.75-Inoh Pipe (0.812 Inoh I. D.) 16.50 6950 136,000 12.90 4490 112,000 7.51 1720 66,600 2.86 308 25,700 1.50 98 13,700 17.70 7520 125,000 16.10 6290 116,000 5200 14.50 108,000 12.00 3660 91,300 3160 11.00 85,000 2650 79,700 9.96 2230 72,500 8.94 1750 65,300 7.95 1370 58,100 6.98 48,600 5.74 980 39,100 4.59 655 29,300 3.39 407
u
0.0077 0.0072 0,0076 0.0074 0.0086 0.0063 0.0064 0.0068 0.0073 0.0046 0.0046 0.0044 0.0050 0.0058 0,0067 0.0046 0.0045 0.0047 0.0047 0.0048 0.0048 0.0047 0.0052 0.0054 a.0064 0.0073 0.0083 0.0045 0,0048 0.0053 0.0066 0.0076 0,0042 0.0043 0,0043 0.0045 0.0046 0.0047 0.0049 0.0049 0.0049 0.0052 0.0055 0.0062
With this apparatus the pressure drops were measured in each of the three pipes at varying flow rates of the two suspensions. The cement rock concentration was varied from 54 to 62 per cent by weight; that of the Filter-Cel, from 21 to 34 per cent. The velocities ranged from 0.3 to 14 feet per second; the pressure
1 Presented as part of the symposium on fluid dynamios a t Carnegie Institute of Technology, Pittsburgh, Penna. Other papers appeared in April, 1939, pages 408-486, and on pages 618-621 of this issue. * Present address, Columbia University, New York, N. Y. a Present address, E. I. du Pont de Nemours & Co., Inc.,Wilmington, Del.
622
