INDUSTRIAL AND ENGINEERING CHEMISTRY
288
Because of this variation i t was considered probable that some of the other values in their data could be made more accurate. Ramsay (3) and Crafts (1) also showed that the boiling point of pure benzene (100 mole per cent benzene in Table I) varies 0.426' C. for a 10.0-mm. variation of pressure in the vicinity of the boiling point of benzene. This would make the boiling point of benzene 80.13" C. a t 760 mm. A large number of references indicate that the average boiling point of pure benzene is 80.01" C. a t 760 mm. This is a variation of 0.12" C. which must be subtracted from Rosanoff, Bacon, and Schulze's boiling point of benzene to give a more accurate value. From these authoritative corrections on the boiling points of 0 and 100 mole per cent benzene in toluene, the writer has endeavored to make the other molar concentrations more accurate by plotting a graph of the benzene molar concentrations in relation to the corrections; a straight line was dra-m through the calculated extreme corrections so that the intermediate corrections may be interpolated as in a thermometer correction graph. The sign of the correction must be recognized. The corrected boiling points for the various molar
.
VOL. 32, NO. 2
concentrations of benzene in toluene are given in the righthand column of Table I. Although these corrected values are close approximations to the absolute values, it is believed that the limit of error has now been reduced to within *0.05" C. Because of the high mathematical and technical ability of
M. A. Rosanoff, C. TV. Bacon, and J. F. W. Schulze and their many valuable contributions to physical chemistry, the writer hopes that his efforts are not misinterpreted as an attempt t o improve their splendid method of mathematical derivation. Their calculations had necessarily to be based on the less accurate vapor pressure data then available.
Literature Cited (1) Crafts, Berl., Ber., 20, 709 (1887). (2) Kahlbaumn, 2.p h y s i k . Chem., 26, 603 (1898). (3) Ramsay, W., and Young, S., 2.Chem., 1, 249 (1887). (4) Regnault, Mim. acad., 26, 339 (1662). (5) Rosanoff, M.A., Bacon, C. W., and Schulze, J. F. W., J . Am. Chem. SOC.,36, 1999 (1914). (6) Young, S.,J . Chem. Soc., 55, 486 (1889).
Free Discharge of Fluids through Small Circular Orifices HIRA LAL ROY AND NIRMAL K. SEN-GUPTA College of Engineering and Technology, Bengal, Jadavpur, India
A
CCORDING to the simplified equation
The purpose of the present investigation is to study the rate of flow with fluids widely varying in their physical properties. With the limited means a t our disposal it has not been possible to exceed the hydrostatic head by more than 6 feet and the orifice diameter by more than 3/* inch.
u =Cfih derived from Bernoulli's theorem, the rate of discharge of different liquids through the same orifice and under the same hydrostatic head (in terms of feet of fluid flowing) Experimental Method through the same cylinder is the same; i. e., other factors The amaratus used is shown in Figure 1. The diameter remaining the same. the rate of discharge is indewndent of of the container is large enough to make the nature of the liquid. Badger and the velocity of the liquid through it McCabe's problem 11 (1) is meant to negligible compared to that through the show this fact, and the authors informed orifice. The orifices are circular and are us (2) that that answer is t o be exmade of 0.1-inch thick brass sheets, pected of students who are given the fitted flush to the wall of the container. simplified equation. We have searched The perforated baffle plates through the literature for experimental data and which the liquid passes when introduced found that a large amount of work from the top eliminate the jerky flow (3, 4, 6, 7) has been done, mostly with that otherwise occurs as a result of the larger orifices and with only water as artificial increase in the hydrostatic the liquid. Variations in the rates of head produced when no baffles are presdischarge were found by most of the inent. The vessel is graduated in feet vestigators, and it is suggested that the from the center of the orifice placed a t factors influencing the rate of discharge the side wall a little above the bottom are hydrostatic head, diameter and shape plate. I n order to keep the level conof the orifice, viscosity, surface tension, stant, outlets are provided a t one-foot and density of the liquid. But it has intervals; they can be closed or opened not been possible to deduce a theoretical as desired and thus fix the hydrostatic equation for the rate of flow involving head under which the flow is to be all these factors; the empirical equameasured. The duration of the flow is tions derived are also of limited applicameasured with a stop watch. With tion, and the liquid used is water. FIGURE 1. DIAGRAM OF APPARATUS
-
FEBRUARY, 1940
INDLSTRIA4L-4ND ENGISEERING CHEIIISTRY
TABLEI. PROPERTIES OF THE LIQCIDS Liquid
so.
Temp.
SP. Gr.
F.
I I1
Water Vacuum oil. D. T . E.. heavy mediumc Vactra oil heax-5- S C Castor oil Super cylinder oil. 600 W C
I11 IT' 1-
Surface Tension Dgneu/ cm.a
T-IBLE111. CRITICAL VELOCITIES~ Viscosity
~ i ~ ~ i J/s d '/le
Centipoisesb 0.75
91.5
0 995
70.7
89.8 87.0 86.5
0.888 0.904 0.956
31.0 32.3 37.9
60 120 435
86 9
0.891
37.9
700
I
I1 111 IT' T'
TABLE 11. ACTLLL VELOCITIESTHROUGH THE ORIFICE
I1
7.91 9.62 11.00 12.25 13.40
Velocity through Orifice Diam. of: in. 1/1 in. 3/16 in. '/P in. Feet per second7 32 7.33 7.49 8 78 8 88 8.88 9.08 10.52 10.19 10.20 10.39 11 90 11.39 11.54 11.34 13 20 12.48 12.46 12.62 14 31
6
8.94 10.45 11.76 12.99 14.10
8.48 10.40 11.06 12.10 13.29
8.22 10.29 11.93 13.28 14.35
8.16 9.79 12.00 13.42 14.86
7.75 9 75 11 41 12 93 14.31
6.36 8.58 10.23 11 48 13 00
2 3 4 5 6
8.84 10.89 12.69 14.13 14.45
8.24 10.24 11.78 13.28 14.47
7.95 9.77 11.37 12.83 14.27
7.84 9.65 11.31 12.76 14.15
6.69 8.59 10.11 11.68 12.86
4.76 6.36 8.02 8.91 10.46
2 3 4 5 6
7.64 9.76 11.34 12.87 14.28
6.91 8.80 10.29 11.74 13.17
6.39 8.16 9.74 11.07 12.38
5.62 7.41 8.94 10.36 11.68
3.94 5.46 6.80 8.13 9.40
1 52 2.35 3.18 4.01 4.84
2 3
7.22 9.14 10.88 12.37 13.i2
6.22 8.27 9.74 11.34 12.46
5.08 7.17 8.65 10.12 11.50
4.39 6.02 7.47 8.85 10.08
2.56 3.77 4.84 5.95 7.05
0.91 1.44 1.94 2.49 2.90
2 3
-1 I11
IV
T'
4 6
--l/r
in.
5/16
1/16
0 29 26.38 51.81 177.63 306.69
'/16
l/'a
0.37 31.97 64.76 222.04 383.36
0.49 43.96 86.34 296.05 511.14
0.73 65.94 129.52 444.08 (66.72
1/18
1.46 131.80 259.04 588.16 1533.44
TABLE IV. COEFFICIESM OF DISCHARGE, C-
I
in.
10.59 13.00 l5,03 16.77 18.37
0.24 21.98 43.17 148.02 255,57
w
'/a
Calculated according t o a n equation of Walker, Lewis, and JIcAdams (91.
4
Liquid
Head over Center of Orifice Feet 2 3 4 5 6
Orifice Diameter, Inches:-
r
a Measured h3- drop method. 6 Measured by U-tube viscometer, according t o the method of t h e Inatitution of Petroleum Technologists. c From Socony-Vacuum Oil Coinpan?, Inc.
Liquid
289
Head over Center of Orifice Feet 2 3 4 5 6 2 3 4 5 6
111
I
3 4 5 6
IT'
2
3
4 6 2 3 4
T'
5
6 0
--
In.
3/s
C a t Orifice Diameter of.---5/10
in.
1/1
in,
3/16
in.
113 in,
1/18
111.
0.698 0 692 0.686 0.683 0.682
0.645 0 640 0.635 0.632 0.635
0,647 0.639 0.635 0.635 0.634
0.660 0.654 0.648 0.643 0 642
0.774 0.757 0.742 0 736 0.728
0.934 0.936 0,938 0.935 0.933
0.788 0.752 0.733 0,724 0.717 0.779 0.784 0.791 0.787 0,735
0.748 0,748 0.689 0.674 0.676
0.724 0.740 0.744 0,740 0.730
0.719 0.205 0.t45 0.748 0 730
0.683 0.702 0 711 0.720 0 728
0.561 0,617 0.638 0.640 0.662
0.727 0.737 0,734 0.740 0.736
0 701 0 703 0.709 0 716 0.726
0.691 0.695 0.705 0.711 0 720
0.590 0.618 0 630 0 651 0 655
0.419 0.458 0.500 0.497 0.532
0.674 0.703 0.707 0.718 0.727
0.610 0.633 0.642 0.655 0.670
0.563 0.587 0 607 0 617 0 630
0.496 0,533 0.558 0.578 0.594
0 348 0.393 0 424 0 453 0.479
0.134 0.169 0.198 0.224 0.246
0.637 0.658 0.678 0.689 0.698
0.548 0.596 0.607 0.632 0.634
0.448 0.516 0 539 0,564 0.585
0.388 0.434 0.466 0.494 0.513
0.226 0.271 0.302 0.332 0.359
0.080 0.104 0,121 0.139 0.148
C = actual discharge/theoretical discharge according t o u =, t'.'yh.
TABLE V. REYSOLDS NUMBERS" Liquid
Head over Center of Orifice
-
~
315 in.
5/16
R e for Orifice Diameter of.in. in. 3/16 in. '/i in.
1/16
ill.
Feel
some practice the opening and closing of the orifice with rubber stoppers can he well regulated. The total discharge is taken from the average of three or four measurements which agree among themselves within 0.1 to 0.2 per cent. Discussion of Results Assuming the flow outside the orifice to be the same as it nould be through a tube of the same diameter, only in the case of water is the critical velocity exceeded, as Tables I1 and I11 shoir . The jets of liquid leaving the orifices are like pencils of circular cross section except that the water jets undergo distortion in case of flow where the critical velocity is exceeded. Table IV s h o w that for liquid I (water), C decreases with increasing head for all orifices except the l/le-inch diameter orifice. This is to be expected if we accept the view that perfect contraction is not possible for low heads and for orifice diameters less than 21i2 inches; the degree of imperfection is more marked as the diameter decreases (3, 6 , 7 ) . Thus, other factors remaining constant, C for water gradually increases with decreasing orifice diameter; with l/lrinch orifice it approaches unity and is almost constant a t all heads. But in the case of other liquids more viscous than water, C decreases with decreasing diameter. I n the case of liquid I1 for '8- and ',Finch orifices, C increases with decreasing
30,180 36,680 41,930 46,680 51,060
23,260 28,230 32,360 36,030 39,650
18,640 22,560 25,900 28,940 31,650
14,280 17,320 19,800 22,010 24,040
11,160 13,3GO 13,110 16,770 18,180
2 3 4 5 6
383 448 504
303 371 395
235 294 341
175 209 257
111 139 163
45 61 73
557 604
432 474
379 410
287 318
188 204
81 03
Ill
2 3 4 5 6
193 238 277 308 315
150 186 214 241 263
116 142 165 187 208
85.5 105 123 139 154
48.7 62.5 73.5 84 9 93.6
23.1 29.2 32.4
IT'
2 3 4 5 6
48.6 62 1 72.2 82 0 90 9
36.7 46.7 54.6 62.3 69.9
27.1 34.6 41.3 46.9 52.5
17.9 23.6 58 3 33 0 37.2
8.4 11.6 14.4 17.2 20.0
1.6 2.5 8.4 4 3 3.1
v
2 3 4
26.: 33.i 40.1 45.6 b0.6
19.1 25.4 29.9 34.8 38.3
12 5 17.6 21.2 24.9 28.3
8.1 11 1 13 8 16 3 18.6
3.2 4.6 3 9
0.6
I
2
3
4
6
I1
J
6
$
6,730 8,260 9,360 10,650 ll,~i70
17.3
38.0
0.9 1 2 1 .j
1.8
Re = D u p / p , where D = diameter of orifice, it.; u = average linear velocity, ft./sec.; p = fluid densits, lb./cu. i t . ; p = absolute viscueity of fluid, lb./sec. f t . a
head just as it does for water, the increment? being greater a t lower heads. For liquid I11 with a 8-inch orifice, C increases with decreasing head down to a 4-foot head; helow this head C decreases. For all other orifices with liquids I1 and I11 and for all orifices with liquids IV and V, C' decreases with decreasing head.
290
INDUSTRIAL AND ENGINEERING CHEMISTRY
There is one peculiarity: The coefficients of discharge for all liquids with x-iscosities higher than that of water are greater above a certain orifice diameter and a certain head. For example, for liquids I1 and I11 with 3/8-3/16-inchorifices, for liquid IV with ”8- and 5/~6-inchorifices, and a t heads higher than 3 and 4 feet, respectively, for liquid V with a 3/’s-inch orifice and at heads above 5 feet, C is greater than the corresponding value of C for water. C for liquid I11 with 3,’s- and 5/lpin~horifices and a t heads above 3 and 4 feet , respectively, are even greater than corresponding values of C for liquid 11. C for liquid IV with a 3/a-inch orifice and at a 6-foot head is greater than the corresponding C values for liquid 11. This means that actual volumes flowing per unit time for the above mentioned orifices and heads increase with increasing L-iscosities. An empirical formula derived by Gourley and Crimp ( 5 ) from experiments on rectangular weirs with end contractions and with only one liquid (water) shows that a 100 per cent increase in viscosity, such as was obtained by altering the temperature of water from 90’ to 40° F., would increase the discharge by 0.6 per cent. I t seems that for every liquid with a certain viscosity and surface tension, a critical diameter and a critical head exist below which C will decrease with decreasing head and diameter and above which it will increase. It is probable that, just as in the case of water (Y),a certain critical diameter and a critical head exist for every liquid above which C is constant. It is possible that there is a relation between the critical velocity and the orifice diameter for any particular liquid, SO that below the point of critical velocity C will decrease with decreasing diameter and the reverse will be true above the
VOL. 32, KO. 2
diameter where critical velocity is exceeded; this is apparent in the case of water, where C is continually increasing with decreasing orifice diameter, whereas in the case of other liquids it is decreasing. To illustrate the probability of the existence of such a tendency, Figure 2 shows the coefficient of discharge plotted against Reynolds number. Extrapolation is avoided for want of further experimental data. Similar graphs are given by Walker, Lewis, McAdams, and Gilliland (8).
Acknowledgment We take pleasure in expressing our gratitude to the authorities and workers of the Governmenb Test House at Alipur for helping us in measur’ng the viscosities of the more viscous liquids, and to S. C. I\ar for discussions of the theoretical aspects of such flow.
Literature Cited Badger and MoCabe, “Elements of Chemical Engineering”, p. 67, New York, McGran-Hill Book Co., 1931. Badger and MoCabe, private communication. Bilton, H. J. I., Eng. News, July 9, 1908. Gibson, A. H., “Hydraulics and Its Applications”, p. 110, London, Constable & Co., 1930. Ibid., p. 119. Judd, Horace, and King, R. S., Eng. S e w s , Sept. 27, 1906. Smith, Demster, and Walker, W. J., PTOC. Insl. Mech. Eng. (London), 1923, 23. Walker, Lewis, McAdams, and Gilliland, “Princlples of Chemical Engineering”, 3rd ed., p. 61, Xew York, McGraw-Hill Book Co., 1937. Ibid., p. 76.
