where V is the velocity at the centre and v the velocity at radius r
in a pipe of radius R with a hydraulic gradient i. Later Bazin
repeated the experiments and extended them (_Mém. de l'Académie des
Sciences_, xxxii. No. 6). The most important result was the ratio of
mean to central velocity. Let b = Ri/U², where U is the mean velocity
in the pipe; then V/U = 1 + 9.03 [root]b. A very useful result for
practical purposes is that at 0.74 of the radius of the pipe the
velocity is equal to the mean velocity. Fig. 84 gives the velocities
at different radii as determined by Bazin.
[Illustration: FIG. 84.]
§ 79. _Influence of Temperature on the Flow through Pipes._--Very
careful experiments on the flow through a pipe 0.1236 ft. in diameter
and 25 ft. long, with water at different temperatures, have been made
by J. G. Mair (_Proc. Inst. Civ. Eng._ lxxxiv.). The loss of head was
measured from a point 1 ft. from the inlet, so that the loss at entry
was eliminated. The 1½ in. pipe was made smooth inside and to gauge,
by drawing a mandril through it. Plotting the results logarithmically,
it was found that the resistance for all temperatures varied very
exactly as v^(1.795), the index being less than 2 as in other
experiments with very smooth surfaces. Taking the ordinary equation of
flow h = [zeta](4L/D)(v²/2g), then for heads varying from 1 ft. to
nearly 4 ft., and velocities in the pipe varying from 4 ft. to 9 ft.
per second, the values of [zeta] were as follows:--
Temp. F. [zeta] | Temp. F. [zeta]
57 .0044 to .0052 | 100 .0039 to .0042
70 .0042 to .0045 | 110 .0037 to .0041
80 .0041 to .0045 | 120 .0037 to .0041
90 .0040 to .0045 | 130 .0035 to .0039
| 160 .0035 to .0038
This shows a marked decrease of resistance as the temperature rises.
If Professor Osborne Reynolds's equation is assumed h = mLV^n/d^(3 -
n), and n is taken 1.795, then values of m at each temperature are
practically constant--
Temp. F. m. | Temp. F. m.
57 0.000276 | 100 0.000244
70 0.000263 | 110 0.000235
80 0.000257 | 120 0.000229
90 0.000250 | 130 0.000225
| 160 0.000206
where again a regular decrease of the coefficient occurs as the
temperature rises. In experiments on the friction of disks at
different temperatures Professor W. C. Unwin found that the resistance
was proportional to constant × (1 - 0.0021t) and the values of m given
above are expressed almost exactly by the relation
m = 0.000311(1 - 0.00215 t).
In tank experiments on ship models for small ordinary variations of
temperature, it is usual to allow a decrease of 3% of resistance for
10° F. increase of temperature.
§ 80. _Influence of Deposits in Pipes on the Discharge. Scraping Water
Mains._--The influence of the condition of the surface of a pipe on
the friction is shown by various facts known to the engineers of
waterworks. In pipes which convey certain kinds of water, oxidation
proceeds rapidly and the discharge is considerably diminished. A main
laid at Torquay in 1858, 14 m. in length, consists of 10-in., 9-in.
and 8-in. pipes. It was not protected from corrosion by any coating.
But it was found to the surprise of the engineer that in eight years
the discharge had diminished to 51% of the original discharge. J. G.
Appold suggested an apparatus for scraping the interior of the pipe,
and this was constructed and used under the direction of William
Froude (see "Incrustation of Iron Pipes," by W. Ingham, _Proc. Inst.
Mech. Eng._, 1899). It was found that by scraping the interior of the
pipe the discharge was increased 56%. The scraping requires to be
repeated at intervals. After each scraping the discharge diminishes
rather rapidly to 10% and afterwards more slowly, the diminution in a
year being about 25%.
Fig. 85 shows a scraper for water mains, similar to Appold's but
modified in details, as constructed by the Glenfield Company, at
Kilmarnock. A is a longitudinal section of the pipe, showing the
scraper in place; B is an end view of the plungers, and C, D sections
of the boxes placed at intervals on the main for introducing or
withdrawing the scraper. The apparatus consists of two plungers,
packed with leather so as to fit the main pretty closely. On the
spindle of these plungers are fixed eight steel scraping blades, with
curved scraping edges fitting the surface of the main. The apparatus
is placed in the main by removing the cover from one of the boxes
shown at C, D. The cover is then replaced, water pressure is admitted
behind the plungers, and the apparatus driven through the main. At
Lancaster after twice scraping the discharge was increased 56½%, at
Oswestry 54½%. The increased discharge is due to the diminution of the
friction of the pipe by removing the roughnesses due to oxidation. The
scraper can be easily followed when the mains are about 3 ft. deep by
the noise it makes. The average speed of the scraper at Torquay is
2(1/3) m. per hour. At Torquay 49% of the deposit is iron rust, the
rest being silica, lime and organic matter.
[Illustration: FIG. 85. Scale 1/25.]
In the opinion of some engineers it is inadvisable to use the scraper.
The incrustation is only temporarily removed, and if the use of the
scraper is continued the life of the pipe is reduced. The only
treatment effective in preventing or retarding the incrustation due to
corrosion is to coat the pipes when hot with a smooth and perfect
layer of pitch. With certain waters such as those derived from the
chalk the incrustation is of a different character, consisting of
nearly pure calcium carbonate. A deposit of another character which
has led to trouble in some mains is a black slime containing a good
deal of iron not derived from the pipes. It appears to be an organic
growth. Filtration of the water appears to prevent the growth of the
slime, and its temporary removal may be effected by a kind of brush
scraper devised by G. F. Deacon (see "Deposits in Pipes," by Professor
J. C. Campbell Brown, _Proc. Inst. Civ. Eng._, 1903-1904).
§ 81. _Flow of Water through Fire Hose._--The hose pipes used for fire
purposes are of very varied character, and the roughness of the
surface varies. Very careful experiments have been made by J. R.
Freeman (_Am. Soc. Civ. Eng._ xxi., 1889). It was noted that under
pressure the diameter of the hose increased sufficiently to have a
marked influence on the discharge. In reducing the results the true
diameter has been taken. Let v = mean velocity in ft. per sec.; r =
hydraulic mean radius or one-fourth the diameter in feet; i =
hydraulic gradient. Then v = n[root](ri).
+---------------+---------+---------+-------+-------+-------+
| | Diameter| Gallons | | | |
| | in | (United | | | |
| | Inches. | States) | i | v | n |
| | | per min.| | | |
+---------------+---------+---------+-------+-------+-------+
| Solid rubber | 2.65 | 215 | .1863 | 12.50 | 123.3 |
| hose | " | 344 | .4714 | 20.00 | 124.0 |
| | | | | | |
| Woven cotton, | 2.47 | 200 | .2464 | 13.40 | 119.1 |
| rubber lined | " | 299 | .5269 | 20.00 | 121.5 |
| | | | | | |
| Woven cotton, | 2.49 | 200 | .2427 | 13.20 | 117.7 |
| rubber lined | " | 319 | .5708 | 21.00 | 122.1 |
| | | | | | |
| Knit cotton, | 2.68 | 132 | .0809 | 7.50 | 111.6 |
| rubber lined | " | 299 | .3931 | 17.00 | 114.8 |
| | | | | | |
| Knit cotton, | 2.69 | 204 | .2357 | 11.50 | 100.1 |
| rubber lined | " | 319 | .5165 | 18.00 | 105.8 |
| | | | | | |
| Woven cotton, | 2.12 | 154 | .3448 | 14.00 | 113.4 |
| rubber lined | " | 240 | .7673 | 21.81 | 118.4 |
| | | | | | |
| Woven cotton, | 2.53 | 54.8 | .0261 | 3.50 | 94.3 |
| rubber lined | " | 298 | .8264 | 19.00 | 91.0 |
| | | | | | |
| Unlined linen | 2.60 | 57.9 | .0414 | 3.50 | 73.9 |
| hose | " | 331 |1.1624 | 20.00 | 79.6 |
+---------------+---------+---------+-------+-------+-------+
§ 82. _Reduction of a Long Pipe of Varying Diameter to an Equivalent
Pipe of Uniform Diameter. Dupuit's Equation._--Water mains for the
supply of towns often consist of a series of lengths, the diameter
being the same for each length, but differing from length to length.
In approximate calculations of the head lost in such mains, it is
generally accurate enough to neglect the smaller losses of head and to
have regard to the pipe friction only, and then the calculations may
be facilitated by reducing the main to a main of uniform diameter, in
which there would be the same loss of head. Such a uniform main will
be termed an equivalent main.
[Illustration: FIG. 86.]
In fig. 86 let A be the main of variable diameter, and B the
equivalent uniform main. In the given main of variable diameter A, let
l1, l2... be the lengths,
d1, d2... the diameters,
v1, v2... the velocities,
i1, i2... the slopes,
for the successive portions, and let l, d, v and i be corresponding
quantities for the equivalent uniform main B. The total loss of head
in A due to friction is
h = i1l1 + i2l2 + ...
= [zeta](v1²·4l1/2gd1) + [zeta](v2²·4l2/2gd2) + ...
and in the uniform main
il = [zeta](v²·4l/2gd).
If the mains are equivalent, as defined above,
[zeta](v²·4l/2gd) = [zeta](v1²·4l1/2gd1) + [zeta](v2²·4l2/2gd2) + ...
But, since the discharge is the same for all portions,
¼[pi]d²v = ¼[pi]d1²v1 = ¼[pi]d2²v2 = ...
v1 = vd²/d1²; v2 = vd²/d2² ...
Also suppose that [zeta] may be treated as constant for all the pipes.
Then
l/d = (d^4/d1^4)(l1/d1) + (d^4/d2^4(12/d2) + ...
l = (d^5/d1^5)l1 + (d^5/d2^5)l2 + ...
which gives the length of the equivalent uniform main which would have
the same total loss of head for any given discharge.
§ 83. _Other Losses of Head in Pipes._--Most of the losses of head in
pipes, other than that due to surface friction against the pipe, are
due to abrupt changes in the velocity of the stream producing eddies.
The kinetic energy of these is deducted from the general energy of
translation, and practically wasted.
[Illustration: FIG. 87.]
_Sudden Enlargement of Section._--Suppose a pipe enlarges in section
from an area [omega]0 to an area [omega]1 (fig. 87); then
v1/v0 = [omega]0/[omega]1;
or, if the section is circular,
v1/v0 = (d0/d1)².
The head lost at the abrupt change of velocity has already been shown
to be the head due to the relative velocity of the two parts of the
stream. Hence head lost
[h]_e = (v0 - v1)²/2g = ([omega]1/[omega]0 - 1)²v1²/2g
= {(d1/d0)² - 1}² v1²/2g
or
[h]_e = [zeta]_ev1²/2g, (1)
if [zeta]_e is put for the expression in brackets.
+--------------+----+----+----+----+----+----+----+----+----+----+----+-----+-----+-----+-----+
| [omega]1/ |1.1 |1.2 |1.5 |1.7 |1.8 |1.9 |2.0 |2.5 |3.0 |3.5 |4.0 | 5.0 | 6.0 | 7.0 | 8.0 |
| [omega]0 = | | | | | | | | | | | | | | | |
| d1/d0 = |1.05|1.10|1.22|1.30|1.34|1.38|1.41|1.58|1.73|1.87|2.00| 2.24| 2.45| 2.65| 2.83|
| | | | | | | | | | | | | | | | |
| [zeta]_e = | .01| .04| .25| .49| .64| .81|1.00|2.25|4.00|6.25|9.00|16.00|25.00|36.0 |49.0 |
+--------------+----+----+----+----+----+----+----+----+----+----+----+-----+-----+-----+-----+
[Illustration: FIG. 88.]
[Illustration: FIG. 89.]
_Abrupt Contraction of Section._--When water passes from a larger to a
smaller section, as in figs. 88, 89, a contraction is formed, and the
contracted stream abruptly expands to fill the section of the pipe.
Let [omega] be the section and v the velocity of the stream at bb. At
aa the section will be c_c[omega], and the velocity
([omega]/c_c[omega])v = v/c1, where c_c is the coefficient of
contraction. Then the head lost is
[h]_m = (v/c_c - v)²/2g = (1/c_c - 1)²v²/2g;
and, if c_c is taken 0.64,
[h]_m = 0.316 v²/2g. (2)
The value of the coefficient of contraction for this case is, however,
not well ascertained, and the result is somewhat modified by friction.
For water entering a cylindrical, not bell-mouthed, pipe from a
reservoir of indefinitely large size, experiment gives
[h]_a = 0.505 v²/2g. (3)
If there is a diaphragm at the mouth of the pipe as in fig. 89, let
[omega]1 be the area of this orifice. Then the area of the contracted
stream is c_c[omega]1, and the head lost is
[h]_c = {([omega]/c_c[omega]1) - 1}²v²/2g
= [zeta]_cv²/2g (4)
if [zeta], is put for {([omega]/c_c[omega]1) - 1}². Weisbach has found
experimentally the following values of the coefficient, when the
stream approaching the orifice was considerably larger than the
orifice:--
+--------------------+-------+------+------+-----+-----+-----+-----+-----+-----+-----+
| [omega]1/[omega] = | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 |0.7 | 0.8 | 0.9 | 1.0 |
| | | | | | | | | | | |
| c_c = | .616 | .614 | .612 |.610 |.617 |.605 |.603 |.601 |.598 |.596 |
| | | | | | | | | | | |
| [zeta]_c = | 231.7 |50.99 |19.78 |9.612|5.256|3.077|1.876|1.169|0.734|0.480|
+--------------------+-------+------+------+-----+-----+-----+-----+-----+-----+-----+
[Illustration: FIG. 90.]
When a diaphragm was placed in a tube of uniform section (fig. 90) the
following values were obtained, [omega]1 being the area of the orifice
and [omega] that of the pipe:--
+--------------------+-------+------+------+-----+-----+-----+-----+-----+-----+-----+
| [omega]1/[omega] = | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |
| | | | | | | | | | | |
| c_c = | .624 | .632 | .643 |.659 |.681 |.712 |.755 |.813 |.892 |1.00 |
| | | | | | | | | | | |
| [xi]_c = | 225.9 |47.77 |30.83 |7.801|1.753|1.796|.797 |.290 |.060 |.000 |
+--------------------+-------+------+------+-----+-----+-----+-----+-----+-----+-----+
Elbows.--Weisbach considers the loss of head at elbows (fig. 91) to be
due to a contraction formed by the stream. From experiments with a
pipe 1¼ in. diameter, he found the loss of head
[h]_e = [zeta]_e v²/2g; (5)
[zeta]_e = 0.9457 sin² ½[phi] + 2.047 sin^4 ½[phi].
+------------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
| [phi] = | 20° | 40° | 60° | 80° | 90° | 100°| 110°| 120°| 130°| 140°|
| | | | | | | | | | | |
| [zeta]_e = |0.046|0.139|0.364|0.740|0.984|1.260|1.556|1.861|2.158|2.431|
+------------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
Hence at a right-angled elbow the whole head due to the velocity very
nearly is lost.
[Illustration: FIG. 91.]
_Bends._--Weisbach traces the loss of head at curved bends to a
similar cause to that at elbows, but the coefficients for bends are
not very satisfactorily ascertained. Weisbach obtained for the loss of
head at a bend in a pipe of circular section
[h]_b = [zeta]_b v²/2g; (6)
[zeta]_b = 0.131 + 1.847(d/2[rho])^(7/2),
where d is the diameter of the pipe and [rho] the radius of curvature
of the bend. The resistance at bends is small and at present very ill
determined.
[Illustration: FIG. 92.]
_Valves, Cocks and Sluices._--These produce a contraction of the
water-stream, similar to that for an abrupt diminution of section
already discussed. The loss of head may be taken as before to be
[h]_v = [zeta]_v v²/2g; (7)
where v is the velocity in the pipe beyond the valve and [zeta]_v a
coefficient determined by experiment. The following are Weisbach's
results.
_Sluice in Pipe of Rectangular Section_ (fig. 92). Section at sluice =
[omega]1 in pipe = [omega].
+--------------------+----+----+----+----+----+----+----+----+----+-----+
| [omega]1/[omega] = |1.0 |0.9 |0.8 |0.7 |0.6 |0.5 |0.4 | 0.3| 0.2| 0.1 |
| | | | | | | | | | | |
| [zeta]_v = |0.00|.09 |.39 |.95 |2.08|4.02|8.12|17.8|44.5| 193 |
+--------------------+----+----+----+----+----+----+----+----+----+-----+
_Sluice in Cylindrical Pipe_ (fig. 93).
+-----------------------+----+-----+-----+-----+-----+-----+------+------+
| Ratio of height of | | | | | | | | |
| opening to diameter | 1.0| 7/8 | 3/4 | 5/8 | ½ | 3/8 | ¼ | 1/5 |
| of pipe | | | | | | | | |
| [omega]1/[omega] = |1.00|0.948|.856 |.740 |.609 |.466 | .315 | .159 |
| | | | | | | | | |
| [zeta]_v = |0.00|0.07 |0.26 |0.81 |2.06 |5.52 | 17.0 | 97.8 |
+-----------------------+----+-----+-----+-----+-----+-----+------+------+
[Illustration: FIG. 93.]
[Illustration: FIG. 94.]
_Cock in a Cylindrical Pipe_ (fig. 94). Angle through which cock is
turned = [theta].
+------------+-----+-----+-----+-----+-----+-----+-----+
| [theta] = | 5° | 10° | 15° | 20° | 25° | 30° | 35° |
| Ratio of | | | | | | | |
| cross |.926 |.850 |.772 |.692 |.613 |.535 |.458 |
| sections | | | | | | | |
| [zeta]_v = | .05 | .29 | .75 |1.56 |3.10 |5.47 |9.68 |
+------------+-----+-----+-----+-----+-----+-----+-----+
+------------+-----+-----+-----+-----+-----+-----+-----+
| [theta] = | 40° | 45° | 50° | 55° | 60° | 65° | 82° |
| Ratio of | | | | | | | |
| cross |.385 |.315 |.250 |.190 |.137 |.091 | 0 |
| sections | | | | | | | |
| [zeta]_v = | 17.3| 31.2| 52.6|106 |206 |486 |[oo] |
+------------+-----+-----+-----+-----+-----+-----+-----+
_Throttle Valve in a Cylindrical Pip_e (fig. 95)
+------------+-----+-----+-----+-----+-----+-----+-----+-----+
| [theta] = | 5° | 10° | 15° | 20° | 25° | 30° | 35° | 40° |
| | | | | | | | | |
| [zeta]_v = | .24 | .52 | .90 | 1.54| 2.51| 3.91| 6.22| 10.8|
+------------+-----+-----+-----+-----+-----+-----+-----+-----+
+------------+------+------+------+------+------+------+------+
| [theta] = | 45° | 50° | 55° | 60° | 65° | 70° | 90° |
| | | | | | | | |
| [zeta]_v = | 18.7 | 32.6 | 58.8 | 118 | 256 | 751 | [oo] |
+------------+------+------+------+------+------+------+------+
[Illustration: FIG. 95.]
§ 84. _Practical Calculations on the Flow of Water in Pipes._--In the
following explanations it will be assumed that the pipe is of so great
a length that only the loss of head in friction against the surface of
the pipe needs to be considered. In general it is one of the four
quantities d, i, v or Q which requires to be determined. For since the
loss of head h is given by the relation h = il, this need not be
separately considered.
There are then three equations (see eq. 4, § 72, and 9a, § 76) for the
solution of such problems as arise:--
[zeta] = [alpha](1 + 1/12d); (1)
where [alpha] = 0.005 for new and = 0.01 for incrusted pipes.
[zeta]v²/2g = ¼di. (2)