v_o² = V_o² - 2V_o u_o cot [phi] + u_o² cosec² [phi];
.: H = {(2 - k²)V_o² - 2kV_o u_o cot [phi] - k²u_o² cosec² [phi]}/2g. (10)
Putting this in the expression for the efficiency, we find a
considerable increase of efficiency. Thus with
[phi] = 90° and k = ½, [eta] = 7/8 nearly,
[phi] a small angle and k = ½, [eta] = 1 nearly.
With this arrangement of pump, therefore, the angle at the outer ends
of the vanes is of comparatively little importance. A moderate angle
of 30° or 40° may very well be adopted. The following numerical values
of the velocity of the circumference of the pump have been obtained by
taking k = ½, and u_o = 0.25 [root](2gH).
[phi] V_o
90° .762 [root](2gH)
45° .842 "
30° .911 "
20° 1.023 "
The quantity of water to be pumped by a centrifugal pump necessarily
varies, and an adjustment for different quantities of water cannot
easily be introduced. Hence it is that the average efficiency of pumps
of this kind is in practice less than the efficiencies given above.
The advantage of a vortex chamber is also generally neglected. The
velocity in the supply and discharge pipes is also often made greater
than is consistent with a high degree of efficiency. Velocities of 6
or 7 ft. per second in the discharge and suction pipes, when the lift
is small, cause a very sensible waste of energy; 3 to 6 ft. would be
much better. Centrifugal pumps of very large size have been
constructed. Easton and Anderson made pumps for the North Sea canal in
Holland to deliver each 670 tons of water per minute on a lift of 5
ft. The pump disks are 8 ft. diameter. J. and H. Gwynne constructed
some pumps for draining the Ferrarese Marshes, which together deliver
2000 tons per minute. A pump made under Professor J. Thomson's
direction for drainage works in Barbados had a pump disk 16 ft. in
diameter and a whirlpool chamber 32 ft. in diameter. The efficiency of
centrifugal pumps when delivering less or more than the normal
quantity of water is discussed in a paper in the _Proc. Inst. Civ.
Eng._ vol. 53.
§ 211. _High Lift Centrifugal Pumps._--It has long been known that
centrifugal pumps could be worked in series, each pump overcoming a part
of the lift. This method has been perfected, and centrifugal pumps for
very high lifts with great efficiency have been used by Sulzer and
others. C. W. Darley (_Proc. Inst. Civ. Eng._, supplement to vol. 154,
p. 156) has described some pumps of this new type driven by Parsons
steam turbines for the water supply of Sydney, N.S.W. Each pump was
designed to deliver 1½ million gallons per twenty-four hours against a
head of 240 ft. at 3300 revs. per minute. Three pumps in series give
therefore a lift of 720 ft. The pump consists of a central double-sided
impeller 12 in. diameter. The water entering at the bottom divides and
enters the runner at each side through a bell-mouthed passage. The shaft
is provided with ring and groove glands which on the suction side keep
the air out and on the pressure side prevent leakage. Some water from
the pressure side leaks through the glands, but beyond the first grooves
it passes into a pocket and is returned to the suction side of the pump.
For the glands on the suction side water is supplied from a low-pressure
service. No packing is used in the glands. During the trials no water
was seen at the glands. The following are the results of tests made at
Newcastle:--
+-------------------------------------+-------+-------+-------+-------+
| | I. | II. | III. | IV. |
+-------------------------------------+-------+-------+-------+-------+
| Duration of test hours | 2 | 1.54 | 1.2 | 1.55 |
| Steam pressure lb. per sq. in. | 57 | 57 | 84 | 55 |
| Weight of steam per water | | | | |
| h.p. hour lb. | 27.93 | 30.67 | 28.83 | 27.89 |
| Speed in revs, per min. | 3300 | 3330 | 3710 | 3340 |
| Height of suction ft. | 11 | 11 | 11 | 11 |
| Total lift ft. | 762 | 744 | 917 | 756 |
| Million galls. per day pumped-- | | | | |
| By Ventun meter | 1.573 | 1.499 | 1.689 | 1.503 |
| By orifice | 1.623 | 1.513 | 1.723 | 1.555 |
| Water h.p. | 252 | 235 | 326 | 239 |
+-------------------------------------+-------+-------+-------+-------+
In trial IV. the steam was superheated 95° F. From other trials under
the same conditions as trial I. the Parsons turbine uses 15.6 lb. of
steam per brake h.p. hour, so that the combined efficiency of turbine
and pumps is about 56%, a remarkably good result.
[Illustration: FIG. 212.]
§ 212. _Air-Lift Pumps._--An interesting and simple method of pumping by
compressed air, invented by Dr J. Pohlé of Arizona, is likely to be very
useful in certain cases. Suppose a rising main placed in a deep bore
hole in which there is a considerable depth of water. Air compressed to
a sufficient pressure is conveyed by an air pipe and introduced at the
lower end of the rising main. The air rising In the main diminishes the
average density of the contents of the main, and their aggregate weight
no longer balances the pressure at the lower end of the main due to its
submersion. An upward flow is set up, and if the air supply is
sufficient the water in the rising main is lifted to any required
height. The higher the lift above the level in the bore hole the deeper
must be the point at which air is injected. Fig. 212 shows an airlift
pump constructed for W. H. Maxwell at the Tunbridge Wells waterworks.
There is a two-stage steam air compressor, compressing air to from 90 to
100 lb. per sq. in. The bore hole is 350 ft. deep, lined with steel
pipes 15 in. diameter for 200 ft. and with perforated pipes 13½ in.
diameter for the lower 150 ft. The rest level of the water is 96 ft.
from the ground-level, and the level when pumping 32,000 gallons per
hour is 120 ft. from the ground-level. The rising main is 7 in.
diameter, and is carried nearly to the bottom of the bore hole and to 20
ft. above the ground-level. The air pipe is 2½ in. diameter. In a trial
run 31,402 gallons per hour were raised 133 ft. above the level in the
well. Trials of the efficiency of the system made at San Francisco with
varying conditions will be found in a paper by E. A. Rix (_Journ. Amer.
Assoc. Eng. Soc._ vol. 25, 1900). Maxwell found the best results
when the ratio of immersion to lift was 3 to 1 at the start and 2.2 to 1
at the end of the trial. In these conditions the efficiency was 37%
calculated on the indicated h.p. of the steam-engine, and 46% calculated
on the indicated work of the compressor. 2.7 volumes of free air were
used to 1 of water lifted. The system is suitable for temporary
purposes, especially as the quantity of water raised is much greater
than could be pumped by any other system in a bore hole of a given size.
It is useful for clearing a boring of sand and may be advantageously
used permanently when a boring is in sand or gravel which cannot be kept
out of the bore hole. The initial cost is small.
§ 213. _Centrifugal Fans._--Centrifugal fans are constructed similarly
to centrifugal pumps, and are used for compressing air to pressures not
exceeding 10 to 15 in. of water-column. With this small variation of
pressure the variation of volume and density of the air may be neglected
without sensible error. The conditions of pressure and discharge for
fans are generally less accurately known than in the case of pumps, and
the design of fans is generally somewhat crude. They seldom have
whirlpool chambers, though a large expanding outlet is provided in the
case of the important Guibal fans used in mine ventilation.
It is usual to reckon the difference of pressure at the inlet and
outlet of a fan in inches of water-column. One inch of water-column =
64.4 ft. of air at average atmospheric pressure = 5.2lb. per sq. ft.
Roughly the pressure-head produced in a fan without means of utilizing
the kinetic energy of discharge would be v²/2g ft. of air, or 0.00024
v² in. of water, where v is the velocity of the tips of the fan blades
in feet per second. If d is the diameter of the fan and t the width at
the external circumference, then [pi]dt is the discharge area of the
fan disk. If Q is the discharge in cub. ft. per sec., u = Q/[pi]dt is
the radial velocity of discharge which is numerically equal to the
discharge per square foot of outlet in cubic feet per second. As both
the losses in the fan and the work done are roughly proportional to u²
in fans of the same type, and are also proportional to the gauge
pressure p, then if the losses are to be a constant percentage of the
work done u may be taken proportional to [root]p. In ordinary cases u
= about 22[root]p. The width t of the fan is generally from 0.35 to
0.45d. Hence if Q is given, the diameter of the fan should be:--
For t = 0.35d, d = 0.20 [root](Q/[root]p)
For t = 0.45d, d = 0.18 [root](Q/[root]p)
If p is the pressure difference in the fan in inches of water, and N
the revolutions of fan,
v = [pi]dN/60 ft. per sec.
N = 1230 [root]p/d revs. per min.
As the pressure difference is small, the work done in compressing the
air is almost exactly 5.2pQ foot-pounds per second. Usually, however,
the kinetic energy of the air in the discharge pipe is not
inconsiderable compared with the work done in compression. If w is the
velocity of the air where the discharge pressure is measured, the air
carries away w²/2g foot-pounds per lb. of air as kinetic energy. In Q
cubic feet or 0.0807 Qlb. the kinetic energy is 0.00125 Qw²
foot-pounds per second.
The efficiency of fans is reckoned in two ways. If B.H.P. is the
effective horse-power applied at the fan shaft, then the efficiency
reckoned on the work of compression is
[eta] = 5.2 pQ/550 B.H.P.
On the other hand, if the kinetic energy in the delivery pipe is taken
as part of the useful work the efficiency is
[eta]2 = (5.2 pQ + 0.00125 Qw²)/550 B.H.P.
Although the theory above is a rough one it agrees sufficiently with
experiment, with some merely numerical modifications.
An extremely interesting experimental investigation of the action of
centrifugal fans has been made by H. Heenan and W. Gilbert (_Proc.
Inst. Civ. Eng._ vol. 123, p. 272). The fans delivered through an air
trunk in which different resistances could be obtained by introducing
diaphragms with circular apertures of different sizes. Suppose a fan
run at constant speed with different resistances and the compression
pressure, discharge and brake horse-power measured. The results plot
in such a diagram as is shown in fig. 213. The less the resistance to
discharge, that is the larger the opening in the air trunk, the
greater the quantity of air discharged at the given speed of the fan.
On the other hand the compression pressure diminishes. The curve
marked total gauge is the compression pressure + the velocity head in
the discharge pipe, both in inches of water. This curve falls, but not
nearly so much as the compression curve, when the resistance in the
air trunk is diminished. The brake horse-power increases as the
resistance is diminished because the volume of discharge increases
very much. The curve marked efficiency is the efficiency calculated
on the work of compression only. It is zero for no discharge, and zero
also when there is no resistance and all the energy given to the air
is carried away as kinetic energy. There is a discharge for which this
efficiency is a maximum; it is about half the discharge which there is
when there is no resistance and the delivery pipe is full open. The
conditions of speed and discharge corresponding to the greatest
efficiency of compression are those ordinarily taken as the best
normal conditions of working. The curve marked total efficiency gives
the efficiency calculated on the work of compression and kinetic
energy of discharge. Messrs Gilbert and Heenan found the efficiencies
of ordinary fans calculated on the compression to be 40 to 60% when
working at about normal conditions.
[Illustration: FIG. 213.]
Taking some of Messrs Heenan and Gilbert's results for ordinary fans
in normal conditions, they have been found to agree fairly with the
following approximate rules. Let p_c be the compression pressure and q
the volume discharged per second per square foot of outlet area of
fan. Then the total gauge pressure due to pressure of compression and
velocity of discharge is approximately: p = p_c + 0.0004 q² in. of
water, so that if p_c is given, p can be found approximately. The
pressure p depends on the circumferential speed v of the fan disk--
p = 0.00025 v² in. of water
v = 63 [root]p ft. per sec.