[eta]GQH = (GQ/g)(w1r1 - w2r2)a;
[eta] = (w1r1 - w2r2)a/gH. (8)
This is the fundamental equation in the theory of turbines. In
general,[7] w1 and w2, the tangential components of the water's motion
on entering and leaving the wheel, are completely independent. That
the efficiency may be as great as possible, it is obviously necessary
that w2 = 0. In that case
[eta] = w1r1a/gH. (9)
ar1 is the circumferential velocity of the wheel at the inlet surface.
Calling this V1, the equation becomes
[eta] = w1V1/gH. (9a)
This remarkably simple equation is the fundamental equation in the
theory of turbines. It was first given by Reiche (_Turbinenbaues_,
1877).
[Illustration: FIG. 185.]
[Illustration: FIG. 186.]
[Illustration: FIG. 187.]
[Illustration: FIG. 188.]
[Illustration: FIG. 189.]
§ 187. _General Description of a Reaction Turbine._--Professor James
Thomson's inward flow or vortex turbine has been selected as the type of
reaction turbines. It is one of the best in normal conditions of
working, and the mode of regulation introduced is decidedly superior to
that in most reaction turbines. Figs. 185 and 186 are external views of
the turbine case; figs. 187 and 188 are the corresponding sections; fig.
189 is the turbine wheel. The example chosen for illustration has
suction pipes, which permit the turbine to be placed above the
tail-water level. The water enters the turbine by cast-iron supply pipes
at A, and is discharged through two suction pipes S, S. The water
on entering the case distributes itself through a rectangular supply
chamber SC, from which it finds its way equally to the four guide-blade
passages G, G, G, G. In these passages it acquires a velocity about
equal to that due to half the fall, and is directed into the wheel at an
angle of about 10° or 12° with the tangent to its circumference. The
wheel W receives the water in equal proportions from each guide-blade
passage. It consists of a centre plate p (fig. 189) keyed on the shaft
aa, which passes through stuffing boxes on the suction pipes. On each
side of the centre plate are the curved wheel vanes, on which the
pressure of the water acts, and the vanes are bounded on each side by
dished or conical cover plates c, c. Joint-rings j, j on the cover
plates make a sufficiently water-tight joint with the casing, to
prevent leakage from the guide-blade chamber into the suction pipes. The
pressure near the joint rings is not very great, probably not one-fourth
the total head. The wheel vanes receive the water without shock, and
deliver it into central spaces, from which it flows on either side to
the suction pipes. The mode of regulating the power of the turbine is
very simple. The guide-blades are pivoted to the case at their inner
ends, and they are connected by a link-work, so that they all open and
close simultaneously and equally. In this way the area of opening
through the guide-blades is altered without materially altering the
angle or the other conditions of the delivery into the wheel. The
guide-blade gear may be variously arranged. In this example four
spindles, passing through the case, are linked to the guide-blades
inside the case, and connected together by the links l, l, l on the
outside of the case. A worm wheel on one of the spindles is rotated by a
worm d, the motion being thus slow enough to adjust the guide-blades
very exactly. These turbines are made by Messrs Gilkes & Co. of Kendal.
[Illustration: FIG. 190.]
Fig. 190 shows another arrangement of a similar turbine, with some
adjuncts not shown in the other drawings. In this case the turbine
rotates horizontally, and the turbine case is placed entirely below
the tail water. The water is supplied to the turbine by a vertical
pipe, over which is a wooden pentrough, containing a strainer, which
prevents sticks and other solid bodies getting into the turbine. The
turbine rests on three foundation stones, and, the pivot for the
vertical shaft being under water, there is a screw and lever
arrangement for adjusting it as it wears. The vertical shaft gives
motion to the machinery driven by a pair of bevel wheels. On the right
are the worm and wheel for working the guide-blade gear.
[Illustration: FIG. 191.]
§ 188. _Hydraulic Power at Niagara._--The largest development of
hydraulic power is that at Niagara. The Niagara Falls Power Company
have constructed two power houses on the United States side, the first
with 10 turbines of 5000 h.p. each, and the second with 10 turbines of
5500 h.p. The effective fall is 136 to 140 ft. In the first power
house the turbines are twin outward flow reaction turbines with
vertical shafts running at 250 revs. per minute and driving the
dynamos direct. In the second power house the turbines are inward flow
turbines with draft tubes or suction pipes. Fig. 191 shows a section
of one of these turbines. There is a balancing piston keyed on the
shaft, to the under side of which the pressure due to the fall is
admitted, so that the weight of turbine, vertical shaft and part of
the dynamo is water borne. About 70,000 h.p. is daily distributed
electrically from these two power houses. The Canadian Niagara Power
Company are erecting a power house to contain eleven units of 10,250
h.p. each, the turbines being twin inward flow reaction turbines. The
Electrical Development Company of Ontario are erecting a power house
to contain 11 units of 12,500 h.p. each. The Ontario Power Company are
carrying out another scheme for developing 200,000 h.p. by twin inward
flow turbines of 12,000 h.p. each. Lastly the Niagara Falls Power and
Manufacturing Company on the United States side have a station giving
35,000 h.p. and are constructing another to furnish 100,000 h.p. The
mean flow of the Niagara river is about 222,000 cub. ft. per second
with a fall of 160 ft. The works in progress if completed will utilize
650,000 h.p. and require 48,000 cub. ft. per second or 21½% of the
mean flow of the river (Unwin, "The Niagara Falls Power Stations,"
_Proc. Inst. Mech. Eng._, 1906).
[Illustration: FIG. 192.]
§ 189. _Different Forms of Turbine Wheel._--The wheel of a turbine or
part of the machine on which the water acts is an annular space,
furnished with curved vanes dividing it into passages exactly or
roughly rectangular in cross section. For radial flow turbines the
wheel may have the form A or B, fig. 192, A being most usual with
inward, and B with outward flow turbines. In A the wheel vanes are
fixed on each side of a centre plate keyed on the turbine shaft. The
vanes are limited by slightly-coned annular cover plates. In B the
vanes are fixed on one side of a disk, keyed on the shaft, and limited
by a cover plate parallel to the disk. Parallel flow or axial flow
turbines have the wheel as in C. The vanes are limited by two
concentric cylinders.
_Theory of Reaction Turbines._
[Illustration: FIG. 193.]
§ 190. _Velocity of Whirl and Velocity of Flow._--Let acb (fig. 193)
be the path of the particles of water in a turbine wheel. That path
will be in a plane normal to the axis of rotation in radial flow
turbines, and on a cylindrical surface in axial flow turbines. At any
point c of the path the water will have some velocity v, in the
direction of a tangent to the path. That velocity may be resolved into
two components, a whirling velocity w in the direction of the wheel's
rotation at the point c, and a component u at right angles to this,
radial in radial flow, and parallel to the axis in axial flow
turbines. This second component is termed the velocity of flow. Let
v_o, w_o, u_o be the velocity of the water, the whirling velocity and
velocity of flow at the outlet surface of the wheel, and v_i, w_i, u_i
the same quantities at the inlet surface of the wheel. Let [alpha] and
[beta] be the angles which the water's direction of motion makes with
the direction of motion of the wheel at those surfaces. Then
w_o = v_o cos [beta]; u_o = v_o sin [beta]
w_i = v_i cos [alpha]; u_i = v_i sin [alpha]. (10)
The velocities of flow are easily ascertained independently from the
dimensions of the wheel. The velocities of flow at the inlet and
outlet surfaces of the wheel are normal to those surfaces. Let
[Omega]_o, [Omega]_i be the areas of the outlet and inlet surfaces of
the wheel, and Q the volume of water passing through the wheel per
second; then
v_o = Q/[Omega]_o; v_i = Q/[Omega]_i. (11)
Using the notation in fig. 191, we have, for an inward flow turbine
(neglecting the space occupied by the vanes),
[Omega]_o = 2[pi]r0d0; [Omega]_i = 2[pi]r_i d_i. (12a)
Similarly, for an outward flow turbine,
[Omega]_o = 2[pi]r_o d; [Omega]_i = 2[pi]r_i d; (12b)
and, for an axial flow turbine,
[Omega]_o = [Omega]_i = [pi](r2² - r1²). (12c)
[Illustration: FIG. 194.]
_Relative and Common Velocity of the Water and Wheel._--There is
another way of resolving the velocity of the water. Let V be the
velocity of the wheel at the point c, fig. 194. Then the velocity of
the water may be resolved into a component V, which the water has in
common with the wheel, and a component v_r, which is the velocity of
the water relatively to the wheel.
_Velocity of Flow._--It is obvious that the frictional losses of head
in the wheel passages will increase as the velocity of flow is
greater, that is, the smaller the wheel is made. But if the wheel
works under water, the skin friction of the wheel cover increases as
the diameter of the wheel is made greater, and in any case the weight
of the wheel and consequently the journal friction increase as the
wheel is made larger. It is therefore desirable to choose, for the
velocity of flow, as large a value as is consistent with the condition
that the frictional losses in the wheel passages are a small fraction
of the total head.