the machine. When the machine is at rest the water issues from
the orifices with the velocity √ (2gH) (friction being neglected). But
when the machine rotates the water in the arms rotates also, and is
in the condition of a forced vortex, all the particles having the same
angular velocity. Consequently the pressure in the arms at the
orifices is H + α2r 2/2g ft. of water, and the velocity of discharge
through the orifices is v = √ (2gH + α2r 2). If the total area of the
orifices is ω, the quantity discharged from the wheel per second is
While the water passes through the orifices with the velocity v, the orifices are moving in the opposite direction with the velocity αr. The absolute velocity of the water is therefore
The momentum generated per second is (GQ/g)(v − αr), which is numerically equal to the force driving the motor at the radius r. The work done by the water in rotating the wheel is therefore
(GQ/g) (v − αr) αr foot-pounds per sec.
The work expended by the water fall is GQH foot-pounds per second. Consequently the efficiency of the motor is
| η = | (v − αr) αr | = | {√ (2gH + α2r 2) − αr } αr | . |
| gH | gH |
Let
| √ (2gH + α2r 2) = αr + | gH | − | g2H2 | ... |
| αr | 2α3r 3 |
then
which increases towards the limit 1 as αr increases towards infinity. Neglecting friction, therefore, the maximum efficiency is reached when the wheel has an infinitely great velocity of rotation. But this condition is impracticable to realize, and even, at practicable but high velocities of rotation, the friction would considerably reduce the efficiency. Experiment seems to show that the best efficiency is reached when αr = √ (2gH). Then the efficiency apart from friction is
η = {√ (2α2r 2) − αr } αr / gH
about 17% of the energy of the fall being carried away by the water discharged. The actual efficiency realized appears to be about 60%, so that about 21% of the energy of the fall is lost in friction, in addition to the energy carried away by the water.
§ 184. General Statement of Hydrodynamical Principles necessary for the Theory of Turbines.
(a) When water flows through any pipe-shaped passage, such as the passage between the vanes of a turbine wheel, the relation between the changes of pressure and velocity is given by Bernoulli’s theorem (§ 29). Suppose that, at a section A of such a passage, h1 is the pressure measured in feet of water, v1 the velocity, and z1 the elevation above any horizontal datum plane, and that at a section B the same quantities are denoted by h2, v2, z2. Then
| h1 − h2 = (v22 − v12) / 2g + z2 − z1. | (1) |
If the flow is horizontal, z2 = z1; and
| h1 − h2 = (v22 − v12) / 2g. | (1a) |
(b) When there is an abrupt change of section of the passage, or an abrupt change of section of the stream due to a contraction, then, in applying Bernoulli’s equation allowance must be made for the loss of head in shock (§ 36). Let v1, v2 be the velocities before and after the abrupt change, then a stream of velocity v1 impinges on a stream at a velocity v2, and the relative velocity is v1 − v2. The head lost is (v1 − v2)2/2g. Then equation (1a) becomes
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| Fig. 184. |
To diminish as much as possible the loss of energy from irregular eddying motions, the change of section in the turbine passages must be very gradual, and the curvature without discontinuity.
(c) Equality of Angular Impulse and Change of Angular Momentum.—Suppose that a couple, the moment of which is M, acts on a body of weight W for t seconds, during which it moves from A1 to A2 (fig. 184). Let v1 be the velocity of the body at A1, v2 its velocity at A2, and let p1, p2 be the perpendiculars from C on v1 and v2. Then Mt is termed the angular impulse of the couple, and the quantity
is the change of angular momentum relatively to C. Then, from the equality of angular impulse and change of angular momentum
or, if the change of momentum is estimated for one second,
Let r1, r2 be the radii drawn from C to A1, A2, and let w1, w2 be the components of v1, v2, perpendicular to these radii, making angles β and α with v1, v2. Then
where the moment of the couple is expressed in terms of the radii drawn to the positions of the body at the beginning and end of a second, and the tangential components of its velocity at those points.
Now the water flowing through a turbine enters at the admission surface and leaves at the discharge surface of the wheel, with its angular momentum relatively to the axis of the wheel changed. It therefore exerts a couple −M tending to rotate the wheel, equal and opposite to the couple M which the wheel exerts on the water. Let Q cub. ft. enter and leave the wheel per second, and let w1, w2 be the tangential components of the velocity of the water at the receiving and discharging surfaces of the wheel, r1, r2 the radii of those surfaces. By the principle above,
If α is the angular velocity of the wheel, the work done by the water on the wheel is
§ 185. Total and Available Fall.—Let Ht be the total difference of level from the head-water to the tail-water surface. Of this total head a portion is expended in overcoming the resistances of the head race, tail race, supply pipe, or other channel conveying the water. Let ɧp be that loss of head, which varies with the local conditions in which the turbine is placed. Then
is the available head for working the turbine, and on this the calculations for the turbine should be based. In some cases it is necessary to place the turbine above the tail-water level, and there is then a fall ɧ from the centre of the outlet surface of the turbine to the tail-water level which is wasted, but which is properly one of the losses belonging to the turbine itself. In that case the velocities of the water in the turbine should be calculated for a head H − ɧ, but the efficiency of the turbine for the head H.
§ 186. Gross Efficiency and Hydraulic Efficiency of a Turbine.—Let Td be the useful work done by the turbine, in foot-pounds per second, Tt the work expended in friction of the turbine shaft, gearing, &c., a quantity which varies with the local conditions in which the turbine is placed. Then the effective work done by the water in the turbine is
The gross efficiency of the whole arrangement of turbine, races, and transmissive machinery is
And the hydraulic efficiency of the turbine alone is
It is this last efficiency only with which the theory of turbines is concerned.
From equations (5) and (7) we get
This is the fundamental equation in the theory of turbines. In general,[1] 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
αr1 is the circumferential velocity of the wheel at the inlet surface. Calling this V1, the equation becomes
This remarkably simple equation is the fundamental equation in the theory of turbines. It was first given by Reiche (Turbinenbaues, 1877).
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| Fig. 185. |
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| Fig. 186. |
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| Fig. 187. |
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| Fig. 188. |
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| 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
- ↑ In general, because when the water leaves the turbine wheel it ceases to act on the machine. If deflecting vanes or a whirlpool are added to a turbine at the discharging side, then v1 may in part depend on v2, and the statement above is no longer true.
