to coincide with the other, it is necessary that the angular velocity of the wheel should have a certain value; but if that cannot be established or maintained, the coincidence cannot be found. For this reason the second case is said to be more frequent than the first. Then in the passage of the water from the one to the other of these two relative velocities, consequent on sudden change, there will be a loss of work.

. When the water begins to pass through the motor, every liquid molecule is endowed with double velocity, the relative velocity with which it flowed into the passages comprised between vane and vane, and the velocity which it has in common with the wheel which carries it. The relative velocity of the efflux is found by applying to the water passing through the motor the theorem of work, and remembering that when the motion occurs in a horizontal direction, it will be convenient to take into consideration the centrifugal force as well. Finally compounding this relative velocity of the efflux with that of translation that the water will have in common with the parts of the wheel from which it flows, there will be obtained the absolute velocity with which the water leaves the wheel. From all that has been said, so far, it will be seen that, if

Q

G

H

V

ར.

W

=

Lm

=

the volume of available water;

the specific weight of the water;

the total height of the fall, which is then divided into two parts, h, the part above the wheel, and h2 the height from which the water descends while it is in the wheel (for centrifugal wheels h2 = 0);

= the velocity of the efflux from the supply chamber;

=

the velocity lost by collision in the entrance of the water into the motor;

= the absolute velocity with which the motor is left;

=

the motive work absorbed by the wheel;

there are obtained accordingly the equations:

The

representing in the first of these equations by 2 that fraction of h1 which is lost during the time that the water still runs in the supply chamber, a fraction which lies somewhere between 0∙10 and 0.15. Thus the principal object to be aimed at, in order to arrive at a useful result, will be to minimise μ, W and V1. values of W and V, can easily be determined, in every actual case, in functions of the dimensions and arrangements of the turbine, as well as of the angular velocity of rotation w. In endeavouring to determine that value of w for which L = 0, the greatest angular velocity which the turbine can attain will be found when it has no

work to do. If on the other hand it be supposed that Lm is a maximum, one will fall upon the angular velocity for which the useful effect becomes greatest. The researches thus indicated are neither impossible, nor of any length. They lead always to equations of higher degrees which it is convenient to resolve by approximation.

These theories are then compared with the results obtained in forty-six experiments made at the hydraulic establishment of the School of Engineering at Turin. It is shown how, in certain points, the coincidence of these results with theory may be regarded as nearly perfect; while in others plausible reason can be assigned for the discrepancy.

From the preceding results there are deduced the following practical rules for the construction of turbines. For turbines of parallel flow, the fall H is divided into two parts, h1 and h2, and the Author believes it is always better to exceed in h; the height of 21 decimètres, at the most, sufficing for the h2. Thus the greater part of the fall acts on the supply chamber. With respect to the supply chamber the following are the principal instructions:(1) In all the passages which lead to the distributing passages, it is necessary to avoid too narrow sections, sudden bends, and rapid changes in the area of the section.

(2) For the distributing passages proper, the entrance should be as far as possible funnel-shaped, and with a vertical direction; the curvature should be gradual in order that at the point of discharge they may have the right direction. Their curvature is of little consequence, so long as it has the desired continuity; 1 decimètre to 2 decimètres will suffice for their vertical height.

(3) The size of the passages in terms of the radius, taken where it is narrowest, i.e. at the orifice of exit, should not exceed the fifth of the radius, and may be less. Circumferentially they should occupy an arc less than half of this circumference-this also upon the upper face, where, in order to give them the form of a funnel, they are more extended. This condition, and the necessity to give free vent to all the available water, give a lower limit for the radius of the turbine.

The total arc occupied by the passages may be in one, or can be divided into two or three parts, symmetrically disposed with reference to the centre. The general practice is to divide it into two parts diametrically opposite. Such an arrangement, in fact, balances the wheel better: restricting the distributing orifices to one place, the useful effect is greater theoretically, and also practically, if the experimental results of Girard are to be relied on.

With regard to the motor, it is to be noted that the width, in the direction of the radius of the circular crown, where the water enters, ought to be that of the distributing zone, increased, if necessary, by a small quantity, in order to avoid the danger of any of the water being lost. The distance between the vanes should be the same as that between the guide-blades of the supply [1877-78. N.s.] 2 A

chamber. The other dimensions will be more simply expressed algebraically. In addition then to the symbols already adopted, let

R the mean radius of the wheel, measured from the centre to the middle of the crown;

α

бо

= the length of the arc (less than ) of the circumference occupied by the distributing passages;

=

the width in terms of the radius of the crown, measured at the top of the wheel, and let do < } R; 81 the same width taken at the inferior extremity of the wheel;

=

=

the angle with the horizontal plane, and therefore with the circumference 2 R of the last element of the supply chamber, taken at the centre of the directrix; 0,= the analogous angle for the first (higher) element of the vane of the wheel;

01 the analogous angle for the lower element;

=

let further

V1

=

the relative velocity with which the water ought to enter into the wheel;

v1 = the relative velocity with which it ought to leave it; the angular velocity of the motor;

W =

Further if, in the outflow of the water from the wheel, a loss of head is admitted, represented by k, it should then be

For centrifugal turbines the rules have much analogy with those given for parallel-flow turbines. In the pipes leading to the distributor passages, it is necessary to avoid all causes diminishing the useful height of the fall, such as friction, and sensible

changes in the direction, or in the area of the section; also for this it will be useful to take care that the entrance into those passages be made funnel-shaped, and that the curvature of the same be continuous. What has been said about the supply chamber should be repeated for the motor, that many directions analogous to those given for the other species of turbine are applicable. Nevertheless the quantities to be determined are different. In the motor there are two radii, the internal R, and the external R1; there are two heights of the wheel, the interior b。 and the exterior b1. α = the extent of the arc of the supply chamber discharging the water; 4, 90, 91, vo, v1, retain the previous significations. k represents the part of the fall which is permitted to be lost, in order to have a certain absolute velocity to the water that leaves the wheel. With these quantities the seven following equations ought to be satisfied :

:

To which it will be useful, but not indispensable, to add

Bearing in mind then that the quantities which may be regarded as available in the construction of the turbine are eight-i.e. the two radii R, and R1, the two heights b, and b1, the three angles, Oo, and 01, and the extent of arc a, and that in solving the equations there ought to be added the three velocities which remain unknown, Vo, v。, and v1, there will be eleven indeterminates, and only eight equations. It is then possible to dispose arbitrarily of three. But it must not altogether be forgotten that there are limits within which some of these quantities ought to be confined; thus, for the sake of example, the first of the eight equations shows that it is better to make b, small, rather than too great; but it must be well understood that this diminution ought never to go lower. than 8 centimètres. It will be more simple to make b1 equal to bo, or rather larger; but if the hypothesis b, b, should lead to a radius R1 too great, it will be useful to take b,>bo. Lastly, the extent of arc a must always be maintained less than T, especially in this second species of turbines, if the distributing passages are to be kept funnel-shaped. Reflecting that the radius of entrance into those passages is less than Ro, a ought to be much less

=

than π; but if this should lead to radii too great, this condition may be surrendered at the cost of having to study another manner of closing for the distributors. From turbines made according to the preceding instructions, it seems to Professor Richelmy that one may always hope for a co-efficient of return superior to 80 per cent.

C. C.

Theoretical Investigation of the Variable Motion of Fluids, and its
Application to the Propagation of the Floods of Rivers.
By M. KLEITZ.

(Annales des Ponts et Chaussées, vol. xiv., pp. 133–193, 1 pl.)

The Author's object is to furnish a theory of the motion of fluids when the height of their surface is constantly altering, with a special view of investigating the motion of floods in rivers.

Taking first the case of an infinitesimal particle, 8 w d s, of the fluid, the equation of equilibrium is

where p is the density of the fluid; g, the component of gravity along the axis designated by 8; P the mean pressure; f, the molecular dynamic force.

Considering next a portion of the current contained between two sections w and w', the equation is

where z represents the ordinates of the surface lines of the sections w and w'. To the above equation must be added the equation expressing the condition of continuity of the fluid,

If the distance between the two sections is finite, equation (2) becomes

dz d s

=

2
1 u2 d w
ds

1/du
g
d t

u d
w dt

g w

+ &.

It is assumed, for the sake of simplicity, that the river is confined to a single channel, and that the flood rises gradually to a maximum, and then gradually sinks to its normal level. A longitudinal section of the surface of a river in flood is liable to irregularities, owing to differences in size of the river bed;