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.
§ 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
f axis of rotation in
c radial flow turbines,
and on a cylindrical
n, surface in axial flow
turbines. At any
lb point c of the path

I the water will have

" some velocity 11, in

' ' the direction of a

- tangent to the path. v' 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 ow turbines. This second component is termed the velocity of flow. Let v., , wo, u., be the velocity of the water, the whirling velocity and velocity of flow at the outlet surface of the wheel, and vt, wg, vt the same quantities at the inlet surface of the wheel. Let a and B be the angles which the water's direction of motion makes with the direction of motion of the wheel at those surfaces. Then /Q, 1 I Fio. 193. w., =v, , cos;8; u., =v., sin B (lo) wi=v;coso.: ui=v; sin a 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 $20, Q; 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 vo =Q/Sb; vi =Q/91. (I I) Using the notation in fig. 191, we have, for an inward flow turbine (neglecting the space occupied by the vanes), Q., =21l'fpdo§ fl; =21l'fidi~ (120) Similarly, for an outward flow turbine, SZ., =21rr., d; SZ; ==2-zrrgd; (1211) and, for an axial flow turbine, Q.. = Qt = 1¥'(7'z2-7't2l- (120) Relative and Common Velocity of the Water and Wheel.-There is another way of resolving the velocity of th€ Waféf- LU be U10 Then the velocity of the water maype resolved into a component V, which the water has V, » in common with the C wheel, and a component of, which is the velocity of the water relatively If (Xw to the wheel. velocity of the wheel at the point c, fig. 194.

I . /I .' ./ » 1, 1 f 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 How, 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. The values most commonly assumed in practice are these:- In axial flow turbines, up = 14i=0' 15 to 0'2/ (2gH); In outward flow turbines, ui =o-25/2glH- E), u, ,=o-21 to 0'I7/ 2g(H-bl; In inward flow turbines, u0=u5=0~125/ lzgl-I). § 191. Speed of the Wheel.-The best speed of the wheel depends partly on the frictional losses, which the ordinary theory of turbines VP FIG. 194. I disregards. It is best, therefore, to assume for V., and V; values which experiment has shown to be most advantageous. In axial flow turbines, the clrcumfercntial velocities at the mean radius of the wheel may be taken V, ,=V, =o-6/-Q-I to o-66/°§§ 'll In a radial outward flow turbine, V; =o-56/ 2g(H - 9) Vo = Vin/fi, where r., rf are the radii of the outlet and inlet surfaces. In a radial inward flow turbine, V4 =o-66~/ 2g1-I, Vo=Vxf»/V+ If the wheel were stationary and the water flowed through it, the water would follow paths parallel to the wheel vane curves, at least when the vanes were so close that irregular motion was prevented. Similarly, when the wheel is in motion, the water follows paths relatively to the wheel, which are curves parallel to the wheel vanes. Hence the relative component, v., of the water's motion at c is tangential to a wheel vane curve drawn through the point c. Let ve, V, , 11, -, , be the velocity of the water and its common and relative components at the outlet surface of the wheel, and oi, Vt, vfi be the same quantities at the inlet surface; and let 0 and 4: be the angles the wheel vanes make with the inlet and outlet surfaces' then 7702 = N/ (vroz 'l“Vn2 ' 2VoUro COS Us =/ (1/fs”-l-Va?-2Vwf; cos 0) ' (13) equations which may be used to determine eb and 0. § 192. Condition determining the Angle of the Vanes at the Outlet Surface of the Wheel.-It has been shown that, when the water leaves the wheel, it should have no tangential velocity, if the eflici- f ency is to be as great as possible; ' / that is, 'w., =0. Hence, , from (IO), cos B=O, , B=90°, u., =v.», and if the direction of the water's motion is normal to the outlet surface of the wheel, radial in radial flow, and axial in axial flow turbines. Drawing vs or u., radial or axial as the case may be, and V, - tangential to the direction of motion, uf., can be found by the parallelogram of velocities. From fig. 195, tan ¢='v»/V»=u»/Vi; (14) but ¢> is the angle which the wheel vane makes with the outlet surface of the wheel, which is thus determined when thf velocity of flow un and velocity of the Wheel V, are known. When ¢ is thus determined, fr I, I 'Ya I I 1 I /, W I I / / U'7'o FIG. 195. v., = u, cosec 4: =V., { (1 -I-14,2/VJ). (14a) Correction of the Angle 41 to allow for Thickness of Vanes.-In determining ¢, it is most convenient to calculate its value approximately at first, from a value of 14, obtained by neglecting the thickness of the vanes. As, however, this angle is the most important angle in the turbine, the value should be afterwards corrected to alllow for the vane thickness. et <t>' = t=1r1”'(“»/Vo) = tan~'(Q/SLV0) be the first or approximate value of 4>, and let t be the thickness, and n the number of wheel vanes which reach the outlet surface of the wheel. As the vanes cut the outlet surface approximately at the angle ¢', their width measured on that surface is t cosec ¢'. Hence the space occupied by the vanes on the outlet surface is For A, fig. 192, ntd.. cosec 4> B, fig. 192, ntd cosec 4> ' (15) C, fig. 192, nt(r2-ri) cosec 4> Call this area occupied by the vanes o. Then the true value of the clear discharging outlet of the wheel is SZ.-ov, and the true value of it., is Q/(90 - w), The corrected value of the angle of the vanes will be ¢=tan [Q/Vaffla-w)]. (16) § 193. Head producing Velocity with which the Water enters the Wheel:-Consider the variation of pressure in a wheel passage, which satisfies the condition that the sections change so gradually that there is no loss of head in shock. When the flow is in a horizontal plane, there is no work done by gravity on the water passin through the wheel. In the case of an axial flow turbine, in which the flow is vertical, the fall d between the inlet and outlet surfaces

should be taken into account.