528 HYDROMECHANICS [HYDRAULICS. flow or axial flow turbines have the wheel as in C. The vanes are limited by two concentric cylinders. Reaction Turbines. 175. Velocity of Whirl and Velocity of Flow. "Lei acb (fig. 192) 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 cylin drical surface in axial flow tur bines. At any point c of the path the water will have some velocity v, in the direction of a tangent to the path. That velo city 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 , 10^ u be the velocity of the water, the whirling velocity and velocity of flow at the outlet surface of the wheel, and v^u ifUi the same quantities at the inlet surface of the wheel. Let a and be the angles which the water s direction of motion makes with the direction of motion of the wheel at those surfaces. Then W{ = v cos a ; ,-= vi 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 ft , 0; 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 Q : r ,=5 (ID- Using the notation in fig. 191, we have, for an inward flow turbino (neglecting the space occupied by the vanes), Similarly, for an outward flow turbine, and, for an axial flow turbine, , . . (126); = I/, = 7T 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. 193. Then the velocity of the water may be re solved into a com ponent V, which the water has in common with the wheel, and a com ponent v r , which is the velocity of the water relatively to the wheel. Velocity of Flow. It is obvious that the frictional losses Fl - of head in the wheel passages will increase as the velocity of flow is gre;it3r, 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. The values most commonly assumed in practice are these : In axial flow turbines, = ?,- = 15 to - 2V 2(/H ; In outward flow turbines, M;=0 25/2{7(H -ft) , _ 176. Speed of the Wheel. The best speed of the wheel depends partly on the frictional losses, which the ordinary theory of turbines disregards. It is best, therefore, to assume for V and A 7 ,- values which experiment has shown to be most advantageous. In axial flow turbines, the circumferential velocities at the mean radius of the wheel may be taken v =v,-=o-6V20H to o-eeVi^H . In a radial outward flow turbine, where r , r f are the radii of the outlet and inlet surfaces, In a radial inward flow turbine, _ to 0-l7V2r/(H-h) ; lu inward flow turbines, 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 rela tively to the wheel, which are curves parallel to the wheel vanes. Hence the relative component, r r , of the water s motion at c is tan gential to a wheel vane curve drawn through the point c. Let y ,
- o " o be the velocity of the water and its common and relative
components at the outlet surface of the wheel, and r,-, V,-, r r i be the same quantities at the inlet surface ; and let and <f> be the angles the wheel vanes make with the inlet and outlet surfaces; then r o 2 = N/( v ro~ + ^> T O~ ~ 2 V iv cos Vi = J(v r ? + Vr - 2 Vii ri cos equations which may be used to determine </> and 6. 177. 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 velo city, if the effi ciency is to be as great as possible; that is, w. = 0. Hence, from (10), cos = 0,0 = 90, 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 r or U Q radial or axial as the case may be, and V tan gential to the direction of motion, Vr can be found by the parallelo gram of velocities. From fig. 194, tan = ^ = but <f> is the angle which the wheel vane makes with the outlet sur face of the wheel, which is thus determined when the velocity of flow UQ and velocity of the wheel V are known. When </> is thua determined, COSCC A = V /T7~<L "(14o) V A 2 Correction of the Angle <f> to allow for Thickness of Vanes. In determining <, it is most convenient to calculate its value approxi mately at first, from a value of M O obtained by neglecting the thick ness of the vanes. As, however, this angle is the most important angle in the turbine, the value should be afterwards corrected to allow for the vane thickness. Let Vo n o v o b the first or approximate value of (f>, 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 <f> , their width measured on that surface is t cosec <J>. Hence the space occupied by the vanes on the outlet surface is For A, fig. 191, ntd cosec $ ) B, fig. 191, ntd cosec <j> V. ... (.15).
C, fig. 191, nt (r.,- r-i) cosec )
