MACHINERY.] HYDROMECHANICS 529 Call this area occupied by the vanes <a. Then the true value of the clear discharging outlet of the wheel is - (a, and the true value of u is . The corrected value of the angle of the vanes will On Q (0 -)V (16). 178. Head producing Velocity with ivlilch the Water enters tJic 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 hori zontal plane, there is no work done by gravity on the water passing 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. Let V,-, V be the velocities of the wheel at the inlet and outlet surfaces, r,-, v the velocities of the water, Uf, u the velocities of flow, Vri, Vr the relative velocities, hi, h the pressures, measured in feet of water, n, r the radii of the wheel, o the angular velocity of the wheel. At any point in the path of a portion of water, at radius r, the velocity v of the water may be resolved into a component V = co equal to the velocity at that point of the wheel, and a relative com ponent v r . Hence the motion of the water may be considered to consist of two parts : (a) a motion identical with that in a forced vortex of constant angular velocity a; (b) a flow along curves parallel to the wheel vane curves. Taking the latter first, and using Bernoulli s theorem, the change of pressure due to flow through the wheel passages is given by the equation The variation of pressure due to rotation in a forced vortex is Vi 2 - V 2 h i-h = Consequently the whole difference of pressure at the inlet and outlet surfaces of the wheel is h i-h = h i + h"i - 7i - 7i" "-+ r ^.. (!7). 20 Case 1. Axial Flow Turbines. V,- = V,,; and the first term on the right, in equation 17, disappears. Adding, however, the work of gravity due to a fall of d feet in passing through the wheel, 7 i. V r0 hi -h = -d Case 2. Outward Flow Turbines. The inlet radius is less than V- 2 - V 2 the outlet radius, and -2- is negative. The centrifugal head diminishes the pressure at the inlet surface, and increases the velocity with which the water enters the wheel. This somewhat increases the frictional loss of head. Further, if the wheel varies in velocity from variations in the useful work done, the quantity . ~ 2_ in creases when the turbine speed increases, and vice versa. Conse quently the flow into the turbine increases when the speed increases, and diminishes when the speed diminishes, and this again augments the variation of speed. The action of the centrifugal head in an out ward flow turbine is therefore prejudicial to steadiness of motion. For this reason r Q : rt is made small, generally about 5 : 4. Even then a governor is sometimes required to regulate the speed of the turbine. Case 3. Inward Flmv Turbines. The inlet radius is greater than the outlet radius, and the centrifugal head diminishes the velo city of flow into the turbine. This tends to diminish the frictional losses, but it lias a more important influence in securing steadiness of motion. Any increase of speed diminishes the flow into the tur bine, and vice versa. Hence the variation of speed is less than the variation of resistance overcome. In the so-called centre vent wheels in America, the ratio rt : r is about 5 : 4, and then the in fluence of the centrifugal head is not very important. Professor James Thomson first pointed out the advantage of a much greater difference of radii. By making r t : r a = 2 : 1, the centrifugal head balances about half the head in the supply chamber. Then the velo- city through the guide-blades does not exceed the velocity due to half the fall, and the action of the centrifugal head in securing steadiness of speed is considerable. Since the total head producing flow through the turbine is H - f>, and of this A,- 7i is expended in overcoming the pressure in the wheel, the velocity of flow into the wheel is . . . (18), where c v may be taken 96. From (14), It will be shown immediately that Vri^Ui cosec 6; or, as this is only a small term, and is on the average 90, we may take, for the present purpose, v r i=ui nearly. Inserting these values, and remembering that for an axial flow turbine V-= V , f) = 0, and the fall d in the wheel is to be added> For an outward flow turbine, For an inward flow turbine, 179. jingle ichich the Guide-Blades make with ihe Circumference of the Wheel. At the moment the water enters the wheel, the radial component of the velocity is Ui, and the velocity is vj. Hence, if 7 is the angle between the guide- blades and a tangent to the wheel This- angle can, if necessary, be corrected to allow for the thickness of the guide-blades. 180. Condition Determining the Angle of the Vanes at the Inlet Sur~ face of the Wheel. The single condition necessary to be satisfied at the inlet surface of the wheel is that the water should enter the wheel without shock. This con dition is satisfied if the direction of relative motion of the water and wheel is parallel to the first ele- 1 Fig. 195. ment of the wheel | vanes. Let A (fig. 195) be a point on the inlet surface of the wheel, and let vi represent in magnitude and direction the velocity of the water entering the wheel, and Vi the velocity of the wheel. Completing the parallelogram, v r i is the direction of relative motion. Hence the angle between v r t and V; is the angle 6 which the vanes should make with the inlet surface of the wheel. 181. Example of the Method of Designing a Turbine. Professor James Thomson s Inward Flow Turbine. Let H = the available fall ftfter deducting loss of head in pipes and channels from the gross fall; Q = the supply of water in cubic feet per second; and , TJ = the efficiency of the turbine. The work done per second is ijGQH, and the horse-power of the turbine is If TJ is taken at 75, an allowance will be made for the frictional losses in the turbine, the leakage, and the friction of the turbine shaft. Then h.p. = -085QH. The velocity of flow through the turbine (uncorrected for the space occupied by the vanes and guide-blades) may be taken
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