§ 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 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 Vi, Vo be the velocities of the wheel at the inlet and outlet surfaces, vi, vo the velocities of the water, ui, uo the velocities of flow, vri, vro the relative velocities, hi, ho the pressures, measured in feet of water, ri, ro the radii of the wheel, α 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 = αr equal to the velocity at that point of the wheel, and a relative component vr. 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 α; (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
h′i + vri2 / 2g = h′o + vro2 / 2g;
h′i − h′o = (vro2 − vri2) / 2g.
The variation of pressure due to rotation in a forced vortex is
h″i − h″o = (Vi2 − Vo2) / 2g.
Consequently the whole difference of pressure at the inlet and outlet surfaces of the wheel is
hi − ho = h′i + h″i − h′o − h″o
= (Vi2 − Vo2) / 2g + (vro2 − vri2) / 2g.
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