Page:EB1911 - Volume 14.djvu/113

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.

Fig. 193.

§ 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 axis of rotation in radial flow turbines, and on a cylindrical surface in axial flow turbines. At any point c of the path the water will have some velocity v, in the direction of a tangent to the path. 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 flow turbines. This second component is termed the velocity of flow. Let v_o, w_o, u_o be the velocity of the water, the whirling velocity and velocity of flow at the outlet surface of the wheel, and v_i, w_i, u_i the same quantities at the inlet surface of the wheel. Let α and β be the angles which the water’s direction of motion makes with the direction of motion of the wheel at those surfaces. Then

wo = vo cos β; uo = vo sin β wi = vi cos α; ui = vi sin α.

${\Big \}}$

(10)

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 Ω_o, Ω_i 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

v₀ = Q/Ω_o; v_i = Q/Ω_i.

(11)

Using the notation in fig. 191, we have, for an inward flow turbine (neglecting the space occupied by the vanes),

Ω_o = 2πr₀d₀; Ω_i = 2πr_id_i.

(12a)

Similarly, for an outward flow turbine,

Ω_o = 2πr_od; Ω_i = 2πr_id;

(12b)

and, for an axial flow turbine,

Ω_o = Ω_i = π (r₂² − r₁²).

(12c)

Fig. 194.

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. 194. Then the velocity of the water may be resolved into a component V, which the water has in common with the wheel, and a component v_r, which is the velocity of the water relatively to the wheel.

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 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,	uo = ui = 0.15 to 0.2 √(2gH);
In outward flow turbines,	ui = 0.25 √2g (H − ɧ),
	uo = 0.21 to 0.17 √2g (H − ɧ);
In inward flow turbines,	uo = ui = 0.125 √(2gH).

§ 191. 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_o and V_i 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_o = V_i = 0.6 √2gH to 0.66 √2gH.

In a radial outward flow turbine,

V_i = 0.56 √2g(H − ɧ)

V_o = V_ir_o / r_i,

where r_o, r_i are the radii of the outlet and inlet surfaces.

In a radial inward flow turbine,

V_i = 0.66 √2gH,

V_o = V_i r_o / r_i.

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_r, of the water’s motion at c is tangential to a wheel vane curve drawn through the point c. Let v_o, V_o, v_ro be the velocity of the water and its common and relative components at the outlet surface of the wheel, and v_i, V_i, v_ri be the same quantities at the inlet surface; and let θ and φ be the angles the wheel vanes make with the inlet and outlet surfaces; then

vo2 = √ (vro2 + Vo2 − 2Vovro cos φ) vi = √ (vr i2 + Vo2 − 2Vivr i  cos θ)

${\bigg \}}$,

(13)

equations which may be used to determine φ and θ.

Fig. 195.

§ 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 efficiency is to be as great as possible; that is, w_o = 0. Hence, from (10), cos β = 0, β = 90°, U_o = V_o, and 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 v_o or u_o radial or axial as the case may be, and V_o tangential to the direction of motion, v_ro can be found by the parallelogram of velocities. From fig. 195,

tan φ = v_o / V_o = u_o / V_o;

(14)

but φ is the angle which the wheel vane makes with the outlet surface of the wheel, which is thus determined when the velocity of flow u_o and velocity of the wheel V_o are known. When φ is thus determined,

v_ro = U_o cosec φ = V_o √ (1 + u_o² / V_o²).

(14a)

Correction of the Angle φ to allow for Thickness of Vanes.—In determining φ, it is most convenient to calculate its value approximately at first, from a value of u_o 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 allow for the vane thickness.

Let

φ′ = tan⁻¹ (u_o / V_o) = tan⁻¹ (Q / Ω_oV_o)

be the first or approximate value of φ, 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 φ′, their width measured on that surface is t cosec φ′. Hence the space occupied by the vanes on the outlet surface is

For A, fig. 192, ntdo cosec φ B, fig. 192, ntd cosec φ C, fig. 192, nt(r2 − r1) cosec φ.

${\bigg \}}$.

(15)

Call this area occupied by the vanes ω. Then the true value of the clear discharging outlet of the wheel is Ω_o − ω, and the true value of u_o is Q/(Ω_o − ω). The corrected value of the angle of the vanes will be

φ = tan [Q / V_o (Ω_o − ω) ].

(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 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.