50. Characteristics of ideal pump. Degree of reaction

Equation (12.8) is inconvenient for calculations as it does not contain the rate of discharge Q. Therefore let us rewrite it to express the head H_ta> as a function of the discharge Q and impeller radius.

From the velocity triangle for the impeller exit (Fig. 133).

where v_2r = projection of absolute exit velocity on radius, i. e., the radial component of vector v₂.

The rate of discharge through the impeller can be expressed in terms of the radial component v_2r and impeller radius as follows:

where b₂ = width of vane slot at exit (see Fig. 132). Hence

Substitution of this expres­sion into Eq. (12.9) yields

Substituting the obtained expression (12.11) for the tangential velocity component

v_2u in Eq. (12.8), we obtain an­other form of the basic ideal pump equation:

This equation can be used to plot the theoretical characteris­tics of an idealised centrifugal pump, i. e., a curve of the head generatedby the pump as a func­tion of the discharge for a con­stant speed of rotation. It is evident from Eq. (12.12) that the characteristic curve of such a pump is a straight line the inclina­tion of which depends on the value of the vane angle p₂.The follow­ing three cases are possible.

1. Angle p₂ < 90°. In this case cot β₂ is positive and the head decreases with the discharge increasing.

2. Angle p₂ = 90°, cot β ₂ = 0, and H_tto does not depend on the discharge and is equal to .

3. Angle P₂>90°, cot β ₂ is negative and the head H_im increases with the discharge.

These three theoretical pump characteristics are shown in Fig. 134. The corresponding vane shapes and velocity parallelograms for the same values of u₂ and y_2r are presented in Fig. 135a, b, с

It thus follows that the optimum head is produced by a forward-curved vane, when β ₂ > 90° and the head is highest. In practice, however, the efficiency of such a pump is low, and the performance of backward-curved vanes at β ₂ < 90° is found to be preferable. Backward-curved vanes are in fact more commonly used, the vane angle usually being about 30°. Radial vanes (p₂ = 90°) are also employed, but the result is lower efficiency and the considerations to be guided by are usually those of size, strength, etc.

In order to understand why pump efficiency falls with t£e angle β ₂ increasing we must examine the components of the head and the way in which the relation between them changes with β ₂.

The head , or what is the same thing, the total increase in the specific energy of a fluid in an impeller, comprises the increase in the specific energy of pressure and the specific kinetic energy, i.e.,

or, introducing another notation,

Expressing the velocities v_i and v₂ in terms of their radial and tan­gential components, we have

Assuming the intake and exit areas of the impeller to be approxi­mately equal, we can consider that v_lr = v_2r. Furthermore, as point­ed out before, there is usually no prerotation of the fluid at the impeller intake, and v_la = 0. Consequently, instead of the fore­going we have

Taking this expression into account, we can now find from Eq. (12.13) the so-called degree of reaction of the pump, which is the ratio of the head imparted to the fluid by the pressure increase to the total head:

Using Eq. (12.8), the latter expression can be rewritten as follows

whence finally, after substituting for v_2u from Eq. (12.9),

It will be observed from this expression that the greater v_2r/v₂ and the smaller angle |3₂, the greater the portion of the head H_t<J} that is produced by the pressure increase, i.e., the higher the de­gree of reaction of the pump. With angle (J₂ increasing the portion of the head #_/00 representing the increase in the kinetic energy be­comes greater. The kinetic energy, in turn, is associated with higher exit velocity of the fluid from the impeller, which results in consid­erable energy losses and lower pump efficiency. That is why it is not expedient to use vanes with large values of |J₂, i.e., forward-bent vanes.

It follows from Eq. (12.15) that for radial vanes (p₂ = 90°) the degree of reaction is 1/2, and at ($₂ < 90° it is more than 1/2 but less than unity.

The way in which the velocity parallelograms change and the increase in the absolute exit velocity v₂ with angle P₂ increasing are illustrated in Fig. 135.