52. Hydraulic losses in pump. Plotting rated characteristic curve

As stated earlier, H_tz is the head that would have been developed by a pump if there were no head losses inside it. The actual head H_pump (see Sec. 48) is less than the theoretical head by the total losses inside the pump:

where Σh_pump = total head loss in pump (at intake, in impeller and volute).

The ratio of the actual head to the theoretical head for a finite number of vanes is called the hydraulic efficiency, denoted η_h. Thus,

Hydraulic efficiency is always higher than total efficiency as it takes into account only head losses inside the pump. It follows from Eqs (12.18) and (12.21) that

where is given by Eqs (12.7) and (12.12).

The hydraulic losses inside the pump H_pump are conveniently treated as the sum of the following two components.

1. Ordinary hydraulic losses, i.e., losses due to friction and partly to eddy formation inside the pump. As turbulent flow is the common regime in a centrifugal pump, this type of head losses increases ap­proximately as the square of the discharge and can be expressed by the equation

where k_{ is a constant depending on the hydraulic efficiency and the dimensions of the pump.

2. Shock losses at impeller and volute entrances. If the relative velocity w_x of the fluid at the entrance to a vane passage is tangent to the vane, the fluid is entering the impeller smoothly, without shock or eddy formation. Shock losses in this case are nil. But this is possible only at some definite rated or normal discharge Q_o and corresponding radial entrance velocity (v_ir)_o (Fig. 138).

If the actual discharge Q is more or less than the rated discharge Q_o and the radial entrance velocity v_1r is more or less than (v_lr)₀ the relative velocity w makes an angle у with the tangent to the vane and the fluid flows past the vane at some positive or negative angle of approach. The effect is that of the fluid impinging on the vane, with eddies forming on the opposite side. Thus, energy is degraded in the impact and eddy formation. The velocity parallelo­grams for the same peripheral velocities corresponding to these ton-rated operating conditions are shown by broken lines in Fig. 138. One of the parallelograms corresponds to the inequality Q > Q_o, the other, to — Q < Q_o.

Shock losses can be assumed to vary as the square of the difference between the actual discharge and the discharge when they are zero, i. e.,

Shock losses at the volute entrance are of the same nature as at the impeller intake, the minimum being at about the same rate of discharge Q_o, and are included in the quantity h₂.

The total loss of head inside the pump is the sum of the two losses considered, i.e.,

The characteristic curves of a pump at uniform rotational speed (n = const) are plotted as follows.

First draw for H as a function of Q atn = const the theoretical characteristic curves for and for a finite number of vanesz. These are inclined straight lines (Fig. 139). Below the Q axis plot the curves for the change in the two components h_i and h₂ of the head losses in the pump. Sum­ming the ordinates of the two curves gives the curve Σh_pump as a function of the discharge. Now, in accordance with Eq. (12.20), sub­tract Σh_pump from H_tz which gives the curve Σh_pump = =f(Q), i. e., the actual characteristic of the pump for a constant rpm.

The curve H _pump = f(Q) in Fig. 139 is typical of a centrifugal pump. The maximum value of the head H_pamp is commonly obtained neither at zero discharge nor at Q= Q_o, but at some intermediate value of Q.

Plotting the characteristic curves by the method described is not very accurate in view of the difficulty of determining the coefficients k_i and k₂ in Eqs (12.22) and (12.23). Therefore the characteristics are commonly plotted by direct experiment, i.e., in testing a pump.

For this a throttling device (some type of valve) is installed atthe outlet pipe of a pump operating at a constant rotational speed. The degree of opening of the valve is gradually changed during a test. For example, at first the valve is wide open. Then the pump is gradually throttled down and the discharge and head are measured. When the valve is completely open, the discharge is maximum and the head is minimum, being equal to the loss of head in the pipeline (point С in Fig. 140). As the valve is throttled, the discharge falls and the head rises to a maximum (point B). As the discharge is further re­duced the head also drops some­what, till at Q = 0 (point A), when the valve is closed completely, the head reaches a value higher than the average but below the maxim­um.

This is the so-called shutoff head. It will be observed that even complete closure of the valve does not present any danger to thepump or pipeline as no further increase in head develops if the¹ num­ber of rpm does not change. For this reason centrifugal pumps, unlike displacement pumps, do not have to be fitted with relief valves.