56. Relation between specific speed and efficiency

Different impeller designs and different specific speeds must evi­dently affect pump efficiency. The degree of these effects, however, is different for hydraulic, volumetric and mechanical efficiency.

Hydraulic efficiency, experiments show, hardly changes with n_s and is much more dependent on the roughness of the flow passages and pump size. Volumetric and mechanical efficiency, on the other hand, is substantially affected when n_s approaches its lower limit.

With the specific speed decreasing the relative loss of power duo to impeller friction increases markedly, with a resulting reduction in the mechanical efficiency of a pump. Internal flow through spaces also increases, relative leakage rises and volumetric efficiency falls.

Thus, the overall efficiency of a centrifugal pump with a low spe­cific speed is relatively lower, decreasing as n_s decreases. This sots a lower limit for n imposed by efficiency considerations. These, of course, will depend on the application and working conditions of the pump.

The formulas for relative energy losses in pumps and correspond­ing efficiencies as functions of specific speed developed below can be used to determine the minimum permissible specific speed and also to facilitate efficiency estimates for different values of n_s.

We shall first investigate relative leakage through pump clearances and de­termine volumetric efficiency.

Flow through clearance spaces (leakage) can be expressed according to the common flow equation:

where μ = coefficient of discharge for flow through clearance space. For com­mon clearances μ = 0.4-0.5, for special labyrinth clearances μ=0.3;

H_ci = head of flow through clearance;

S = clearance space area. For open (unshrouded) impellers

where D_ci = diameter of clearance space;

δ = clearance, assumed proportional to D_cl .

The value of H_cl can be found as the difference between the pressure head developed by the impeller (i. e., tho head H for a finite number of vanes, with hydraulic losses taken into account) minus the pressure drop in the clearance space between impeller and casing due to fluid rotation. As one of the wearing surfaces is at rest and the other is moving, it is commonly assumed that the fluid is rotating at half the speed of the impeller.

Using Eq. (3.3) for the pressure in the fluid at relative rest and taking into account the foregoing considerations, we have

The head H_cl can be assumed approximately proportional to the delivered head and expressed in the form

where k_c, = 0.6-0.85.

The clearance space diameter is approximately equal to the eye diameter D₉ which, as will be shown in the following sections, must be made equal to

where k₀ = 4.2-4.5.

Substituting these values into the leakage equation, determine the ratio of the leakage to the normal discharge as a function of the specific speed. We have

Taking into account the expression for specific speed (12.38), we obtain

where

At m = 300, u. = 0.5, k_cl = 0.8 and k₀ = 4.5, the constant A ≈ 1. For this value of A there is plotted on the diagram in Fig. 147 the dependence of the relative leakage in an open impeller as a function of the specific speed.The curve illustrates graphically how the importance of leakage grows with the specific speed decreasing. For a closed impeller A is twice as much.

The volumetric efficiency of a pump can be expressed in terms of the specific speed by the formula

The volumetric efficiency curve is also given in Fig. 147.

Friction between impeller shroud and fluid usually occurs when the flow is turbulent, and the shearing stress т can therefore be assumed proportional to the product of the specific weight of the fluid times the velocity head, just as in turbulent pipe flow. In this case, however, the velocity head must be ex­pressed in terms of the peripheral speed of the impeller, which is proportional to the radius. Thus,

where cj == dimensionless proportionality coefficient called the friction coef­ficient.

The loss of power due to friction between the impeller shroud and the fluid can be computed by integrating the expression for the differential friction moment multiplied by the angular velocity of rotation of the impeller:

where dS = differential area equal to 2nrdr, r = running radius; к = coefficient taking into account the proportion of thp total shroud area exposed to friction; usually 1 < к < 2.

Substituting the previous expression for x and taking into account that и = ωr, and assuming that to a first approximation the coefficient of friction cj in turbulent flow is constant for the whole shroud surface, the integration can be carried out in the form

Or

where С = constant including all numerical coefficients as well as k, and other constants.

Investigations show that for approximate calculations for highly finished surfaces it can be assumed that С = 1.2 X 10-⁶ in the MKS system of units; Nj in this case is expressed in horsepower.

Determine the ratio of the friction horsepower to the hydraulic horsepower of the pump (see Sec. 53):

Now express the impeller diameter D in terms of the peripheral velocity us and the speed of rotation n, and the peripheral velocity in terms of head from the expressions

whence

Substituting into the basic equation, we obtain

or, introducing the specific speed n_s and using the foregoing expression for η_v in terms of n_s

(12.41)

where

At k₁ =1.2, η_h =0.8 and C=1.2x10-⁶ B = 377.

The curve of as a function ofn_s for the determined values of В and for A ≈1 is presented in Fig.148, which shows graphically how the relative horsepower loss due to friction between the fluid and the impeller shrouds increases when the specific speed decreases.

To go over from to the mechanical efficiency we regard the latter as aproduct

where η_m = efficiency taking into account power loss due to friction between impeller shroud and fluid, whence

(12.42)

η``_m = efficiency taking into account friction in packings and bearings and equal to

In the latter expression is the loss of power due to friction in the packings and bearings and N_o is the total shaft power of the pump:

Using the dependence of on the specific speed presented above, it is possible to plot η`_m as a function of n_s for the given values of the constants A and B.

The factor η``_m can be assumed to be independent of n_s and equal to about 0.95.

The curve showing the drop in η`_m with n_s decreasing is presented in the diagram in Fig. 148.

It should be borne in mind that by judicious design of the passages leading from the clearance between the impeller and casing wearing surfaces to the volute (of the order of 3 per cent of the impeller diameter on each side) a part of the energy dissipated in friction can be restored by utilising the kinetic energy of the fluid carried along by the impeller shroud. The actual mechanical efficiency may therefore be slightly higher than the rated value.

In using the plotted curves of and η_m as a function of n_st their approximate nature should be borne in mind. To obtain more accurate values of these quantities the values of the constants A and В must be computed each time for the specific data.

It is practically impossible to offer an analytical expression tor the total hydraulic losses inside a pump, and consequently for η_h, as this quantity depends on a number of factors the effects of which are poorly known. For high-head (low-speed) impellers the hydraulic efficiency may vary from 0.7 to 0.9, the lower limit referring to small values of n_s and small impeller diameters of the order D = 100-200 mm, and the upper limit corresponding to n_s = 90-120 and D = 500-600 mm.