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The Ideal Euler Head
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The ideal pressure that a pump impeller can develop is called the Euler pressure. Consider the flow through a radial impeller between two radii R1 and  R2. The impeller is rotating at an angular speed   ѡ (in rad/s) so that the peripheral speeds are respectively:

                       U1 = R1 ·  ѡ                            (8-3a)
                       U2 = R2 ·  ѡ                            (8-3b)
The liquid flows radially at a meridional velocity Cm, perpendicular to the peripheral velocity U. The value of Cm is determined from continuity equation, It is necessary to take into account the local area of the flow, which is a function of the radius and the width of the channel, minus the blockage area due to the finite thickness and angle of inclination of the blades.

   

           FIGURE 8-6 Pressure distribution in an impeller versus radius for condition of flow and no flow.


The channels between the impeller vanes follow a certain profile. At the intersection with the radius under consideration, the angle between the vane and the tangent to the radius is defined as . A component of velocity is in the direction of  and is called the relative velocity W.
 
The vector addition of U and W result in the absolute velocity V. Both V and W share the same component of meridional speed Cm; a vector representation is shown in Figure 8-8.
 
The Euler “total” head between radii R1 and R2 is defined as
   

Furthermore because the curvature of the front and back shrouds of an impeller, are different, the meridional velocity is not uniform and may be higher toward the back shroud. For a linear variation of the meridional velocity between the front and back shrouds (Figure 8-7), Stepanoff (1993) derived the following equation for theoretical head:
 
   


   


The term 1 + [(V2V1)2/12 Cm2 ] is greater for the wide impellers encountered in mining slurry pumps. For slurry pumps, the value of _1 at the tip diameter of the eye of the impeller is between 14 and 30 degrees. The value of 2 at the tip diameter of the vanes is typically between 25 and 35 degrees. Stepanoff (1993) has indicated that inlet angles as high as 50 degrees are used on water pumps. This is, however, not the case with slurry pumps, as prerotation causes tremendous wear of the throat bush.

The vast majority of modern pumps have a discharge angle 2 smaller than 90 degrees. They are called impellers and have backward curved vanes. Expellers are often designed with radial vanes (i.e., 2  = 90 degrees). Forward vanes with2  larger than 90 de- grees are restricted to very low flow and high-head slurry pumps and to some expellers. Theoretically, an impeller with forward vanes would give a higher static head rise. Unfortunately, it is also the largest consumer of power and is considered to be inefficient.
 
Clay and other slurries can be very viscous. Herbrich (1991) has suggested using discharge angle 2  as high as 60 degrees on impellers for very viscous slurries but did not produce data to support such a suggestion. Stepanoff (1993) recommended the following design procedures for special pumps. These pumps would be suited to pump viscous liquids, but their performance may be impaired on water.

1. Use high impeller discharge angles up to 60 degrees to reduce the impeller diameter necessary to produce the same head and effectively reduce disk friction losses. Consequently, the impeller channels become shorter and the impeller hydraulic friction is reduced.
 
2. Eliminate close-clearance wide sealing rings at the impeller eye and provide knifeedge seals (one or two) similar to those used on blowers. Leakage loss becomes secondary when pumping viscous liquids.
 
3. Provide a similar axial seal at the impeller outside diameter to confine the liquid between the impeller and casing walls. This in turns raises the temperature of the liquid in the confined space (due to friction) well above the temperature of the remaining liquid passing through the impeller. Due to the temperature effects, viscosity is artificially reduced and disk friction losses are trimmed down. In fact, Stepanoff (1993) goes as far as suggesting injecting a light or heated oil in the confined space to reduce power loss due to friction.
 
4. Provide an ample gap (twice the normal) between the casing tongue or cutwater and the impeller outside diameter. Otherwise, the shrouds of the impeller would produce head by viscous drag at low capacities, and would decrease the efficiency of pumping.
 
5. High rotational speed and high specific speed lead to better efficiency and more head capacity output than low specific speed pumps on viscous liquids.
 
These recommendations were written with very viscous fluids in mind. Obviously, points 2 and 3 would not apply to a slurry pump. However, slurry pumps may use pumpout vanes, which effectively are dynamic seals. These recommendations can be modified to suit the design of special pumps for viscous slurries. The field of slurry pumps for very viscous slurries and difficult flotation frothy slurry associated with the oil sands industry is continuously evolving.
 
In some cases of pumping oil sand froth, it has been found that injecting 1% of water or a light oil as a lubricant just at the suction of the pump can improve the efficiency of the pump.

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