## Operating Characteristics

The operating characteristics covered in this section assume the fluid is a true incompressible liquid with a viscosity independent of the rate of shear (shear strain). Common fluids used in testing are either a light lubricating oil or cool water with small amounts of soluble oil added for lubricity.

Displacement The theoretical or geometric displacement Qd of a rotary pump is the total gross fluid volume transferred from the OTI volume to the OTO per unit time. A standard unit of displacement is gallons per minute (cubic meters per hour). For any given pump, the displacement depends only upon the physical dimensions of the pump elements and the pump geometry and is independent of other operating conditions. In those pumps designed for variable displacement, the pump usually is rated at its maximum displacement.

Slip Flow slip, Qs, is an important aspect of rotary pump performance. It is defined as that quantity of fluid that slips from the OTO volume to the OTI volume per the unit of time. Slip is a function of the clearances between the rotating and stationary members, the differential pressure between the OTO volume, the OTI volume, and the fluid viscosity. Hydraulically, it is equivalent to a bypass line from the pump outlet back to the inlet. The most common unit of measuring slip is U.S. gallons per minute (cubic meters per hour).

The major slip paths through the pump in the presence of a positive differential pressure are the clearances between the end faces of the rotors and the endplates of the pump cham ber, and those between the outer radial surfaces of the rotors and the inner radial surfaces of the chamber. The width, length, and height of the apertures formed vary considerably with different positions of the rotor as the drive shaft turns through a complete revolution. If the differential pressure across the pump remains constant during a revolution, then the instantaneous slip rate usually varies throughout the revolution. This variation in the slip is caused by the same effect that would be produced if the physical dimensions of the equivalent bypass around the pump were varied as a function of the angular rotation of the drive shaft. This is also one of the common causes of flow pulsation in rotary pumps. It is particularly dominant when pumping low-viscosity fluids at high pressures.

The average slip for any set of operating conditions can be found by measuring the flow rate from the outlet port (assuming an incompressible liquid) and subtracting that flow rate from the theoretical displacement flow rate Qd that would otherwise be expected at those operating conditions. Most slip paths are constant in width but may vary in height with runout of the outside diameter of the rotors or wobble of the end faces of the rotors as they rotate. The paths can also vary considerably in length with the changing positions of the rotary and body-sealing surfaces during rotation.

The effect of pressure on the slip is complex. The primary effect is direct in that slip increases in direct proportion to pressure. However, several secondary effects should be considered as well. The first is the effect of pressure differences across the pump on the dimensions of the slip path. This occurs because of the deflection of pump elements as a function of pressure. This is relatively small in rigid element pumps but can be significant in flexible vane pumps where the pressure can cause the vanes to flatten out and move away from the body walls. In addition, while the slip may increase in flexible member pumps at high pressures, in rigid rotor pumps it can actually decrease as clearances close, due to the high-pressure deflection of the rotors.

Another secondary consideration is the indirect effect of pressure on the fluid velocity through the slip paths. At any given viscosity, the flow through these paths can have the characteristics of turbulent, laminar, or slug flows. The majority of practical applications would require that the slip be a minor percentage of the pump displacement. To remain so, the velocity of fluid flow through slip paths would normally be in the laminar flow region, and the slip would then be directly proportional to the pressure difference. A pressure increase could cause a change to turbulent flow and a corresponding change in the slip as a function of pressure.

Also, an indirect effect of pressure exists on the effective compression ratio of compressible fluids. The compression ratio reduces the amount of net volume flow through the outlet port relative to the displacement of the pump. Although not a true slip in the sense discussed up to this point, the type of slip caused by this effect reduces the net volume delivered through the outlet port and consequently affects the volumetric efficiency. This effect is a secondary effect in most liquids but can become a large component of the slip in aerated or compressible liquids. An increase in the compression ratio caused by an increase in the pressure difference causes an increase in slip from this effect.

Flow Rate and Displacement The flow rate or capacity Qc of a rotary pump is the net quantity of fluid delivered by the pump per the unit of time through its outlet port or ports under any given operating condition. When the fluid is incompressible, the flow rate is numerically equal to the total volume of liquid displaced by the pump per the unit of time minus the slip, all expressed in the same units. When a rotary pump is operating with zero slip, the theoretical or geometrical displacement Qd of the pump becomes the flow rate Qc. A common unit of flow rate is U.S. gallons per minute (cubic meters per hour):

The theoretical displacement D per revolution (where Qd = ND and N = revolutions per unit time) can be found by integrating the differential rate of a net volume transfer over one shaft revolution with respect to the angular displacement of the drive shaft through any complete planar segment taken through the pump chamber between the inlet and outlet ports. Most pump rotors have constant radial dimensions in the axial direction in the body cavity and sweep a right circular cylinder of volume while rotating. This means in single-rotor pumps or in multiple-rotor pumps where no sealing contact exists between rotors (all dynamic seals are formed between rotor elements and body surfaces), the volume transfer computation can be based on polar coordinates centered on each rotor axis and the contribution to the net volume transfer found for each rotor independently.

In general, the axial dimension of the rotor in the body cavity can be most simply expressed if the planar segment is taken through the rotor axis, or at least parallel to the rotor axis. Also, for most types of rotary pumps, the computation is simplified if the intersections of the plane with the body cavity occur in a CTIO region, usually midway between the inlet and outlet ports of the pump. This is particularly true for those rotary pumps that pump equally well in either direction of rotation and are generally symmetric. In many cases, the computation can be further simplified by separating the differential statement for volume transfer through the plane from the inlet to the outlet from that for volume transfer through the plane from the outlet to the inlet and expressing the results as a difference. Examples of this method of computation for some commonly used types of rotary pumps follow.

A section through a vane pump is shown in Figure 14. Let z be the axial distance toward the front endplate from the rotor end surface next to the rear endplate, and let Z be the total axial length of the rotor. Let r be the radial distance from the rotor axis. Let R1 be the minimum radial dimension of the rotor elements at the intersection of the plane with the minor cam radius of the pump chamber in the CTIO zone. Let R2 be the maximum radial dimension of the rotor elements at the intersection of the plane with the major cam radius of the pump chamber in the CTIO zone. Let f be the angular displacement of the drive shaft (assumed to be direct-coupled to the rotor with no gear increase or decrease). Then, the general equation for D is where k is a constant used to convert D to desired units (k = 1 if z and r are in feet and D is in cubic feet per revolution). For vanes with non-zero thickness, the actual value of D will be slightly smaller than this.

The equation describing the transition of the major radius cam surface to the minor radius cam surface is not used or needed in Equation 3, because the planar segment is entirely in the CTIO zone of the pump. Also, the integration limits for r were chosen by noting that the net volume transfer for all r < R1 cancels and equals zero. The same result is obtained if the integral is expressed as the difference of the positive contribution of the integration limits 0 to R1.

The same computations and formula apply to flexible vane pumps with the vanes on the rotor and to any vane-in rotor pump where the surface creating the pumping action is

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