Fig.E-1 Simplest integration of the acceleration term in the constant
cross section, specified acceleration case.
Integrating the new term:
- does not now bring any substantial problems; it is a path integration from to exactly like those treated in previous chapters.

The forced motion case:
A particularly simple situation arises if the accelereration is directly given as an input variable. This is, in fact, a situation of practical importance - as shown in Figs.E-1 to E-3, it represents an acceptable simplification of the piston type (or, generally, positive-displacement type) pump with acceleration given by the kinematic of the crank mechanism driving the piston. For didactic purposes, Fig.E-1
Fig.E-2 A more difficult situation with nonconstant cross section is converted to two
cases corresponding to Fig.E-1 by separating the integration into constant-area
segments. This approach has been used in one-dimensional,
approximate calculations of positive displacement pumps.

introduces an oversimplified case in which the acceleration is constant along the integration path and it is possible to extract it in front of the integral, which then reduces into the difference in lengths.
In Fig.E-2, there is a somewhat more difficult case which represents a better approximation to real piston pump situations: there is a change in cross-sectional area between and which corresponds to unequal area of cylinder cross section and of the pipeline connected to it. Since it is possible to divide the integration path into two segments with constant cross sections, each of them corresponding to the case from Fig.E-1, the only problem here is how to calculate acceleration in the pipe if it is the piston acceleration which is specified. As shown in Fig.E-2,
Fig.E-3 Piston pump and the main concern in its hydraulic design: avoiding
decrease of pressure below the vapour pressure limit.
the relation
between accelerations in various pipe
segments is obtained by differentiating Castelli Theorem equation with respect to time - and integrating it along the spatial path. Although only approximate, this approach suffices for explanation of the basic hydraulic problems of positive delivery pumps. The main problem of these pumps, from the hydraulic point of view, is the fact that at higher speeds the piston acceleration becomes so high that the inertial term, after
integration of the equation, is substantially higher than other terms. It then exerts a decisive influence upon the pressure under the piston (or other displacement component - Fig .E-4). The most dangerous situation arises during the suction phase, where pressure at the piston may easily decrease below the vapour pressure . This brings about the danger of cavitation - formation of vapour bubbles. Liquid then ceases to follow the piston motion and implosive collapse of bubbles generates high local force action upon the metallic structure of walls. This effect can rapidly spall and erode the surfaces. The vapour pressure curve in Fig.E-3 represents a projection of the equilibrium surface from Fig.A-9 along the specific volume axis.
Fig.E-4 Most important principles of positive-displacement pumps.
In the accompanying table, there are values for water. It is apparent that the values increase eather rapidly with temperature. This is why in pumps that are designed to pump warm liquids (such as, e.g. supply pumps for steam boilers, pumping water recovered from condensig steam), it is necessary to choose negative suction height - the hydrostatic pressure serves there to counter the generated low suction pressure.

Positive displacement pumps: The basic cuase for the problems associated with fluid acceleration is the fact that pumps base upon displacement of liquid from some volume fluid occupies cannot operate continuously - this is evident in the reciprocating piston type Fig.E-3: the volume change in the cylinder of the piston-type M/F transducer according to Fig.B-7 must end after finite distance is travelled by the piston. The piston then must return to allow re-occupation of the volume by new fluid. The required one-directional flow is then achieved by application of fluidic diodes (check valves) connected so as tu for a rectifier. In Fig. E-3 above the rectifier is basically one half of the Gratz bridge. The generated flow pulsates - to smooth it, the pump involves some capacitance. In Fig.E-3 the capacitance is that of a vessel with trapped air cushion in addition to liquid level change according to Fig.B-24. Instead of using the diode valves, it is possible to obtain the rectification by mechanically driven valves, such as e.g. rotating sleeve valves. In some arrangem,ent shown in Fig.E-4, the rotating sleeve valve is substituted directly by the rotating pistons or piston-like components: the teeth of the gear pump or the sliding vanes serve simultanepously to open and close suction and delivery orifices.

Calculation of the pressure at the piston - Fig.E.3 above:
Integrating the unsteady-flow equation between the surface and piston crown - and then inserting the expressions for piston travel distance, piston velocity and piston acceleration leads to the following expression for the pressure at the piston:

Differentiating this expression and putting the derivative equal to zero provides the condition for minimum pressure , which is:

This equation has two solutions: one for corresponds to pressure maximum, the other, is the sought minimum.


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This is page Nr. E02 from textbook Vaclav TESAR : "BASIC FLUID MECHANICS"
Any comments and suggestions concerning this text may be mailed to the author to his address tesar@fsid.cvut.cz

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