chapter 12 hydraulic transient design for pipeline systems

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12.10 Chapter Twelve HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS • Flow rate. Capacity Q at operating point in chosen units. • Total dynamic head (TDH). The total energy gain (or loss) H across pump, defined as H � ⎛ ⎜� ⎝ Pd � � zd γ ⎞ ⎟ ⎠ � ⎛ ⎜� ⎝ Ps γ � � z s ⎞ ⎟ ⎠ 2 2 d Vs � �V � � �� (12.11) 2g 2g where subscripts s and d refer to suction and discharge sides of the pump, respectively, 12.4.2 Homologous (Affinity) Laws Dynamic similitude, or dimensionless representation of test results, has been applied with perhaps more success in the area of hydraulic machinery than in any other field involving fluid mechanics. Due to the sheer magnitude of the problem of data handling it is imperative that dimensionless parameters be employed for transient analysis of hydraulic machines that are continually experiencing changes in speed as well as passing through several zones of normal and abnormal operation. For liquids for which thermal effects may be neglected, the remaining fluid-related forces are pressure (head), fluid inertia, resistance, phase change (cavitation), surface tension, compressibility, and gravity. If the discussion is limited to single-phase liquid flow, three of the above fluid effects—cavitation, surface tension, and gravity (no interfaces within machine)—can be eliminated, leaving the forces of pressure, inertia, viscous resistance, and compressibility. For the steady or even transient behavior of hydraulic machinery conducting liquids the effect of compressibility may be neglected. In terms of dimensionless ratios the three forces yield an Euler number (ratio of inertia force to pressure force), which is dependent upon geometry, and a Reynolds number. For all flowing situations, the viscous force, as represented by the Reynolds number, is definitely present. If water is the fluid medium, the effect of the Reynolds number on the characteristics of hydraulic machinery can usually be neglected, the major exception being the prediction of the performance of a large hydraulic turbine on the basis of model data. For the transient behavior of a given machine the actual change in the value of the Reynolds number is usually inconsequential anyway. The elimination of the viscous force from the original list reduces the number of fluid-type forces from seven to two–pressure (head) and inertia, as exemplified by the Euler number. The appellation geometry in the functional relationship in the above equation embodies primarily, first, the shape of the rotating impeller, the entrance and exit flow passages, including effects of vanes, diffusers, and so on; second, the effect of surface roughness; and lastly the geometry of the streamline pattern, better known as kinematic similitude in contrast to the first two, which are related to geometric similarity. Kinematic similarity is invoked on the assumption that similar flow patterns can be specified by congruent velocity triangles composed of peripheral speed U and absolute fluid velocity V at inlet or exit to the vanes. This allows for the definition of a flow coefficient, expressed in terms of impeller diameter D I and angular velocity ω: Q CQ � �� 3 (12.12) ωDI The reciprocal of the Euler number (ratio of pressure force to inertia force) is the head coefficient, defined as Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
A power coefficient can be defined gH CH � � ω2D� 2 I (12.13) CP � �� 5 (12.14) ρω For transient analysis, the desired parameter for the continuous prediction of pump speed is the unbalanced torque T. Since T � P/ω, the torque coefficient becomes CT � �� 5 (12.15) ρω Traditionally in hydraulic transient analysis to refer pump characteristics to so-called rated conditions—which preferably should be the optimum or best efficiency point (BEP), but sometimes defined as the duty, nameplate, or design point. Nevertheless, in terms of rated conditions, for which the subscript R is employed, the following ratios are defined; Q Flow: v � �� speed: � � � ω Q ω � � N �� head: h � H �� torque: β � T �� N H T R R R Next, for a given pump undergoing a transient, for which DI is a constant, Eqs. (12.12–12.15) can be written in terms of the above ratios v CQ �� C � � � α QR ω � �� Q ω h CH �� H �� α2 C H ω2 R � ω2 β CT � α2� � �� T �� C T ω 2 R �2 ω QR HYDRAULIC TRANSIENT DESIGN FOR PIPELINE SYSTEMS R Hydraulic Transient Design for Pipeline Systems 12.11 HR P 3DI T 2DI 12.4.3 Abnormal Pump (Four–Quadrant) Characteristics The performance characteristics discussed up to this point correspond to pumps operating normally. During a transient, however, the machine may experience either a reversal in flow, or rotational speed, or both, depending on the situation. It is also possible that the torque and head may reverse in sign during passage of the machine through abnormal zones of performance. The need for characteristics of a pump in abnormal zones of operation can best be described with reference to Fig. 12.5, which is a simulated pump power failure transient. A centrifugal pump is delivering water at a constant rate when there is a sudden loss of power from the prime move—rin this case an electric motor. For the postulated case of no discharge valves, or other means of controlling the flow, the loss of driving torque leads to an immediate deceleration of the shaft speed, and in turn the flow. The three curves are dimensionless head (h), flow (v), and speed (α). With no additional means of controlling the flow, the higher head at the final delivery point (another reservoir) will eventually cause the flow to reverse (v � 0) while the inertia of the rotating parts has maintained positive rotation (α � 0). Up until the time of flow reversal the pump has been operating in the normal zone, albeit at a number of off-peak flows. To predict system performance in regions of negative rotation and/or negative flow the analyst requires characteristics in these regions for the machine in question. Indeed, any peculiar characteristic of the pump in these regions could be expected to have an influence on the hydraulic transients. It is important to stress that the results of such analyses are critically governed by the following three factors: (1) availability of complete pump characteristics in zones the pump will operate, (2) complete reliance on dynamic similitude (homologous) laws during transients, and (3) assumption that steady-flow derived pumpcharacteristics are valid for transient analysis. In vestigations by Kittredge (1956) and Knapp (1937) facilitated the understanding of Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. R R TR R R
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