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.Some additional losscoefficients from specific valve manufacturers and coefficient values as a function of theamount of the valve opening can be found in Appendix C.Table 2.5 Loss Coefficients for FittingsFittingKLGlobe valve, fully open10.0Angle valve, fully open5.0Butterfly valve, fully open0.4Gate valve, fully open0.23/4 open1.01/2 open5.61/4 open17.0Check valve, swing type, fully open2.3Check valve, lift type, fully open12.0Check valve, ball type, fully open70.0Foot valve, fully open15.0Elbow, 45o0.4Long radius elbow, 90o0.6Medium radius elbow, 90o0.8Short radius (standard) elbow , 90o0.9Close return bend, 180o2.2Pipe entrance, rounded, r/D < 0.160.1Pipe entrance, square-edged0.5Pipe entrance, re-entrant0.8An abrupt contraction has first a region of accelerating flow, followed by a region ofdecelerating flow caused by flow separation.Though the region of accelerating flow maybe larger, the head loss is attributable principally to the deceleration and separation whichoccurs immediately downstream from the contraction.The local loss coefficient for a pipecontraction is given in Fig 2.3.© 2000 by CRC Press LLC0.50.4KL0.30.20.10.20.40.60.81.0DD21Figure 2.3 Local loss coefficient for a sudden contraction as a function of diameter ratio.2.3 PUMP THEORY AND CHARACTERISTICSThe addition of mechanical energy hm = hp per unit weight to a fluid stream is accom-plished by pumps, as was mentioned with Eq.2.3.Although positive displacementpumps sometimes play a role, by far the more important class of pumps contains arotating impeller to inject energy, in the form of an increased pressure head, into theflowing fluid in the pipe.The characteristic shape of the impeller varies with the operatingregime of the pump.The energy addition is called the net head hp of the pump.Thewater power Pw that is delivered to the fluid stream is the product of the net head, thedischarge, and the unit weight of the fluid, or Pw = Qγ hp.The mechanical power tooperate the pump must be larger; it is called the brake horsepower or bhp = Tω, inwhich T and ω are the torque and angular velocity of the pump drive shaft.The ratio η= Pw/ bhp is the pump efficiency, which may be larger than 0.8 for large and/or efficientpumps that are operating near their best efficiency point (bep), also called the designpoint, but which may be much lower for small, old or worn pumps.Pumps are sufficiently complex that they cannot be designed on the basis of theoryalone.To refine a new or revised design, model experiments are first conducted, and aftersuccess is achieved with the model, then the full-scale or prototype pump is built.Theresults of dimensional analysis are used to relate the model and prototype to each other.First we assume that the model and prototype are similar in shape, called geometricsimilarity, and second that the velocity fields also have a similar shape, called kinematicsimilarity.Devices satisfying these requirements are called homologous.Thenondimensional parameters that are used to complete the scaling process are called affinityor scaling laws.They are three in number and are called the head, discharge, and powercoefficients CH, CQ, and CP, respectively:ghpCH =;CQ =Q ; CP =P(2.28)N 2 D 2ND 3ρ N 3 D 5The diameter of the rotating impeller is D.These coefficients may be computed in anyconsistent set of units.If plots of one nondimensional coefficient vs.another are© 2000 by CRC Press LLCconstructed, homologous units having different sizes and/or rotative speeds can be related toeach other.Or one can say for homologous units thathph=p Q P;=Q;=P(2.29) N 2 D 2 1N 2 D 2 2ND 3 1ND 3 2 ρ N 3 D 5 ρ N 3 D 5 12In a way these relations are more versatile than Eqs.(2.28) because the units no longermust lead to a truly nondimensional group so long as each variable is measured in thesame units.Thus rotative speed can be in rad/s, rev/s or rev/min.If pumps 1 and 2have the same diameter, Eqs.2.29 show how hp, Q, and P respond to changes in N,or for fixed N we see how the variables scale with the diameter D.The specific speed NS is a parameter for homologous pumps that contains theimportant pump variables, the discharge Q and head hp, without containing the unit sizeD; different ranges of this parameter therefore capture the essential differences in shape, notmere size, that separates the performance of one kind of pump from another type of pump.The nondimensional form of pump specific speed, with N in rad/s, isNS = NQ 1 / 2(2.30)gh( p)3 / 4In the United States, however, for many years it has been customary instead to use()( gal / min)1 / 2N' = rev / min(2.31)Shp ( ft)[]3 / 4which is clearly far from dimensionless.Based on specific speed, pumps can be classifiedinto three categories, based on impeller shape, as given in Table 2.6.Table 2.6Pump Type vs.Specific SpeedRadial FlowMixed FlowAxial FlowNSNS < 1.461.46 < NS < 3.73
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