charts for water hammer in low head pump discharge

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harts for water ham mer in low head pum p discharge lines resulting from watercolumn separation and check valve closureEUGENRUUS,B R YAN AR NEY,N D F AR OUK . EL-FITIANY

Depczrtmerlr oJ Civil Engineering. UniversiQ of ritish Colurnbin. V ancouver, B.C ., Ccznadn V6T W 5Received January 13, 1984

Revised manuscript accepted June 13, 1984Maximum pressure head rises resulting from water column separation and check valve closure are calculated and plottedfor a simple low head pum p discharge line with one well-defined high point. Basic parameters suc h as pipeline con stant, pipewall friction, complete pump ch aracteristics, pump, inertia constant, and the relative location of the high point ar e accountedfor in the analyses. Th e results of this paper can be used to determine (a) when w ater column separation is expec ted, (b) howto avoid water column separation, and (c) the necessary wall thickness in cascs where no protection against water columnseparation is provided. C omputer studies indicate that both the vertical an d horizontal location of the high point a s well as thepipe friction, the pipeline con stant, and the pump inertia have a major effect on pressure head rises. W ater column separationdoes not always constitute a danger to the pipeline.Key words: waterhammer. water column separation, check valve closure, pressure rise, pump discharge line, chart.

Les surpressions maximales resultant de la separation de la colonne d'eau et de la fermeture d'un clapet de retenue sontcalculCes et mises en graphique s dans le cas d'une conduite simple, com portant un point haut bien dkfini, alimentee par unepompe developpant une faible hauteur de charge. L'analyse tient compte des parametres fondamentaux tels que la constantede la con dui te, le frottcmcnt la paroi. I'cnscmblc dc s caractcristiqucs dc la pom pc. la cons tantc cl'incrtic dc la pompc ct laposition rclativc du point haut dc la conduitc. Les rcsultats contcnus dans cct article pcuvcnt Ptrc utilisks pour detcrmincr: (a)sous quelles conditions la separation d e la colonne d'eau est susceptible de se produire, (b) com ment Cviter la separation dela colonne d'eau et (c) 1'Cpaisseur de paroi requise dans les cas oh il n'existe aucune protection con tre la separation d e la colonned'eau. Les etudes montrcnt que les coordonnees verticale et horizontale d u point haut de m me que le frottement dans laconduitc, la constantc dc l conduitc ct I'incrtic dc la pompc joucnt un r61c important dans la valcur dc la surprcssion. Laseparation de la colonnc d'cau ne constituc pas toujours un dangcr pour l conduitc.Mots cle s: conduitc de refoulement d 'un e pom pe, cou p de bitlier, diagrammcs, fermcture de clapet, separation d' un e colonned'eau, surpression. [Traduit par la revue]

IntroductionWater column separation is likely to occur whenpumps fail in an unprotected low head installation thathas a long pipeline carrying water at high velocity.Man y pipe failures in low head installations have beendirectly ascribed to wa ter column separation as the maincause or as the aggravating condition that led to thefailure. It is therefore of interest to know which circ um-stances produce water column separation and what theresulting max imum pressure rise would be. In low headinstallations the steady state pressure head is often onlya fraction of the allowable pressure head, the m inimum

pipe wall thickness being determined by considerationsof wear, corrosion, handling, etc. I t may therefore bemore econom ical in some cases to make the pipe strongenough to withstand the consequences of water colum nseparation, rather than making provisions to avoid thecolumn separation.When the minimum total pressure head at a pointalong the pipe falls to vapour pressure head, watercolumn separation is initiated. The minimum total head

is obta ined by subtractin g the largest pressure head dropfrom the initial steady state pressure head. Th e pressurehead drop is caused by the pump slowdown and thecorresponding discharge reduction during the down-surge condition that is initiated by power failure. Itfollows that for a given velocity reduction and the re-sulting pressure head dro p, the vapour pressure is morelikely reached with a lower initial head than with ahigher one. Therefore low head installations are moresusceptible to water column separation t han high headinstallations.Wat er colu mn separation is most likely to occ ur in aknee in the profile in an unprotected pipeline. A sketchof a typical pump installation involving a high point isshown in Fig. 1Upon power failure, the speed of the pump, the dis-charge through the pump, and the pumping head de-crease rapidly. When the vertical distance between thefalling hydraulic grade line EDFW and the point Areaches approximately 9.6 m (31.50 ft) water columnseparation is initiated at this point. Th e cavity forming


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Ste a d y . to ts hy dr a ul ic gr a de l ine7 10

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FIG. 1 Sketch of installation.in the top portion of the pipe knee is filled mainly withwater vapour. In the bottom portion of the pipe waterstill flows for a while in the forward direction see detailA in Fig. I).After the cavity has formed at A, the column APdecelerates rapid ly, while the hydraulic grade line at thepump drops from E to S and eventually the check valvecloses upon flow reversal through the pump. The de-celeration of the column AB continues at approximatelythe same rate as that at the instant when water columnseparation starts. The hydraulic grade line DFW risesgradually to DGW and remains unchanged during theexistence of the cavity. After a while the column ABcomes m omentarily to a stop, while the cavity reachesits maximum volume. Thereafter the column AB re-verses its motion and the cavity reduces in size. Sub-stantial velocity in reverse can be reached by the col-umn AB before the cavity collapses. The sudden pres-sure head rise, at the instant of cavity collapse at A, isequal to a/2 g)A VThe collapse of the cavity causes two pressure wavefronts to form in the pipeline, one travelling upstreamand the other downstream from point A. T he maximumpressure head rise AH,,, usually occurs at the pumpend when the upstream travelling wave front is reflectedfrom the closed check valve. Ho wever, these events canbe further complicated by the closu re of the check valv eitself, which causes its own pressure wave to be super-imposed on the waves resulting from the cavity col-lapse. If the reflection of the upstream travelling wavefront coincides with the check valve closure, a pressurepeak of even greater magnitude can occur.

In the past, semigraphical proc edure s Berg1961; Gandenberger 1950) were used to determpressure rises resulting from water column separaMore recently, computerized techniques WylieStreeter 1983; Chaudhry 1979) have been develand attempts have been made to allow for the presof air during water column separation.Distributed air bubbles in the initial steady stawell as air release during the existence of cavitysubstantially reduce the pressure peak caused byjoining water columns at the instant of cavity collaThis reduction is, however, not assured. First, thcontent in the pipeline cannot yet be reliably prediIn addition, during an extended shutdown air bubrise to the crown and coalesce there into air pocwhich travel along the pipeline and collect at the mits if air release valves are not provided. The mpart of the cushioning effect of the distributed air bles is therefore lost; indeed, the air pockets atsummits can now substantially increase the traneffects. Present design practice is to prevent the tence o f air pockets by installing air release valves asummits. Similarly, the effect of the air release intcavity can vary widely. When the cavity extends a considerable length of pipeline, the active watervapour pocket interface becomes substantial. Wsuch a condition occu rs, a major amou nt of air relcould result. However, a very localized high pointline with short cavity duration could yield only a mair release. In view of these uncertainties, it is advisto ignore any reduction of the pressure peak causeeither distributed air bubbles or air release until mo


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RUUS ET AL. 7 1known about their existence and behaviour.In this paper a systematic study is presented com-bining the effects of water column separation and checkvalve closure on the maximum pressure head rise, bothinitiated by power failure.

ain parametersThe main parameters of the pipeline are given belowin classical nondimensional form (Allievi 1925). Thepipeline constant

in which a = water hammer wave velocity, V, = initialsteady state velocity in the pipe, g = acceleration ofgravity, and H , = rated pumping head. H, = H , H iH I H , , w he re H o = static head, H I = head loss due FIG. 2 . Pump characteristics. normal p u m p operation.to pipe wall friction and viscosity in the initial steadystate , H I= sum of local losses (check valve, gate valve,pump header junction, e tc.), and H , = velocity head in

ratioinitial steady state.Th e pipe wall friction, local losses, and velocity headare lumped together and considered in the non- The complete characteristics of a pump (Donsk1961;Wylie and Streeter 1983) with a specific speed N

= 147 (7600) rpm are used in the zone of normal pumoperation (see Fig. 2).To adapt the pump characteristics fo r computer usethe dimensionless ratios

dimensional form

in which friction H, = f ( L / D ) ( ~ 8 / 2 ~ )yDarcy-Weisbach, where = friction factor, L =length of pipe, D = pipe diamete r, local loss H I =K V ; / ~ ~ ,h5re K = local loss coefficient, velocityhead H , = V,/2g, and h, = total relative head loss.The pump constant is

are emplo yed. In these ratios H , T, Q, and N are thtransient head, torque, discharge, and speed. The subscript r indicates these quan tities at the rated conditionWith the dimensionless ratios for homologous pumpsand with the transformation proposed by Marshal et a(19 6 3 , the expressionsto be used in the equation of angular deceleration

in which w = unit weight of water, Q, = rated dis-charge through one pump, WR2= moment of inertia ofrotating parts of one unit including water in the im-peller, q, = pump efficiency at the rated con dition, N,= rated pump speed, A a = (N , N2)/Nr is the non-dimensional speed reduction during a sm all time inter-val A t N l and N2 = transient pump speeds at thebeginning and at the end of the time interval A t and P= average relative torque.The nondimensional time is

rr tan- lare obtained for the relative head W H and the torquWB . T he data are available in tabular form (Wylie anStreeter 1983 ) for equal intervals of = .rr/44 rad, anare used to plot the characteristics shown in Fig. 2.Th e relative horizontal location of the high point A iFig. 1 is designated byin which X = distance from the pump end, measurealong the pipe.The relative elevation of the high point is designateby

in which t = time in seconds and p = 2L la is the timeconstant.The pump constant K l and the time constant an beconveniently combined into a single dimensionless


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FIG. 3. Sketch of high point locationin which Y vertical distance from the elevation ofwater surface in the sump to the elevation of the highpoint see Figs. and 3).

Basic assumptionsIn the anal yses, it is assumed that a) the pipe di-ameter and wall thickness are constant, b) the checkvalve is located near the pump, c) the closure of checkvalve occurs suddenly at the instant of flow reversalthrough the pum p, d) only one localized high pointexists see Fig. 3) in which water column separationoccu rs, e) no dispersed air bubbles exist in the initialsteady state, no air pocket exists in the high point priorto or during water column sep aration, and no air releaseoccurs in the cavity during water column se paration, f)the velocity head and local losses are lumped togetherwith pipe wall friction, g) the difference between theatmospheric pressure head and the vapour pressure headis equal to 9.6 m 31.50 ft), e.g. water column sepa-ration starts at a pressure head of -9. 6 m , h) thesuction piping is short and can be disregarded, i) thepump is working in the rated condition in the initialsteady state, i.e. at the point of best efficiency.

Calculations and resultsTw o partial differential equations describing the un-steady flow are transformed into four total differentialequations by the method of characteristics. Two sepa-rate computer programs are written to solve these totaldifferential equations with appropriate boundary condi-tions for the ran ge of p T and h f ormally encounteredin pumping systems using the approach of specifiedtime intervals Wylie and Streeter 1983). In the calcu-lations for the maximum pressure head rise, the pipe isusually divided into 40 equal reaches of Ax In caseswhere no column separation occurs, runs with 20reaches are made. A time interval At Ax/a is usedthroughout the calculations.For each separate combination of r r 2 hf and Tone maximum pressure head rise is calculated. This

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max imum , which usually occurs at the pum p end, musbe considered for the entire pipeline.The first of the two programs analyzes the basisystem consisting of the pump, the check valve, thpipe, and the reservoir for a power failure conditiowith the check valve closing upon the flow reversathrough the pump. No high point exists and no watecolumn separation is considered.This program calculates firstly the max imum downsurge at the pump end, at the midpoint, and at ththree-quarter point near the reservoir. The results arshown in Figs. 4-6, which give the downsurge in termof the rated pumping head H , on the T-p plane fovarious pipe friction values h at these locations. Separate curves are drawn in these figures for h f 0 . 0 , 0 . 20. 4, and 0.6 . Intermediate values of p ipe wall frictiocan be considered by interpolating between the curvesThe curves are only slightly smoothened computeplots. Th e numbers on the curves indicate the maximumrelative pressure head dro p and are measured downwarfrom the initial steady state pressure head at the pumpUsing the plotted values of maximum pressure headrop at the pump end, midpoint, and three-quarter poinalong with the value at the reservoir end, which has pressure head drop of zer o, an enveloping curve can bdrawn. This curve will show the maximum pressurhead drop along the entire pipeline. This curve, wheplotted on the profile of the pipeline, will enable thdesigner to see in one glance, and with good accuracwhether or not water column separation is expected any point along the pipeline.Secondly this program calculates the maximum upsurge at the pump end. Th e results are shown in Fig. 7which gives the maximu m pressure head rise in terms othe pumping head H,. Th e numbers on the curves indcate the maximum relative pressure head rise and ameasured upward from the initial steady state pressurhead at the pum p. In this case the smoothe ned computplots are quite accurately shown as straight lines. Thmaximum relative pressure head rise at the pump approaches 1.0 for low T and high p values for the case frictionless flow. For c ases of large friction , this rise drasticalIy reduced.The results for frictionless flow were compared at thpump end and at the mid-length with those reported bDonsky 1961). obtained using the same pump charateristics and the graphical method of water hammanalysis. Very good agreement was observed at bodownsurge and upsurge conditions. Because of the sloand gradual variation of the maximum pressure hearise obtained from this program, only a limited numbof points needs to be calculated.The second program includes the analysis of watcolumn separation, which occurs in the high point. Thprogram is identical with the first one in all other r


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spects. In the second program, each of the relativevalues of pipe friction h 0 .0 , 0 .2 , 0 .4 , and 0 .6 i sconsidered together with up to 97 values of the pipelineconstant p ranging from 2.0 to 16.0 and with 33 valuesof the inertia parameter T ranging from 0.125 to 2.0.Over 10 00 0 extreme pressure rises are calculated foreach of the 13 high points considered (F ig. 3). Theresults of the second program are superimposed onthose of the first program and presented as curves ofequal maximum pressure head rise on the double loga-rithmic graphs in Figs. 8- 20 . In this way the pressurehead increase due to water column separation can beeasily identified.The curves shown in these figures are hand-drawnenveloping curves based on the computer plots. Asample of the computer plot with the correspondinghand-drawn enveloping curves for a case includingcolumn separation is shown in Fig. 21.Th e graphs in Figs. 8-20 have been derived for atotal pumping head H and a barometric pressure headHh equal to 12 ni (39.3 7 ft) and 9. 6 m (31 .50 ft) re-spectively. The numbers on the curves indicate themaximuni relative pressure head rise in terms of thispumping head and are measured upward from this ini-tial steady state pressure head at the pump.In pipelines with large friction h ,> 0.4). the up-surge may never reach the initial steady state pressurehead at the pump. In such cases a zero pressure headrise is indicated on the graphs.

A large-scale analysis of water colunln separation,using the data of the same punip and the graphicalmethod of water hamnier analysis was carried out by th ewriters to spot check the validity of the results obtainedby the coniputer program. Very good agreement wasobtained.Discussion of results

Smooth and gradual variation in the maximum pres-sure head drop and rise at the pump end is noticed onthe double logarithmic plot in the studied range of vari-ables p T and, h r when no water column separationoccurs (see Figs. 4-7). On the other hand, large localpressure head changes can occur whenever water col-umn separation is present. The largest pressure rises insuch cases usually occur at high p values combined withlow T values. Large deviations, howe ver, appear in thisgeneral pattern, depending on the horizontal locationand the elevation of the high point and on pipe friction.Little or no water column separation is experiencedwhen low p values are combined with high T values.The check valve closure at the instant of flow reversaldetermines the maximum pressure head rise in the lattercases.With the graphs presented in this paper, one caneasily see which combinations of p T and h lead to

T AL 7water column separation in a given high point and whis the resulting pressure head rise. The graphs also giva clear indication of how to change p and T to avocolumn separation.Th e maxim um pressure head change depends largeon the pipe friction. Linear interpolation for friction sufficient in many cases, but parabolic interpolatioshould be used for the regions of rapid pressure heavariations. Parabolic interpolations in the vertical anhorizontal direction a re also needed wh en considerinhigh point locations other than the ones examined in thpaper.The graphs give the pressure head rises quite accurately for the basic data used in the analyses. Howeveone must bear in mind that p T a n d h values ausually not accurately known in field conditions. Iaddition, the characteristics of the pump niay be sustantially different from those used in derivation of thgraphs because of different specific spe ed' and differedesign o f the pump. ~ o r e o ~ e rhe dou ble interpolatiorequired for the high point (i n the vertical and horizontdire ctio n) and the interpolation required for friction itroduces additional inaccuracies. Therefore a margfor possible variation for variables p T and h must allowed, particularly in areas where large pressupeaks occur. For both p and values a ma rgin of at lea? 10 is suggested.The graphs. in particular those with r 0.75 anr 0. 50 , show that the largest p value (correspondinto the maxinium velocity) does not necessarily result the maximum pressure head rise. A p value substatially less than p ? 10 could be criti cal, and therfore partial flows must be considered whenever suflows ca n occur for any considerable duration of pumoperation. When doing this, the change in K I , causeby changes in Q H and qr needs to be considerePipe friction 1 2 will also change for reduced flows ansome consideration should be given to the loss of tvalve at the outlet (if oresent).Th e grap hs are intended primarily for the prelitninadesign for large pipelines having only o ne well-definehigh point, where only a localized column separatiooccurs. Th e graphs should be sufficient for final desigof small pipelines, if an ample margin is applied.com pute r program should be used to analyze a pipeliwith two or more high points and in cases where tcavitating flow extends over a substantial length opipe. s uc h programs shou ld include so m e consideratioof air release.

'According to Kinno and Kennedy (Kinno and Kenned1965) the pressure drop in the downsurge condition is rathinsensitive to the specific speed and design of thc pumTherefore only a small inaccuracy is expected in downsurgbecause of these reasons.


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W o l e r columns r e j o i n 4I

;v v v v v\low reverses I01 column ACV

ot c o lu m n AB+

T I M E IN I N T E R V A L S OF A F T E R POWER F A I L U R EFIG.22. Vclocitics of upstrcam and downstrcam watcrcolumns aftcr scparation.

ressure head peaksWhen no water column separation occurs, the vari-ation in pressure head rise is fairly smooth and gradualin both directions on the and p plane (see Figs. 4-7).Whenever water column separation does occur. thewhole area on computer plot that is influenced by watercolumn separation is covered by individual, more orless sharp pressure head peaks, separated by valleys oflower pressure head rise. Most of these peaks are muchsharper and more pronounced than those shown inFig. 21. These pressure peaks are usually representedby envelop ing curves in a fashion indicated in Fig. 2 1.The distribution of the peaks is rather comple x. De-pending on the location of the high point, many individ-ual pressure head peaks appear in groups, which areoriented as straight lines predominantly in the directionof the T-p axis. For a given value, there can be up to16 distinct pressure head peaks. Figure 18 show s mostof these peaks for h - 0.2. A few of the groups areoriented in some intermediate direction, and for thehighest point investigated at three-quarter length of pipewith h 0.4, the pressure peaks are lying on a curve(Fig. 18). In numerous case s, pressure peaks also occurat random.The compu ter plots indicate clearly that friction hasa large effect on the location and magnitude of thepressure head peaks. In general, a relative friction

h r 0.2 results in highest pressure head rises. Thepressure head peaks shown in Figs. 8-20 are indicatedindividually where these are pronounced and located farapart on the computer plot; enveloping curves aredrawn when these peaks are less pronounced and closetogether.

A detailed explanation for the number, location, the orientation of the pressure head peaks is diffito give because many variables are involved. A general comments follow.Figure 22 shows the typical relative velocities ofwater columns d uring a vapour cavity. After the bening of column separation, the velocity of the dostream water column AB reduces until the colucomes to rest. then reverses its motion and reachsubstantial velocity in reverse at the instant of cacollapse. During this time, the velocity of the wcolumn ACV is also reduced to zero and this colucontinues to oscillate around zero veloc ity as indicin Fig. 22. The period of this oscillation depends onvalue and the amplitude on the p value.T he imm ediate pressure head peak at the collapsa vapo ur cavity depend s on the velocity difference tween the water columns on either side of the cavFor a case where the high point is located near reservoir end ( r , 0. 75 ), the column AC V is relativlong and therefore the vapour cavity could exist fonly a fraction of the period to may be 2-3 periodthe oscillation of column ACV. The magnitude ofoscillating velocity may reach a substantial fractiothe initial steady state velocity. Depending on the ration of vapour cavity, the rejoining of the depawater columns can occur at any velocity betweenmaximum and the minimum velocity of the oscillacolumn ACV. From Fig. 22, it appears that the mimum (positive) velocity will result in much greimme diate pressure head rise than the minimum (netive) velocity. This explains to a large extent why,r l 0.7 5, many shar p and pronounced pressure hpeaks occur in the area influenced by water coluseparation on the T-p plane.In the other extreme, for cases where the high pand the water column separation occurs very nearpump (r , 0.05) the column ACV is short. Hemany velocity oscillations of short period and smamplitude occur during the existence of vapour cavTherefore it makes no substantial difference whethe velocity of the colu mn ACV is positive or negaat the instant of cavity collapse. Consequently onfew less pronounced pressure head peaks occur inundulating pressure head surface for r , 0.05.High points at the quarter length, r , 0.25, anthe mid-leng th, r l 0. 5, cause more pronounced psure head undulations. At these points, the preshead peaks are more frequent and substantially shathan those at r, 0. 05 , although they are more mthan the peaks at the three-quarter length with r0.75.Whe reas the numb er and the intensity of the preshead peaks increase as the location of the high pmoves in the downstream direction, the general psure head .intensity increases when the location of


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high point moves in the upstream direction.The pressure peaks discussed above are modified bycheck valve closure, which is nearly independent ofcavity collapse and can occ ur before, after, or sirnulta-neously with a cavity collapse. Most of the pressurepeaks resulted not from the first cavity collapse butfrom the first cavity collapse that followed check valveclosure.In cases where the high pressure head rise resultsfrom the second , third, or a later cavity collapse, someof the basic assumptions do not hold rigidly any longer.Thus, in reality, the cavity may not be localized to apoint anymore, but could extend over a considerabledistance. Therefore in such cases the actual pressurehead rise is probably somewhat smaller than that indi-cated-by the curves.

R U U S ET A L . 74

How to avoid water column separationToo high an elevation of the high point is the primarycause for water column separation. Such a high point ismost critical near the pump end of the pipeline.Pipeline constant p is the most important of pipeparameters when water column separation is a problem .A high p value resulting from a high steady state watervelocity in the pipe usually causes a high pressure headrise. When calculating the economical pipe diameter,water hammer should therefore always be considered.This would result in a lower initial steady state velocitythan otherwise would be the case.Of pump param eters, the inertia of the motor can bevaried without undue change in cost. Ultralight modernelectric motors should be avoided whenever wa ter col-umn separation could occur. Column separation is elim-inated at the majority of the points studied when theinertia parameter T of the pumping unit is 0.50 or more .In addition, a lower steady state speed of the unit couldbe conside red. This results in a heavier pump motor anda larger T value.T o eliminate the danger of particularly high pressurehead rises resulting from column separation, avoid theareas of sharp pressure head peaks that occur in highpoints located at the mid-length and at the three-quarterpoint of pipelines with extremely low T values.Finally, the economical analysis will indicate wheth-er or not i t is advisable to lower or to eliminate the highpoint, increase the pipe diameter, increase the intertia

of the motor, strengthen the pipe to withstand the pres-sure head rise, provide protective devices, or to use acombination of these provisions.Pumping heads other than 2 m

The graphs Figs. 8-20) have been drawn for a totalpumping head of H , 12.0 m 39.3 7 ft) and a bar-ometric pressure head of H h 9.6 m 31.50 ft). Thesegraphs can be used for other pumping heads H , andbarometric heads H b in cases where the variables p h r

T BLE. Relative height of high point

For H , 9 m 29.53 f t)1.47 13.2 43.3 1)1.27 11.4 37.40)1.07 9.60 31.50)0.87 7.80 25.59)For H , 12 rn 39.37 ft)1.20 14.4 47.24)1 .oo 12.0 39.37)0.80 9.60 31S O0.60 7.20 23.62)

For H , 15 rn 49.21 ft)1.04 15.6 51.1 8)0.84 12.6 41.34)0.64 9.60 31S O

T , and r I are changed accordingly. The cha nge s to p, I? and T do not need any explanation. The new value forz is calculated as follows:

in which ri-. relative elevation of the high point ia pumping system with pumping head H , and barometric pressure head H , equal to 12 m 39.3 7 ft) an9. 6 m 31.5 0 ft) respectively, rF.Hh actual relativelevation of the high point in a pumping syste m with thpumping head H , other than 12 m and a barometripressure head Hb other than 9.6 m.The graphs are based on relative high point elevatiorI values of 0 .6 ,0 .8 ,0 .9 , l .O, and 1.2. Th e corresponding r z values for 9 m 29. 53 ft) and 15 m 49.2 1 ft) heapumping systems are shown in Table 1. Below the rvalues in this table are printed the corresponding higpoint elevations Y measured in feet from the sumlevel. Intermediate values for both r , and Y can binterpolated.

ExampleConsider the installation shown in Fig. 1 having thfollowing data: H , 15 m, H , 9 m , H,. HIH , 3 m , h r 0.20, p 5.0, T 0 . 4 7 , r , 0.25and r f 5 . L 0.7 .Find a) the relative elevation of high point ri2 .h tenter the graphs of Figs . 8-20, which are based o

H , 12 m and H , 9. 6 m; and b) the maximumpressure head rise caused by water column separationThe relative elevation of high point is


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742 C A N . J . CIV EN G .For the max imum pressure head rise, a relative valueof 1.50 is read from Fig. 12 for h = 0 .2 , p = 5 .O and

T = 0.47. Therefore a maximum pressure head rise of1.50(15) = 22.5 m must be allowed for when calcu-lating the required strength of the pipe.Conclusions

( I ) Water column separation occurring in a highpoint can yield a dangerous pressure head rise in thepipeline. In the studied range of p and T water columnseparation occurs in all high points when low T valuesare combined with some p value between 2 and 16.(2) The vertical elevation of the high point has theprimary influence on water column separation and onth e resulting maximum pressure head rise.(3) Sharp pressure head peaks occur at nearly allhigh point locations studied. The magnitude and thelocation of the pressure head peaks on the p-T planedepend on the elevation and the horizontal location ofthe high point and on pipe friction and pump inertia.Pressure head rises higher than nine times the initialsteady state pumping head can occur in these cases.(4) Very little or no water column separation occursin the high points considered when low p values arecombined with high T values.(5) Water column separation does not alway s signala danger for the pipeline. In many case s, column sepa-ration only marginally increases the pressure head riseabove that produced by check valve closure. However,the hissing and thumping noise that may occur may beobjectionable.6) Large friction in general reduces the pressurehead rise caused by water column separation and some-times eliminate s the separation.(7) Low head installations are more susceptible towater column separation than high head installations.

AcknowledgementsThe a uthors are grateful t o the Natural S cienc es andEngineering Research Council of Canada for financialsupport of the research, and to Mr. R. Brun, who didthe large amount of drafting involved.

ALLIEVI, . 1925. Theory of water hammer (translated byE. E. Halmos). Riccardo Garoni, Rome.BERGERON,. 1961. Water hammer in hydraulics and wavesurges in electricity. John Wiley So ns , New York, NY.CHAUDHRY,. H. 1979. Applied hydraulic transients. VanNostrand Reinhold, New York, NY.DONSKY, . 1961. Complete pump characteristics and theeffects of specific speeds on hydraulic transients. Journal ofBasic Engineering, 83, pp. 685-699.GANDENBERGER,. 1950. Grundlagen der graphischenErmittlung der Druckschwankungen in Wasser versorgung-

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sleitungen. Verlag von R. O ldenbourg, Munich, GermKINNO, ., and KENNEDY,. F. 1965. Water hammer chfor centrifugal pump systems. ASCE Journal of Hydraulics Division, pp. 247-270.MARCH AL, ., FLESCH, ., and SUTE R, . 1965. The calation of water hammer by means of the digital compInternational Symposium on Water Hammer in PumStorage Projects, ASME, pp. 168-200.WYLIE, E. B., and STREETER,. L. 1983. Fluid transiFEB Press, Ann Arbor, MI.

List of symbolswater hammer wave velocity (m/ s)

tt

voWR'Y

aa

lrIJ

pipe diameter (m)friction factoracceleration of gravity (m/s2)transient pumping head (m)barometric pressure head (m)head loss due to wall friction (m)sum of local losses (m)static head (m)rated pumping head (m)velocity head (m)relative transient pumping headrelative head losslocal loss coefficientpump constantpipe length (m)transient pump speed (rpm)rated pump speed (rpm)transient pump discharge (m3/s)rated pump discharge (rn.'/s)relative horizontal location of high pointrelative elevation of high poin ttransient p um p torque kN m )rated pump torque (kN.m)time (s)relative timetime interval (s)relative transient pump dischargeinitial steady state velocity (m /s)moment of inertia of rotating parts ((kg.m3)unit weight of water (kN/m3)distance f rom pum p end (m)vertical distance from water surface to point (m)relative pump speedrelative spe ed reductionrelative pump torqueaverage relative torquepump efficiency at rated conditiontime constant (s)p pipeline consta nt

T inertia constan t

charts for water hammer in low head pump discharge

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