On the Use of the Water Hammer Equations with Time Dependent Friction during a Valve Closure, for Discharge Estimation

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1 Journl of Applied Fluid echnics, Vol. 9, o. 5, pp , 26. Avilble online t ISS , EISS DOI:.8869/cdpub.jfm On the Use of the Wter Hmmer Equtions with Time Dependent Friction during Vlve Closure, for Dischrge Estimtion G. Dunc, R. G. Iovănel, D.. Bucur, nd. J. Cervntes 2,3 University OLITEHICA of Buchrest, Romni 2 Luleå University of Technology, Luleå, Sweden 3 orwegin University of Science nd Technology, Trondheim, orwy Corresponding Author Emil: dmbucur@yhoo.com (Received July 2, 25; ccepted October 28, 25) ABSTRACT The pper presents new method for in site dischrge estimtion in pressured pipes. The method consists in using the wter hmmer equtions solved with the method of chrcteristics with n unstedy friction fctor model. The differentil pressure hed vrition mesured during complete vlve closure is used to derive the initil flow rte, similrly to the pressure-time (Gibson) method. The method is vlidted with numericl experiment, nd tested with experimentl lbortory mesurements. The results show tht the proposed method cn reduce the dischrge estimtion error by.6% compred to the stndrd pressure-time (Gibson) method for the flow rte investigtion. Keywords: ressure mesurement; ethod of chrcteristics; Unstedy friction fctor; Dischrge evlution; Hydropower. OECLATURE A cross-section re of the pipeline pressure wve speed B pressure mplitude corresponding to the fundmentl hrmonic of the free pressure oscilltion C - negtive chrcteristic C + positive chrcteristic C* vrdy s sher decy coefficient D pipe dimeter dh pressure hed difference dh stedy stte pressure hed difference E pipe wlls Young modulus e bulk modulus f friction fctor fq qusi-stedy friction fctor g ccelertion due to grvity H pressure hed h the oscilltion dmping decrement k brunone friction coefficient L length of the mesuring segment q Q Q Qref T t t t2 t3 t4 V x Δp ε μ ω ζ ρ lekge flow stedy-stte dischrge initil guess for dischrge reference vlue for dischrge the pressure wve period independent temporl vrible beginning of the nlysed time-history end of the initil stedy stte end of the trnsient stte corresponding to the forced flow rte chnge end of the nlysed time-history bulk velocity independent sptil vrible differentil pressure pipe wll thickness rbitrry vlue the circumferentil wve frequency pressure drop due to viscous losses density of the flowing liquid. ITRODUCTIO In the field of efficiency mesurement with ppliction to hydrulic mchinery, the dischrge is the most difficult prmeter to determine ccurtely. The more precise is its estimtion, the more relible the turbine hydrulic efficiency will be.

2 G. Dunc et l. / JAF, Vol. 9, o. 5, pp , 26. For in site dischrge mesurement, there re severl methods presented in the interntionl stndrd IEC 64 (99) which cn be pplied for dischrge mesurement in pipes: velocity-re method, pressure-time or trcer methods. The thermodynmic nd ultrsound methods my lso be used. All of these methods hve good ccurcy if used in the recommended conditions. The pressure-time method is the simplest in terms of costs nd requirements when pressure tps re vilble. The pressure-time method, Gibson (923), is bsed on the second lw of ewton, conservtion of momentum, nd consists in mesuring the pressure difference between two pipe cross-sections, during flow stop due to vlve closure for exmple. The dischrge is computed using the following eqution: t, () A Q p dt q L where: Q is the unknown dischrge before closure, A is the pipe cross-section re, ρ is the liquid density, Δp is the mesured pressure difference, ζ is the pressure loss between the two mesuring crosssections, L is the length between the two mesuring sections nd q is the lekge flow fter the vlve closure, present when some gps exist fter complete closure. The method requires certin conditions for good flow estimtion: the distnce between the two mesuring cross-sections (the mesuring length) must be greter thn m (L > m), the product between this distnce nd the stedy stte flow velocity must be higher thn 5 m 2 /s (V L >5 m 2 /s). The improvement of the dischrge determintion method hs concerned the reserchers in the lst yers. Jonsson et l. (22) developed the unstedy Gibson method by implementing n unstedy friction fctor, using the Brunone model. The obtined procedure ws tested nd vlidted in situtions outside the stndrd limittions, using numericl nd lbortory dt. Lter, the method ws vlidted with on-site experimentl tests by Dunc et l. (23). Admkowski nd Jnicki (2), Admkowski (22) showed tht the stndrd pressure-time eqution for dischrge estimtion doesn t consider the residul pressure oscilltions tht pper fter the vlve closure. In order to ccomplish tht, he introduced term tht modifies the integrtion upper limit for the estimtion of the dischrge in Eq. (). The results obtined by pplying this procedure were shifted systemticlly towrds lower vlue of the dischrge. This led to shift in the dischrge estimtion error, but did not increse the estimtion precision in ll nlysed cses. Further, Admkowski nd Jnicki (23) developed nother procedure tht considers both the liquid compressibility nd the pipe wlls deformbility vi the speed of sound,. It uses the wter hmmer equtions in which the quntity represented by the pressure hed H ws replced by the pressure hed difference dh. The method of chrcteristic (OC) ws used to solve the equtions. In the present work, development bsed on the method described by Admkowski nd Jnicki (23) using the wter hmmer eqution is proposed, by using n unstedy model for the friction fctor insted of constnt one. The numericl implementtion of the model is mde considering the most pproprite numericl scheme (explicit derivtive scheme or implicit derivtive scheme). The resulted procedure is tested on numericl set of dt nd vlidted with experimentl dt. The method ccurcy is compred to those chieved with the stndrd pressure-time nd unstedy pressure-time methods. 2. ETHOD The equtions describing the fst vrition of the flow in -dimensionl pressurized pipes re the continuity eqution nd the momentum eqution (Eq. (2) nd Eq. (3)). F2 V H H V (2) g x t x FV V H V V 2 V g f (3) t x x 2D where: x is the xil distnce, t is the time, V is the men flow velocity, H is the pressure hed, g is the ccelertion due to grvity, f is the friction fctor, D is the pipe dimeter nd is the pressure wve speed. The prmeter depends on the pipe wlls Young modulus E, liquid density ρ, nd bulk modulus e, pipe wll thickness, ε nd dimeter, D, ccording to the reltion: D ee. These equtions cn be solved using vrious numericl methods (Krney nd cinnis, 992, Chudlhry, 987). The method of chrcteristics (OC) is one of the most frequently used methods to solve this system. It consists in writing the two equtions s liner combintion of them, s: F F F 2 (4) where the prmeter μ hs n rbitrry vlue. Thus, Eq. (4) becomes: H t V g H x 2 V V V V V f t g x 2D (5) where two vlues for the prmeter μ re selected to obtin the totl derivtives of H nd V between the brckets. These re μ = ±/g nd then, ccording to Wylie nd Streeter (993), Eqs. (2) nd (3) cn be replced by two ordinry differentil equtions, i.e., Eq. (6) vlid long the positive chrcteristic, C + ( dx dt ), nd Eq. (7) vlid for the negtive 2428

3 G. Dunc et l. / JAF, Vol. 9, o. 5, pp , 26. chrcteristic, C - ( dx dt ), respectively: dh dt dh dt g g dv V V f dt 2gD dv V V f dt 2gD (6) (7) In this wy, reltion between the flow prmeters, V nd H, during the wter hmmer phenomenon is determined. To solve Eqs. (6) nd (7), n explicit numericl scheme is employed with dimond grid scheme for interpoltion, s recommended by Vitkovsky et l. (2). Becuse V <<, on grid s in Fig., the following discrete expressions re obtined long C + ( dx dt ) from Eq. (6), nd long C - ( dx dt ) from Eq. (7), respectively ccording to Bergnt et l. (2): H H t H H g g f x 2g D V V V V f x 2g D (8) V V V V (9) dh dh g A f x 2 2g D A Q Q Q Q () The model developed by Admkowski nd Jnicki (23) ims to obtin the dischrge flowing through pipe using the pressure hed difference mesured between two cross-sections, s the pressure-time method. In the method, the effects of liquid compressibility nd pipe wlls deformbility re considered using the Eq. () nd () vi the speed of sound,. The computtionl procedure implies defining certin moments in time, which chrcterize the wter hmmer trnsient phenomenon (Fig. 2): - t beginning of the nlysed time-history - t2 end of the initil stedy stte - t3 end of the trnsient stte corresponding to the forced flow rte chnge - t4 end of the nlysed time-history. Differentil pressure t t 2 t 3 t 4 Δt C + C - Time Δt Fig.. Chrcteristics in the plne xot. By solving the two equtions, the flow prmeters V nd H in the current point cn be obtined. The model proposed in the present pper is bsed on rewriting the wter hmmer clssicl equtions in the form presented by Admkowski nd Jnicki (23). The pressure hed H nd the flow velocity V re replced with the pressure hed difference, dh, between two cross-sections nd the dischrge, Q (Eq. nd Eq. ): - long the positive chrcteristic C + dh dh g A Δx f x 2 2g D A Δx Q Q Q Q - long the negtive chrcteristic C - x () Fig. 2. Differentil pressure vrition nd time definition. The vlues of t2 nd t3 correspond to the time intervl in which the flow is completely stopped. The time corresponding to the end of the trnsient stge, t3, is difficult to define being present reserch subject. In the clssicl pressure time method for dischrge determintion, the IEC 4 stndrd presents wy to estimte this finl integrtion time. Admkowski (22) proposed wy to determine this integrtion time by solving the definite integrl: ht B e cos( t) dt B h e h cos( ) sin( ) h 2 2 h (2) where B, 2 T h T ln B i Bi nd T re the pressure mplitude corresponding to the fundmentl hrmonic of the free pressure oscilltion, the circumferentil wve frequency, the oscilltion dmping decrement nd the pressure wve period (Fig. 3). The solution of Eq. (2) is the vlue of the time corresponding to the end of the trnsient stge, τ =t3., 2429

4 G. Dunc et l. / JAF, Vol. 9, o. 5, pp , 26. p B i T Bi Fig. 3. Free pressure oscilltion fter vlve closure. In the method presented by Admkovski, the friction fctor f is considered constnt. This hypothesis is cceptble for pipes with high roughness nd qusi-stedy-trnsient phenomenon, i.e., slow trnsient. For unstedy flows, there re severl friction models presented in the literture (Krney nd cinnis, 992, Bergnt et l., 2, Bhrni nd our, 24). Bergnt et l. (2) nlysed some of the friction models obtining the best results with the Brunone model. This model gve good results in other studies s Jonsson et l. (22) nd Dunc et l. (23). In the present work, the model is implemented in the method proposed by Admkowski to evlute possible improvement in the error ssocited with the dischrge estimtion. The Brunone model is described by Bergnt et l. (2). It consists in expressing the friction fctor f s: k D V V f f q (3) V V t x where fq is the qusi-stedy friction fctor, k is the Brunone friction coefficient, V t is the instntneous locl ccelertion nd V x is the instntneous convective ccelertion. The coefficient k cn be determined either by tril nd error method or nlyticlly using the Vrdy s coefficient (Vrdy s sher decy coefficient C * ), * k C 2, empiriclly clibrted. Coefficient C * is.476 for lminr flows while for turbulent flows is computed using the eqution: * C (4).5 log 4.3 Re Re 7.4 The qusi-stedy prt of the friction fctor, fq, is computed using Drcy eqution for lminr flow ( f q 64 Re ) nd the Hlnd eqution for turbulent flow: f q. D log 3.7 Re t (5) The Brunone model ws numericlly implemented by solving the time derivtive (locl instntneous ccelertion) nd the spce derivtive (convective instntneous ccelertion) with dimond grid in n explicit derivtive scheme Vitkovsky et l. (2). Using the nottions from Fig., the locl instntneous ccelertions from Eq. 3 re evluted by V t V V ' / t nd V t V V' / t, while the convective ccelertions by V x V ' V / x nd V x V V' / x. In order to evlute the flow rte with the proposed evlution procedure, the following informtion is needed: - pressure hed difference, dh, mesured between two cross-sections. An initil dischrge vlue is imposed s initil guess for this code. - definition of the moments t, t2, t3 nd t4 bsed on the pressure hed difference dh. The time vlue t3 is determined by solving the Eq. (2). - geometricl chrcteristics of the pipe (D dimeter, E pipe wlls Young modulus, L distnce between the pressure hed mesuring sections), nd the liquid properties (ρ density, e bulk modulus). A discretiztion grid is generted in the spce-time domin, xot (Fig. ), in order to pply the method of chrcteristics. The procedure is itertive. An initil guess for the stedy stte dischrge, Q, is firstly mde, nd vlue for f before vlve closure is obtined by 2 Q f 2 g D dh L (6) A where dh is the mesured pressure difference in the stedy stte regime. Strting with these vlues for Q, nd f, in ll grid points t the time t, the OC is pplied during t t4 time-intervl, using the boundry conditions: - t upstrem end (first mesuring section): dh (t) =, while Q(t) results from Eq. () long C - - t downstrem end (second mesuring section): dh (t) ccording with mesured dt, while Q(t) results from Eq. () long C + For ny grid points in-between, the Eqs. () nd () re used, to obtin the time vritions of the pressure hed difference dh nd the dischrge Q. In ll computtions, the Brunone unstedy friction fctor, Eq. (3), is tken into ccount by the presented explicit scheme. A new vlue of the stedy stte flow rte Q, is then derived s the verge vlue of the dischrge trce during the stedy stte t t2 time-period. The obtined dischrge vlue is compred with the previous one nd if the difference between them is less thn n imposed vlue the computtion stops. If this condition is not ccomplished, the computtion 243

5 G. Dunc et l. / JAF, Vol. 9, o. 5, pp , 26. resumes with the new Q. 3. UERICAL VALIDATIO The proposed model ws developed in the ATLAB softwre nd vlidted using numericlly generted cse from which time vrition of pressure hed difference during vlve closure were obtined. In order to test the efficiency of the proposed method outside the limittions stipulted in the IEC 4 stndrd for the pressure-time method, the time vritions of the pressure hed re numericlly creted considering two cses with distnce of.9nd 9 m between the two mesuring sections. The dt were generted considering the Brunone unstedy friction fctor model nd vlve closing function derived from experiments mde by Jonsson et l. (22). The normlized vlve closure with time durtion of 4 s is presented dimensionless in Fig. 4 nd nmed here experimentl closure. - the method developed by Jonsson et l. (22) unstedy Gibson - the method developed by Admkowski nd Jnicki (23) stedy Admkowski - the proposed method here nmed unstedy Admkowski The results re presented in figures 5 nd 6 s reltive errors of dischrge obtined with ech method compred to the reference vlue Qref, for the numericl generted dt. % Q Q ref Q ref (7) Figure 5 shows tht regrdless of the distnce between the mesuring sections for pressure hed difference the methods stedy Gibson nd stedy Admkowski provide pproximtely equl results:.5% error for the dischrge vlue.6 m 3 /s, -.5% for.3 m 3 /s nd -.2% for.4 m 3 /s. ormlized velocity Dischrge estimtion error - [%] Stedy Gibson Unstedy Gibson Stedy Admkowski Unstedy Admkowski ormlized time Fig. 4. Vlve closure function. Cse is 4 m long pipe, with n inner dimeter D =.3 m. The wter is supplied from tnk with m hed. The differentil pressure hed were obtined by solving the wter hmmer equtions with OC in two cross-sections locted t.9 m one from the other. The two mesuring crosssections re locted t.4 nd.3 m from the downstrem vlve, respectively. Cse 2 is pipe with length of 4 m, n inner dimeter D =.3 m. The wter is supplied from tnk with m hed. The considered upstrem nd downstrem sections were locted t 9 m one from the other. The two mesuring cross-sections re locted t 4 nd 3 m from the downstrem vlve, respectively. In both cses, the pressure wve speed ws considered equl to 9 m/s. Three vlues of dischrge.6,.3 nd.4 m 3 /s were considered to obtin pressure hed vritions. For both vlues of length nd the three dischrge vlues, 4 different methods for dischrge evlution were pplied: - stndrd pressure-time method stedy Gibson Dischrge Q - [m 3 /s] Fig. 5. Dischrge evlution error for Cse - mesuring length of.9 m. Both unstedy evlution procedures (unstedy Gibson nd unstedy Admkowski) gve errors close to zero, s the differentil pressure hed dt were obtined using the sme unstedy friction model. In this wy, the methods were in fct vlidted. With the two unstedy evlution procedures, the errors were of.6% for the dischrge vlue.6 m 3 /s,.3% for.3 m 3 /s nd.2% for.4 m 3 /s. The difference between the stedy nd unstedy methods points out the potentil error induced by ssuming constnt friction fctor. In Fig. 6, the stedy evlution procedures provide lso similr results. Some differences pper between the two unstedy evlution procedures results. The two unstedy methods re similr, beside the liquid compressibility nd pipe wlls deformbility effects tken into ccount in the proposed method. A sensibility nlyse ws performed to highlight this difference. A new simultion ws performed in order to obtin differentil pressure hed trce with shrper closure of the vlve (Fig. 7). The results from the four evluting procedures using the new generted dt re presented in Tble. 243

6 G. Dunc et l. / JAF, Vol. 9, o. 5, pp , 26. Dischrge estimtion error - [%] Stedy Gibson Unstedy Gibson Stedy Admkowski Unstedy Admkowski significntly influenced. For Cse 2, the correct choice of t3 hd n importnt effect over the error by doubling it. The time t3 is thus of importnce when the compressibility effects becomes significnt. 4. EXERIETAL VALIDATIO OF THE ODEL ormlized velocity Dischrge Q - [m 3 /s] Fig. 6. Dischrge evlution error for Cse 2 - mesuring length of 9 m Shrper closure Experimentl closure ormlized time ormlized velocity Shrper closure Experimentl closure The second step of the study ws testing the proposed method using experimentl dt. The vilble pressure trces were mesured t the Wterpower Lbortory t TU (Jonsson, 2) for three dischrge vlues:.6,.3 nd.4 m 3 /s. The test rig (Fig. 8) consisted in pipeline system supplied from tnk. The hydrulic hed of the system ws 9.75 m. The testing section of the setup hd length of m nd n inner dimeter of.3 m. The pressure wve speed ws determined from the experiment mesurements to 9 m/s. The mesuring sections for the differentil pressure trnsducer were locted t 9 m one from nother. The first section ws 4 m from the vlve. The differentil pressure trnsducer hs rnge of ±.5 br nd the ccurcy of.25%. A mgnetic flowmeter with n ccurcy of.% ws mounted on the rig, so the dischrge vlues mesured could be used s reference for the ccurcy nlysis of proposed method. For ech of the three dischrge vlues, severl tests were mde, so the repetbility of the mesurements could be ssessed ormlized time Fig. 7. Shrper vlve closure function. A shrper vlve closure leds to n increse of the difference between the unstedy methods. This emphsises the influence of liquid compressibility nd pipe wlls deformbility over the method precision. Tble Dischrge estimtion errors for new simulted dt with shrper closure ethod Shrper closure Experimentl closure Stedy Gibson Unstedy Gibson Error [%] Stedy Admkowski Unstedy Admkowski Another importnt spect of the method is the correct choice of the time when the free pressure oscilltions begin to occur, t3. In his study, Admkowski nd Jnicki (23) emphsized tht his method is very sensitive to it. Different vlues for t3 were tested with Admkowski method in Cse nd the dischrge estimtion error ws not Fig. 8. CAD drwing of the test rig (Jonsson, 2). The dischrge ws estimted using the four procedures nlysed in Section 3 with the numericl experiment. The results re presented s vlues of the reltive error of the obtined dischrge compred to the reference vlues in Fig. 9. The stedy Admkowski method gives the sme result s the stndrd Gibson method for the lower dischrge vlue (Q =.6 m 3 /s), but for the other two dischrge vlues, the first method is less precise. Compring the unstedy Gibson method nd the unstedy Admkowski method, the second one hs smller errors in dischrge evlution. The error vlues of the proposed method (unstedy Admkowski) do not exceed ±.% for the rnge of investigted dischrge vlues. In cse of the lower 2432

7 G. Dunc et l. / JAF, Vol. 9, o. 5, pp , 26. vlue of dischrge,.6 m 3 /s, implementing the unstedy friction fctor in Admkowski method leds to decrese of the dischrge evlution error by lmost.6%. The proposed method gives better results thn the stndrd Gibson nd the unstedy Gibson methods, the dischrge estimtion error being decresed with.35% compred with the unstedy Gibson method error. This is due to the importnce of liquid compressibility nd pipe wlls elsticity, which re not considered in the two mentioned methods. Dischrge estimtion error - [%] Stedy Gibson Unstedy Gibson Stedy Admkowski Unstedy Admkowski Dischrge Q - [m 3 /s] Fig. 9. Dischrge evlution error using lbortory dt. For the other two dischrge vlues the errors obtined using the proposed method re comprble with those obtined with the unstedy Gibson method:.3% for.3 m 3 /s nd -.8% for.4 m 3 /s. The influence of the correct choice of the moment when the free pressure oscilltions occur, t3, ws considerble in this phse of the study. Choosing nother moment then the one indicted by Admkowski nd Jnicki (23) in his study doubles the dischrge evlution error. 5. COCLUSIOS The pper presents new model developed to evlute the dischrge bsed on pressure hed time vrition mesurements during trnsient regimes, similr to the stndrd pressure-time method (lso known s Gibson method). The model consists in solving the wter hmmer equtions in which n unstedy friction fctor model is implemented. As boundry condition, the differentil pressure hed mesured between two sections, during the entire trnsient regime is used. The novelty of the method resides into tking into ccount both the liquid compressibility nd the pipe wlls deformbility, nd the unstedy chrcter of the hydrulic losses, unlike the previous methods. The model ws vlidted in numericl experiment nd used to evlute the dischrge bsed on lbortory mesurements. In the numericl experiment two lengths of the pipe were considered (4 nd 4 m) nd the differentil pressure hed ws extrcted between two sections situted t.9 nd 9 m one from the other (both mesuring lengths being lower thn the stndrd limit of m). The results showed tht the effect of liquid compressibility nd pipe wlls deformbility influences the dischrge evlution procedures ccurcy. Also, tking into ccount the correct time for the end of the trnsient stge, the error cn be considerbly decresed. Using the lbortory experimentl dt, the proposed method obtined dischrge estimtion error between ±.8% for the entire nlysed dischrge vlues rnge. Implementing the unstedy friction fctor together with the correct choice of the end of the trnsient stge led to n estimtion error reduction of bout.6%, which cn be very importnt in site efficiency tests. In future works, the proposed method for dischrge evlution will be used on in site mesured detin order to test its efficiency for different dischrge vlue rnges nd in different evlution conditions. ACKOWLEDGEETS The work hs been funded by the Sectorl Opertionl rogrmme Humn Resources Development of the inistry of Europen Funds through the Finncil Agreement OSDRU/59/.5/S/ The reserch presented ws crried out s prt of "Swedish Hydropower Centre - SVC". SVC hs been estblished by the Swedish Energy Agency, Elforsk nd Svensk Krftnät together with Luleå University of Technology, KTH Royl Institute of Technology, Chlmers University of Technology nd Uppsl University. REFERECES Admkowski, A. (22). Dischrge mesurement techniques in hydropower systems with emphsis on the pressure time method, Hydropower-prctice nd ppliction (Edited by Hossein Smdi-Boroujeni.), Chp.5, InTech. Admkowski, A. nd W. Jnicki (2). Selected problems in clcultion procedures for the Gibson dischrge mesurement method. roceedings of the 8th Interntionl Conference on Hydrulic Efficiency esurement - IGHE 2, 73-8, Rookie, Indi. Admkowski, A. nd W. Jnicki (23). A new pproch to clculte the flow rte in the pressure-time method ppliction of the method of chrcteristics. roceedings of HYDRO 23, Innsbruck, Austri. Bhrni, S. A. nd C. our (24). Intermittency in the Trnsition to Turbulence for Sher- Thinning Fluid in Hgen-oiseuille Flow. Journl of Applied Fluid echnics 7, -6. Bergnt, A., J.. Simpson nd A. R. Vitkovsky (2). Developments in unstedy pipe flow friction modelling, Journl of Hydrulic Reserch 39,

8 G. Dunc et l. / JAF, Vol. 9, o. 5, pp , 26. Chudhry,. H. (987). Applied hydrulic trnsients, Vn ostrnd Reinhold, ew York. Dunc, G., D.. Bucur,. J. Cervntes, G. roulx nd. Bouchrd Dostie (23). Investigtion of the pressure-time method with n unstedy friction. roceedings of HYDRO 23, Innsbruck, Austri. Gibson,. R. (923). The Gibson method nd pprtus for mesuring the flow of wter in closed conduits. ASE ower Division 45, IEC 64 (99). Interntionl stndrd field cceptnce tests to determine the hydrulic performnce of hydrulic turbines, storge pumps nd pump turbines, 3rd ed. vol. 4, Interntionl Electrotechnicl Commision, Genev, Switzerlnd. Jonsson,.. (2). Flow nd ressure esurements in Low-Hed Hydrulic Turbines, h. D thesis, Lule University of Technology, Lule, Sweden. Jonsson,.., J. Rmdl nd. J. Cervntes (22). Development of the Gibson method Unstedy friction, Flow esurement nd Instrumenttion 23, Krney, B. W. nd D. cinnis (992). Efficient clcultion of trnsient flow in simple pipe networks. Journl of Hydrulic Engineering ASCE 8, 4-3. Vitkovsky, J..,. F. Lmbert, A. R. Simpson nd A. Bergnt (2). Advnces in unstedy friction modelling in trnsient pipe flow. The 8th Interntionl Conference on ressure Surges, BHR, The Hgue, The etherlnds. Wylie, E. B. nd L. V. Streeter (993). Fluid Trnsients in Systems. rentice-hll, Inc., Englewood Cliffs,.J. 2434