Abstract. 1 Introduction

Transcription

1 Numercal models for unsteady flow n ppe dvdng systems R. Klasnc," H. Knoblauch," R. Mader* ^ Department of Hydraulc Structures and Water Resources Management, Graz Unversty of Technology, A-8010, Graz, Austra ** Vorarlberger Illwerke AG, A-6901 Bregenz, Austra Abstract Parts of the pressure of an exstng hgh-head power development are beng reconstructed. Ths ncludes the adapton of the connecton to the surge tank. The flow condtons n ths arrangement were to be studed for varous loadng cases occurrng durng operaton. The pressure and flow condtons occurrng durng these processes were calculated for the overall system usng a undmensonal numercal program based on the characterstcs method. The unsteady flow processes n the regon of the branch pont were calculated. The unsteady processes were seen to proceed relatvely slowly so as to allow the assumpton of steady flow condtons for the subsequent calculatons. In order to accomplsh a 3-D analyss of the flow patterns n the regon under study, we chose a numercal model usng the Fnte Volume Method (FVM).We ncluded n our studes certan loadng cases expected to gve rse to maxmum loadngs wth respect to velocty peaks and local rsks of subatmospherc pressure. The results of the computaton showed that under the gven condtons there s no rsk of local zones of subatmospherc pressure to develop so as to exclude the rsk of potental cavtaton. 1 Introducton In an exstng hgh-head power development, two above-ground penstocks are beng replaced by a one pressure shaft. The fve two-jet Pelton turbnes n the power house wll reman unchanged. Total power staton capacty s approxmately 155 MW. Ths alteraton requres the adaptaton of the juncton wth the surge tank, whereas the surge tank tself wll not be changed n desgn. For geologcal

2 372 Hydraulc Engneerng Software reasons, the new desgn provdes for the approach ppe to make an angle of 70 wth the surge tank. As ths unusual arrangement nvolves a potental cavtaton rsk, flow condtons were studed for dfferent loadng cases: turbne start-up - rapd turbne shut-down - resumpton of turbne operaton - rapd turbne shut-down. The pressure and flow condtons occurrng durng these processes were calculated for the overall system usng a undmensonal numercal program based on the characterstcs method. These unsteady flow processes were plotted for the surge-tank juncton and served as a bass for a more detaled nvestgaton nto the local flow and pressure condtons. To allow 3-D analyss of flow n ths regon, a numercal model usng fnte volume elements (FVM) was used. We evaluated three loadng cases expected to nvolve maxmum levels of velocty and local subatmospherc pressure (cavtaton rsk). Ths allowed accurate localsaton of zones subject to the potental rsk of subatmospherc pressure, and actually proved the juncton to be suffcently safe from cavtaton. Also, the magntudes and ponts of occurrence of the maxmum velocty levels were determned. 2 Descrpton of the System The overall system of the hydro development conssts of the followng man components: reservor - penstock - surge tank - pressure shaft - power house wth turbne. Constructon of the pressure shaft called for the desgn of a new connecton between pressure condut and surge tank, whch led to the followng new stuaton n the regon of the surge-tank juncton (see fg. 1). / surge tank reservor K3 power house a f Fgure 1: Connecton between pressure condut and surge tank The analyses amed at (a) determnng the dstrbutons n space of pressures and veloctes to obtan the mnmum pressures and maxmum veloctes n the regon of the juncton

3 and Transactons on Ecology and the Envronment vol 7, 1994 WIT Press, ISSN Hydraulc Engneerng Software 373 (b) determnng the velocty dstrbuton n the drecton of flow at a pont some 12 m downstream of the juncton, where an ultrasonc dscharge measurng nstrument s to be nstalled. 3 Calculaton of the Surge Tank Program The whole system comprsng reservor - penstock - surge tank - pressure shaft- power house wth turbne was subjected to a undmensonal water hammer analyss based on the characterstcs method wth allowance beng made for frcton, so as to allow the defnton of maxmum loadng cases for the condtons present n the surge-tank juncton. The analyss was based on the two frst-order lnear homogeneous dfferental equatons after Allev (contnuty equaton and energy equaton), makng allowance for the frcton slope. The analyss yelded the followng soluton for a pre-determned water level n the reservor (see fg. 2): ' Fgure 2: Results of the surge tank programm Tme n [sec] These results were used to determne 3 loadng cases nvolvng extreme condtons for use as a bass for the spatal analyss of pressure and velocty condtons n the juncton. Case Tl; tme Tl = s Case T2; tme T2 = s Case T3; tme T3 = s In order to save computng tme, steady-flow condtons were assumed for the detaled calculatons whch followed.

4 374 Hydraulc Engneerng Software 4 Numercal Model The program used, solves the conservaton equatons of mass, mpulse (three Cartesan components), thermal energy, turbulence and passve transport equatons n general nonorthogonal movng systems of coordnates. Spatal dscretsng was accomplshed by use of the fnte volume element method [4]. Mathematcal treatment of the turbulences was based on the k - e turbulence model. Ths s a two-equaton model usng the prncple of dynamc vscosty for turbulent flow and a transport equaton for the rate of dsspaton, e, assocated wth a certan length. Ths s the wdely-used computer program [5]. The k - e turbulence model employs addtonal partal dfferental equatons for the turbulent knetc energy k defned by «' * - k* k = - plus ts dsspaton: = c 2 L In the above equaton L stands for a turbulent length scale and c^ s an emprcal coeffcent. There are two mportant features that dstngush near-wall regons from other portons of the flow feld: Frst there are steep gradents of most of the flow propertes, and secondly, the turbulent Reynold's number s low so that the effects of molecular vscosty can nfluence flow energy [2]. Actually, these problems could also be solved by means of the above method provded a very dense mesh of elements s establshed near the boundary of the model. But t s easer to approxmate the boundary layer by sem-emprcal relatonshps. Thus, the near-wall velocty (u<. - pont C) varaton s descrbed by the logarthmc relatonshp [5]: K where K and C are expermentally determned constants. The velocty, u^, can be wrtten as: (wth r^ = wall shear stress) The computer program FIRE [2] s based on the assumpton of a hydraulcally rough ppe (roughness k$ = 0,23 mm).

5 Hydraulc Engneerng Software 375 surge tank &^ K3 Kl Fgure 3: Geometry and generated volume elements Before undertakng the computaton, we had frst to determne the geometry of the bfurcaton and then generate a mesh of fnte volume elements (wth about elements, Fg. 3). The boundary of the model s decded by the geometry of the ppe. Inlet and outlet openngs must be defned and boundary condtons have to be found out for the nlet openng. These are head, flow and temperature. In the case under dscusson, a temperature of 10 Celsus was adopted. Adabatc processes do not play a decsve role n case where water s used as a flud. In the dstrbuton of elements over the cross sectonal area, there are certan crtcal elements for whch calculaton presents problems. The deal condton for the calculaton s that all the angles of an element are approxmately 90 degrees. The more the angles dffer from 90 degrees, the larger are the resultng naccuraces n the calculaton. The problem can be mtgated by establshng a closer mesh n these areas. 5 Results The analyss proper was performed on a Type DEC AXP - Open VMS Dgtal workstaton. Three cases (Tl, T2, T3) were analysed. T 1 T 2 T 3 K 1 29,4 mvs 14,65 mvs 0 K1 K3 K 2-29, 4 mvs 14,65 mvs 24,6 mvs Fgure 4: Scheme of K ,3 mvs 24,6 mvs bfurcaton

6 376 Hydraulc Engneerng Software Ths three flow rates used n the analyses were taken from the undmensonal surge tank calculaton for a certan reservor water level. For each loadng case, a number of tme steps wth 50 maxmum possble teratons each was assumed as shown below. (These values resulted from an optmsaton analyss relatng to the convergence of solutons.) Tme steps Maxmum number of teraton per tme step Total tme CPU-tme Tl T T J1 76,8. sec. j1 95 sec. j 76,8 sec. 9,66 h 6,52 h 9,45 h Ths analyss was based on the assumpton of sothermal and ncompressble flow condtons for a water temperature of 283 K. As ndcated above, answers had to be found to the followng questons: (A) Lowest pressure reducton n the regon of the acute-angle juncton, ncludng maxmum veloctes, both as to magntude and locaton. Tme Tl: Maxmum relatve pressure reducton (flow around the acute angle) Turbne shut-down wth the surge tank n operaton, extreme values QK, = 29.4 mvs and Q^ = 0; vjnax = 9.5 m/s mn. pressure Fgure 5: Tme T2: Pressure dstrbuton (Iso-lnes) and local velocty peak Maxmum relatve pressure reducton h = 6,85 m Turbne start up wth the surge tank n operaton, extreme values QK, = Q%2 = mvs and Q^ = 29.3 mvs;

7 Hydraulc Engneerng Software 377 \, mm. pressure Fgure 6: Pressure dstrbuton (Iso-lnes) Maxmum relatve pressure reducton h = 5,04 m Tme T3: Turbne start-up, startng from an operatng surge tank, extreme values Q%: = 0x3 = 24.6 mvs and QK, = 0; As a result of the smaller flow, ths case gves smaller values than case T2. (B) Most unfavourable velocty dstrbutons n the regon of the flow meter n the valve chamber. Tme Tl: Q^ = 0, hence no analyss; Tme T2: Turbne start-up wth the surge tank n operaton, extreme values QKI = Q%2 = mvs and Q^ = 29.3 mvs; gvng more favourable condtons; M = 5.1 m/s v = 4.2 m/s v_, = 6.9 m/s Fgure 7: Velocty dstrbuton (Iso-lnes), seen n flow drecton

8 378 Hydraulc Engneerng Software Tme T3: Turbne start-up, startng from an operatng surge tank, extreme values 0x2 = 0*0 = 24.6 mvs and 0%, = 0; Fgure 8: Velocty dstrbuton (Iso-lnes), seen n flow drecton For queston (B), the most unfavourable case s ndependent of the reservor water level. 6 Summary For the gven case of an acute-angle juncton wthn the system of a hydro development, a undmensonal water hammer program was used for calculatng the boundary condtons for a local analyss of flow condtons. The results obtaned served as a bass for the exact analyss of the juncton usng a three-dmensonal turbulent-flow program based on the prncple of fnte volume elements (FVM). The results showed good agreement between the two numercal programs used and n addton gave exact values relatve to the local pressure and velocty pattern n the juncton. In ths way t was possble to prove that under the prevalng condtons local subatmospherc pressures dd not develop and, hence, that there was no cavtaton rsk. In the future, the numercal as well as physcal studes are to be focussed on locatons subject to potental cavtaton rsks. For ths purpose, a juncton model wll be studed on the cavtaton test stand as part of a research project. References [1] Klasnc, R., Knoblauch, H.; Dum, Th.;."Power losses n dstrbuton ppes", Hydrosoft '92, Span [2] AVL; "Flow n recprocatng engnes (FIRE)", Program manual, Vers. 5.2 [3]Bollrch, G.; PreBler, G.; "Technsche Hydromechank", Band 1, Verlag fur Bauwesen, Berln 1992 [4] Hnze, O.J.; "Turbulence", McGraw-Hll Classc Text- book, 1987 [5] Rod, W.; "Numersche Berechnung turbulenter Stromungen n Forschung und Praxs", Karlsruhe 1992 [6] Schonung, B.E.; "Numersche Stromungsmechank", Sprnger Verlag 1990