Prcticl exercise 7. Surge tnk Introduction Surge tnk is used t hydro power plnts for reduction of wter hmmer tht occurs t closing of the turbine inlet vlve. In this exercise we will mesure mss osciltions t vlve closing on surge tnk model. We will do mesurements t two different vlve closing speeds nd t set flow rte nd hed. igure: culty of civil nd geodetic engineering, bortory for hydrulics. The surge tnk is cylinder mde of plexi glss, seen on the left imge. The right imge shows flow settling tnks with weirs (left nd center of the imge, blue color). Surge tnk Bsic fetures nd vlues of the surge tnk cn be summed in the following lines: - vlue of the surge tnk: t longer hed rce or til rce chnnels/tunnels, the hed rce tunnel cn be mde of thin wlls, which mens cheper, - the surge tnk is n expensive structure, - mss osciltions re present in the surge tnk, - the surge tnk is importnt when mss flow is vried (closing of vlves), - there is no wter hmmer in the hed rce tunnel, or the wter hmmer in the hed rce tunnel is within the rtio of cross sections of the hed rce tunnel nd surge tnk, - wter hmmer is still present in the penstock, - simple surge tnk is the surge tnk where the cross section vries continuously with its height. When the turbine vlve is closed, the wter level in the surge tnk rises due to the inflowing wter from the hed rce tunnel. When wter level in the surge tnk becomes higher thn wter level in the ccumultion lke, the wter strts to flow through the hed rce tunnel bck to the lke or t lest the flow rte in the hed rce tunnel is decresed. After few oscilltions the flow nd the height settle. 31
surge tnk penstock hed rce tunnel lke turbine igure: Surge tnk scheme. Surge tnks re used in power plnts with ccumultion lkes tht re distnt from the turbine (long hed rce tunnels). control vlve for flow pumps: 20 l/s 50 l/s 100 l/s slow settling tnk 1 (weir) flow settling tnk 2 (weir) mnul setting of height surge tnk vlve ngle meter closing vlve tnk in the bsement 60 m 3 pressure trnsducer pressure tp tringulr notch weir computer for dt cquisition view from the front side, dischrge to the bsement tnk igure: bortory experiment, digrm. low settling tnks serve for ccurte setting of hed, becuse wter level is exctly set by the weir. The second flow settling tnk enbles mnul setting of height, which llows simultion of lke height bove the surge tnk height. Assumptions, necessry for derivtion of equtions When deriving equtions for mss oscilltions in the surge tnk we use the following ssumptions: - cross section S S(z), mening tht the surge tnk cross section chnges continuously with respect to height, - incompressibility of fluid or wter, = const, - in ny instnt the flow rte Q in the hed rce tunnel does not vry long its length, - inerti of wter mss in the surge tnk nd penstock is neglected (smll mss compred to the mss in the hed rce tunnel), - kinetic energy of wter in the surge tnk is neglected, becuse the speed of wter level oscilltions in the surge tnk is smll nd we ssume hydrosttic pressure gh t surge tnk entry, - wter level in the lke is not chnging. We will write the dynmic nd continuity equtions for surge tnk. Derivtion nd nomenclture re prtilly tken from the book Hidrvlik nestlneg tok, Rjr R., Univerz v jubljni, kultet z grdbeništvo in geodezijo, jubljn, 1980. 32
z level S=S(z) cross-section hj lke -w,-q w,q -z g b Qt h hb hdin hstt turbine river igure: Derivtion of equtions for the surge tnk, symbols. Derivtion of dynmic eqution We proceed from the Euler eqution. Chnge of velocity in the hed rce tunnel is consequence of the pressure difference tht ccelertes the fluid flow (2nd term on the right) nd inerti due to mss forces m (1st term on the right, index m mens force per unit mss). Mss forces m consist of grvittionl force nd friction force. dw 1 p m dt x The sign of the pressure term (2nd term on the right) is negtive becuse flow ccelertes from the loction of smller pressure (loction ) to the loction of higher pressure (loction b). m g, m fr, m Sin() is written s rtio of sides (rtio of heights/), becuse for smll ngles sin(). Slope of hed rce tunnel is usully smll. g, m g sin m g h hb riction force is proportionl to drg, tht is squre of velocity or flow rte. E fr mg fr E fr fr 2 Q Q Q mge fr mgq Q Absolute vlue in the bove eqution serves for preservtion of friction force sign. riction force in the bove eqution hs positive sign, becuse the eqution for mss forces hs friction written with negtive sign. The friction force per unit mss is written with the following expression. 33
fr, m fr m mgq Q m gq Q We express prtil derivtive of pressure with respect to x s difference of pressures t the beginning nd the end of the hed rce tunnel, the pressure difference is further written with heights p p x p b p g h z h h h j b j g z h We insert ll the bove written terms into the Euler eqution. w h Q Q hb z h hb g g g t w Q Q z g g t We replce the chnge in velocity with the chnge in flow rte. Index hrt mens hed rce tunnel. 1 S hrt 1 Shrt gs hrt Q Q Q g g t z / g h Q Q Q z g g t Q z Q Q 0 dynmic eqution for surge tnk t b Derivtion of the continuity eqution We write the continuity eqution on the bsis of the following considertion: flow rte to the turbine equls flow rte through the hed rce tunnel minus flow rte to the surge tnk (if the surge tnk is emptying, the flow rte is negtive, which mens tht the flow rte to the turbine is higher thn the flow rte through the hed rce tunnel). Qs is flow rte through the surge tnk, Qhrt is flow rte through the hed rce tunnel nd Qt is flow rte through the turbine. Ss is the surge tnk cross-section nd Shrt is the tunnel cross-section. Symbol v mens velocity in the surge tnk nd w mens velocity in the tunnel. Q Q t Q S v S t dz v dt s hrt Q s hrt w Agin we write the flow rte Q in the hed rce tunnel without the index, the sme s in the dynmic eqution. We copy the dynmic eqution. 34
dz Qt Ss Q continuity eqution for surge tnk dt gshrt Q z Q Q 0 dynmic eqution for surge tnk t We vry the flow rte to the turbine by using vlve, therefore the flow rte to the turbine is known vrible. The unknown vribles re flow rte in the hed rce tunnel nd the wter level in the surge tnk. Q=Q(t) z=z(t) flow rte in the hed rce tunnel wter level in the surge tnk We hve two equtions with two unknowns if we don't tke into ccount the friction coefficient nd geometricl dimensions. Initil nd boundry conditions for solution of the system of equtions ) initil condition 2 z( t 0) Q 0 b) left side boundry condition, the lke wter level is not chnging hj=const c) right side boundry condition, we ssume closing of the vlve ccording to the following eqution 1 t Q t t Q0 T0 time of vlve closing t the turbine T0 Mesuring of flow rte with tringulr notch weir The tringulr notch (V-notch) weir is intended for mesuring of fluid flow rte in open chnnels. The wter height t the notch is used to determine the flow rte. The figure below shows two tringulr notch weirs with ngles of =90 nd =45. low rte through the weir is sum of flow through infinitesimlly thin surfce res (htched re) cross the whole wter height H. Since the wter height for ech surfce is different, the outflow velocity is lso different. We ssume n element of height h being on height h. We replce the tringulr weir with two right tringles. We write the element width b with the following eqution: b 2 2 H htn The surfce re A of the htched element on the figure equls the product of height nd width. 35
A 2 2 H htn h igure: Tringulr notch weir intended for mesuring of fluid flow rte in open chnnels. Above is the weir with =90 ngle, below is the weir with =45 ngle. We write the ouflow velocity in the sme wy s the outflow velocity for fluids in tnk. The outflow velocity increses proportionl to the squre root of the wter height bove the outflow surfce. v 2gh low rte through the infinitesimlly thin element of the notch is product of its surfce re nd flow velocity. Q 2 2 2 H htn gh h If we integrte the bove expression between h=0 nd h=h, we get the following expression. Q 2 tn 2 8 15 tn 2 2g H 0 2g H Hh 5/ 2 1/ 2 h 3/ 2 dh 2 tn 2 2 2g H 3 5/ 2 2 5 H 5/ 2 The theoreticlly derived flow rte is not exctly the sme s the mesured one, therefore we introduce dischrge coefficient Cd, which mkes the ctul flow rte to be 8 Q Cd tn 2g H 15 2 5/ 2. 36
The dvntge of the tringulr notch over the rectngulr notch weir is tht the shpe of the nppe (the body of wter dischrging over the notch of the weir) is not chnging significntly. Tht mens tht the dischrge coefficient Cd is not chnging significntly for different flow rtes Q. The tringulr notch weir llows us to mesure wide rnge of flow rtes. Use the eqution bove when determining the operting point for this exercise. Description of mesurement loctions nd mesurement equipment Mesurements include the following vribles: - mesurement of wter height in the flow settling tnk, - mesurement of wter flow rte in the supply pipe, - mesurement of wter flow rte through the surge tnk t the tringulr notch weir, - mesurement of wter level in the surge tnk by using pressure trnsducer, - mesurement of vlve ngle, - determintion of integrl vribles or system dimensions, - recording of mesured vlues on computer. Wter height in the flow settling tnk is mesured by using tpe meter. Height in the second settling tnk is tken s reference. Wter flow rte in the supply pipe is mesured by the ABB Wtermster electromgnetic flowmeter. The flowmeter is wired to the DAQ crd nd the computer, which enbles disply nd recording of mesured vlues. Wter flow rte through the surge tnk is mesured by using 45 tringulr notch weir locted downstrem of the surge tnk nd vlve. It is not possible to mesure nd record the wter flow rte electroniclly. Use equtions tht were derived bove. Determine the dischrge coefficient Cd from digrm tht is locted in the lbortory. Wter level in the surge tnk is mesured by using the Endress Huser pressure trnsducer with rnge between 0 nd 4 br. Mesurement with this trnsducer is performed reltive to the tmospheric pressure. Tht mens tht only one pressure port hs to be connected, the other one is open to the tmosphere. Air hs to be removed from the pressure trnsducer nd connecting pipes before the mesurement. The pressure trnsducer is wired to the DAQ crd nd the computer. The voltge signl from the trnsducer hs to be clibrted on the computer to disply the ctul pressure. Since reltion between pressure nd wter height in the tnk is liner, the signl cn be directly clibrted to the wter level height in the surge tnk. Note: Electricl connection of the trnsducer is done by two wires. Supply voltge for the trnsducer is DC 24 V. or two-wire connection it is necessry to use voltge divider in order to cquire the signl by DAQ crd. or this purpose pproprite resistors re connected ccording to the figure below. The DAQ crd rnge is from 0 to 5 V. 37
pressure trnsducer power supply 24 V igure: Electricl connection of the pressure trnsducer to voltmeter or DAQ crd. Mesurement of vlve ngle will be performed by using precise rottionl resistor. Electricl resistnce of the rottionl resistor is dependent on the ngle of shft. The rottionl resistor hs three terminls for connection to the power supply nd to the DAQ crd. The rnge of the DAQ crd is from 0 to 5 V. Determine which ngle corresponds to individul voltges on DAQ crd nd mke clibrtion curve for the rottionl resistor. Integrl vribles re: cross sections of the supply pipe nd surge tnk nd time of vlve closing. Before or fter the mesurements you hve to determine the system dmping in stedy stte. Mesure the flow rte by using the tringulr notch weir. Assume tht the vlve closing is liner. Recording of flow rte, pressure nd vlve ngle to computer will be mde by using 16 bit DAQ crd NI 6036 nd NI bview softwre. Choose suitble time nd frequency of dt cquisition for recording to the hrddrive. The softwre will be prepred by the ssistnt. Assignment Assemble mesuring system tht will llow mesuring of the wter height in the surge tnk, the flow rte in the supply pipe nd the vlve ngle. Clibrte ll signls for the electronic meters by clculting equtions of line. Mesure flow rte nd wter height oscilltions in the surge tnk model. Mke mesurements t two different vlve closing speeds. In stedy stte compre flow rte indicted by the flowmeter to the flow rte mesured by the tringulr notch weir. 38