ANALYSIS OF PUMPED HYDROELECTRIC STORAGE FOR A WIND/PV SYSTEM FOR GRID INTEGRATION

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "ANALYSIS OF PUMPED HYDROELECTRIC STORAGE FOR A WIND/PV SYSTEM FOR GRID INTEGRATION"

Transcription

1 ANALYSIS OF PUMPD HYDROLCTRIC STORAG FOR A WIND/PV SYSTM FOR GRID INTGRATION G. Notton, V. Lzro, L. Stoyno Abstrct: This pper dels with the pumping storge system nlysis. The wter reseroir seres for dily nd sesonl energy storge, thus bsiclly soling the problem of energy storge, which is the biggest problem of wider use of renewble energy sources. The electricl energy produced in excess by the renewble energy system is conerted in potentil energy by pumping wter to higher eletion where it cn be stored indefinitely nd then relesed to pss through hydrulic s nd generte electricity. The estimtion of the stored energy, the nominl electricl power of the hydroelectric plnt nd the eportion rte of the wter reseroir is presented. Keywords: solr energy; wind energy; hybrid systems; pumped storge. 1. INTRODUCTION In smll electricl grids with limited or no connectiity to lrge scle power system some difficulties pper s to keep control of oltge nd frequency. In such n isolted system, we note numerous network filures nd limittion of the renewble energy prt due to the rndom nture of the solr nd/or wind source. Thus the inititie to diersify the energy sources is hindered. This is the cse with the Corsic Islnd in the Mediterrnen Se, which is not connected to the French minlnd electricl grid (nd only smll connection with Srdini). The combintion of renewble resources with energy storge cn be solution becuse it cn increse the lue of photooltic (PV) nd windgenerted electricity, mking supply coincident with periods of pek consumer demnd. nergy storge systems he different ppliction s to follow lod, stbilize oltge & frequency, mnge pek lods, improe power qulity, defer upgrde inestments, nd support renewbles. Fig. 1. sho the power nd dischrge time requirements for riety of storge pplictions in the utility industry. Storge time (minutes) Commodity 1 h Storge T&D Fcility Deferrl Customer nergy T. Voltge Mngement Rpid Resere 1 h Regultion Are Control & Frequency Renewble nergy Mngement Responsie Resere 1 mn Power Qulity & Relibility Trnsmission 1 s System Stbility 1 ms Storge power requirements (kw) Fig. 1. Power nd dischrge time requirements for riety of storge pplictions. OVRVIW The pumped storge is system of generting electricity, lso known s hydroelectric storge, which uses wter tht hs been pumped into n eleted reseroir during the hours of low consumption to generte electricity during hours of pek demnd. This type of hydroelectric system is used by some power plnts for lod blncing. The method stores energy in the form of wter (potentil mechnicl energy) from lower eletion reseroir to higher eletion. In conentionl electricl system, the low-cost off-pek electricl 64

2 power is used to run the pumps nd the stored wter is relesed during periods of high electricl demnd, generlly with cost benefit. In Renewble energy system, t time of low electricl demnd, excess genertion power produced by the renewble energy system (wind nd/or photooltic system) is used to pump wter into higher reseroir nd then, when there is higher demnd, the wter is relesed bck in the lower reseroir through, generting electricity. There re two possibilities: - two wter wys: one for the pumping, the other for the ; - one unique wter wy : reersible mchinery (usully Frncis Turbine design) (Fig. ) is used for both pumping nd generting; it is designed s motor nd pump in one direction nd s nd genertor in opposite rottion (Fig. 3) Fig.. A Frncis Turbine Fig. 3. Schemtic of conentionl pumped-storge deelopment with one reersible pump. The sttion is locted underground if the geologicl conditions re fourble, otherwise it is situted on the lower reseroir. Vrious pumped storge plnts exist worldwide with power from bout 1 MW (RH Lepenic, 1985, Croti) to 7 MW (Knngw, 5, Jpn) [1]. Pumped hydro is ilble t lmost ny scle with dischrge times rnging from seerl hours to few dys. The dntges of such systems re []: - More thn 1 yers of experience; - High efficiency: in the 7% to 85% rnge. - Multipurpose fcilities; - nironmentl friendly; - CO - oiding; - Highest ilbility compred to other technologies; - Quick response to lod rition (some seconds) nd resere cpcity. Tking into ccount eportion losses from the exposed wter surfce nd conersion losses, pproximtely 7% to 85% of the electricl energy used to pump the wter into the eleted reseroir cn be regined. A comprison of rious storge mens is shown in Fig. 4 [3, 4] in term of rtings. We note tht hydropumped system (PSH) is dpted for high rted power (4 to 4 MW) with high dischrge time between 1 nd 1 hours. 65

3 P S H Екологично инженерство и опазване на околната среда, No 1, 11, с The cpitl cost is n importnt economic prmeter, so the totl ownership cost (including the impct of equipment life nd O&M (Opertion & Mintennce) costs) is n interesting index for complete economic nlysis. A comprison from n economic point of iew of the energy storges is shown in Fig. 5 [3]. The cost of PSH system is between 1 nd $/kw. 1 CAS Compressed ir N-S Sodium-sulfur L/A Led cid Ni-Cd Nickel-cdmium VR Vndium redox Li-Ion Lithium Ion Zn-Br Zinc-bromine Ni-MH Nickel-metl hybride FW Flywheels DLC Dbl-lyer cpcitors N-S Sodium-sulfur PSH Pumped hydro 1 VR N-S CAS Dischrge Time (hr) Dischrge Time (hr) Ni-MH Li-Ion Zn-Br FW N-S Ni-Cd L/A.1 DLC Rted Power (MW) Rted Power (MW) Fig. 4. Storge systems rtings [3] Cpitl cost per unit energy ($/kwh) High power C Cpcitor High power FW Li-ion Long durtion Ni-Cd FW btteries Zinc-ir Led-cid N-S btteries btteries btteries Long durtion Flow btteries C cpcitor PSH CAS Metl-ir btteries Cpitl cost per unit power ($/kw) Fig. 5. Storge systems costs [3]. fficiency nd cycle life re two importnt prmeters to consider long with other prmeters. Both of these prmeters ffect the oerll storge cost. Low efficiency increses the effectie energy cost s only frction of the stored energy could be utilized. Low cycle life lso increses the totl cost s the storge deice needs to be replced more often. The present lues of these expenses need to be considered long with the cpitl cost nd operting expenses to obtin better picture of the totl ownership cost for storge technology. Fig. 6. illustrtes this influence for rious storge technologies. The lifetime (for depth of dischrge (DOD) 8%) for such pumped hydro system is one of the highest ilble with storge mens. Per-cycle cost cn be the best wy to elute the cost of storing energy in frequent chrge/dischrge ppliction, such s lod leelling. 66

4 Fig. 7 sho the cpitl component of this cost, tking into ccount the impct of cycle life nd efficiency. For more complete per-cycle cost, one needs to lso consider O&M, disposl, replcement nd other ownership expenses, which my not be known for the emerging technologies. We note tht the cpitl cost per cycle is lrgely lower thn for other existing storge systems. There is oer 1 GW of pumped storge in opertion world wide, which is bout 3 % of globl genertion cpcity (3 GW in urope, 1 GW in Jpn, 19.5 GW in USA, nd others in Asi nd Ltin Americ) [5-7]. Pumped storge plnts re chrcterized by long construction times nd high cpitl expenditure. 1% 9% Li-ion NS.C cpcitor Fly Wheels fficiency 8% 7% 6% Led-Acid Flow Bt Ni-Cd PSH CAS 5% Metl -ir 4% Lifetime Lifetime t 8% t 8% DOD - Cycles Fig. 6. Life efficiency of rious storge systems [3]. Fig. 7. Per-cycle cost for rious storge systems [3]. Pumped storge is the most widespred energy storge system in use on power networks. Its min ppliction is for energy mngement, frequency control nd proision of resere. 67

5 3. APPLICATION WITH RNWABL NRGY SYSTM Using hydro-pumped storge llo to improe the qulity of the proided electricity nd to reduce the pek power of the other energy generting systems. This system flttens out the lod rition on the power grid, nd permits therml power sttions tht proide bse-lod electricity to continue operting t pek efficiency while reducing the need for peking power plnts tht use costly nd polluting fuels. Moreoer, pumped storge system helps control electricl network frequency nd proides resere genertion. Therml plnts re much less ble to respond to sudden chnges in electricl demnd, potentilly cusing frequency nd oltge instbility. The hydro-pumped system cn respond to lod chnges within seconds. These mchines generte in synchroniztion with the network frequency, but operte synchronously (independent of the network frequency) s motorpumps. A new use for pumped storge is to leel the fluctuting output of intermittent power sources. The pumped storge bsorbs lod t times of high output nd low demnd, while proiding dditionl pek cpcity. In certin jurisdictions, electricity prices my be close to zero or occsionlly negtie (Ontrio in erly September, 6), indicting there is more genertion thn lod ilble to bsorb it; lthough t present this is rrely due to wind lone, incresed wind genertion my increse the likelihood of such occurrences. It is prticulrly likely tht pumped storge will become especilly importnt s blnce for ery lrge scle photooltic genertion. The principle is illustrted in Fig. 8. Fig. 8. Principle of coupling between renewble energy sources nd hydro-pumped system. The typicl oerll efficiency of hydro-storge systems is between 65% nd 77% [4, 8] with mximum depth of dischrge up to 95%. Kldellis et l [4] showed tht this type of system for islnd utonomous electricl networks is the optiml solution for big size islnds nd for smll nd medium sized islnds; it ppers to be the best solution for utonomy of 4 hours. 4. STORD NRGY AND MACHINS POWR STIMATION The reltie low energy density of pumped storge system requires either lrge body of wter or lrge rition in height. For exmple, 1 kg of wter (1 m 3 ) t the top of 1 m tower hs potentil energy of bout.7 kwh. The only wy to store significnt mount of energy is by hing lrge body of wter. We consider the system presented in Fig. 9. The deeloped excel softwre estimtes the stored energy nd the nd pump pek powers for gien cse study. The input dt re: - upper tnk ltitude: h upper - lower tnk ltitude: h lower ; - pump or ltitude h pump or h ; - the wter tnks olume V; - the length nd dimeter of upstrem nd downstrem tubes: L 1, L, D 1, D ; - the mximum wter flow rte into the nd the pump (or the reersible pump) Q pump, Q ; - the roughness height of the tube ε (equl to.1 mm) 68

6 h upper pump,1 L 1, D 1 V turb,1 Pump h pump Turbine pump, h turb V turb, L, D h lower Fig. 9. Schemtic presenttion of the studied system. We clculte: - the mximum wter speed in ech tube: pump,1,,1 : pump ( π ) ( π ), 1 = 4Q pump D 1 nd, 1 = 4Q D 1 (1) - the Reynold number nd the flow regime with υ is the kinemtic iscosity tken equl to m/s²: R e.d ν nd pump = pump R e.d ν () = - the Drcy friction fctor λ clculted using the Swmee-Jin eqution [9] which estimtes Colebrook eqution with -5% error nd gien by : ε λ =. 5 Log (3).. D Re - the Drcy-Weisbch coefficients J 1 nd J (m) gien by : L1 1 L J1 = λ 1 nd J = λ (4) D1 g D g - the mnometric height t ech tnk H upper nd H lower : ptm H upper = hupper + nd ρg ptm H lower = hlower + (5) ρg where p tm is the tmospheric pressure ( P) nd ρ the wter density (1 kg.m -3 ) - the mnometric heights upstrem nd downstrem of the pump nd the re gien by : H down, up, = H J (6) lower upper 1, H = H + J (7), H down,pump up,pump = H J (8) upper lower pump, H = H + J (9) pump,1 - the hydrulic power of the pump nd the re gien by : P,hydrulic P P pump,hydrulic ( H down, H up, ) Q ( H up,pump H down,pump ) Q pump = ρ g (1) = ρ g (11) - the electricl powers re gien by :,elec P pump,elec = P η η (1),hydrulic pump,hydrulic ( η η ) pump motor lterntor = P (13) - the gross stored energy, generted by the reseroir, is [1]: gross ( H H ) = ρ g (14) upper lower - the totl dily hydrulic energy (kwh) which the renewble system cn produce t the outlet of the pumping system is obtined by [1]: H, dily ( H down, H up, ) 1 g = ρ (15) 3 Q ( m / dy) 5. WATR TANK VAPORATION RAT STIMATION Fig. 1 sho the principle digrm of the renewble plnt with chrcteristic elements. For clculting the hydrulic blnce of the pump/ system, seerl physicl dt re needed or must be clculted s: - eportion rte - wtershed flow rte - infiltrtion To determine the eportion rte, which cn be importnt prticulrly in rid or sunny res, the min prmeters to clculte re: 69

7 - the conectie het trnsfer coefficient h c ; - the mss trnsfer coefficient K - the wter surfce temperture T - nd finlly the eported wter rte. A study mde by Srtori [11] sho tht rious methods re ilble to clculte the eportion flow rte. It hs been shown tht the reltie humidity must be tken into ccount for better ccurcy. A similr more recent work [1] confirms the ribility of the results obtined by the rious models. In this work, the eportion rte estimtion must sty simple becuse its influence on the energy system behiour is not importnt enough to necessitte ery thorough work. Power Sttion Q WSU (i) Q PRU (i) Q VU (i) nergy consumption LOAD (i) PV plnt Φ SUN (i) Wind energy plnt (i) PS (i) PSD (i) PSP (i) T (i) PV (i) Upper reseroir Q INFU (i) Q P (i) Q WSL (i) Turbine/ QPRL (i)q VL (i) lterntor Q T (i) PVD (i) PVP (i) WTD (i) WT (i) WTP (i) Pump P (i) Q INFL (i) Lower reseroir Fig. 1. Principle digrm of the renewble plnt with chrcteristic elements Wtmuss et l [13] suggested to use s conectie het trnsfer coefficient the following expression: h c = (16) with the wind speed. The uthors conerted this eqution tking into ccount tht the wind speed is generlly mesured t 1 m height. Their fitted eqution becme: h c = (17) q. (17) is used in this work. The mss trnsfer coefficient, s expressed by the exct form of Chilton- Colburn nlogy my be written s follo [14]: K hc = β with ρc p.67 pt Pr β = (18) P Sc P nd P T re respectiely the prtil pressure of ir nd the totl pressure. The simplified form of Chilton-Colburn Anlogy ssumes the rtio of Pr to Sc is equl to one nd ignores the effect of wter pour pressure on the prtil pressure of ir (P =P T ). Under such condition, β will be equl to 1 nd eqution (18) simplifies to: K c ( ρc ) = h (19) p This simplifiction is clled the Lewis hypothesis, only lid for the ir nd used by Belrbi nd Sïghi [15] to estimte the eportion rte of wter surfce. The sturted nd prtil wter pour pressures (in P) re gien by [1]: P, w 177. ( ) ( T ) T = exp ( T ) P ( T ) h P ( T ) s () = (1) r s, w The wter density nd its specific het cpcity depend on the bsolute humidity X nd re gien by: P( T) X =. 61 () P P T [ T ( )] ( T ) p ( T ) pt p ρ = + (3) 87. 5T T C = C + XC (4) p p,w p,dryir 7

8 The eportion from wter body is estimted s the difference between energy inputs nd outputs mesured t site. The energy loss through eportion represents mjor component of the energy blnce in typicl wter body. The temperture of the wter surfce is clculted from n energy blnce [15-17]: R = L + H G (5) n w, + A prt of the rdition R n is used to eporte the wter ilble t the wter surfce nd cretes the ltent het flux L w, where is the eportion rte nd L w, the ltent het of wter poriztion. The rest is dissipted in het into the ir by conection (H sensible het) or stored into wter (G conductie het flux). The wter ltent het flux for poriztion is expressed from Hrrison formultion [18]: 3 w L, = T (6) The storge of the het into the wter during one hour period is generlly smll compred with the other het fluxes [15-16] nd G cn be neglected. Then q. () becomes: R R = L H (7) n w, + n 4 4 ( α ) R + εσ ( T 6) εσt = (8) 1 g with R g, the globl solr rdition, α the lbedo of the wter surfce (.8) [19], ε the wter surfce emissiity (.98) [] nd σ the Stefn-Boltzmnn constnt, T sw nd T re the wter surfce nd ir tempertures. c ( T T ) H = h (9) The ltent het flux is clculted by the Stefn method (simplified expression of the Fick theory) bsed on the mss trnsfer or bulk trnsfer theories. L w, L K w, w = [ P ( T ) P ( T )] (3) RT M s,w with M w, the moleculr mss, P s,w sturted wter pour pressure nd P wter pour prtil pressure. Thus, in replcing equtions (8), (9) nd (3) into eqution (7), the energy blnce becomes: 4 4 ( 1 α) R + εσ( T 6) εσt h ( T T ) L w, K RT g M w [ P ( T ) P ( T )] s, w c = (31) Thus, we cn clculte, using method of dichotomy, the wter surfce temperture t ech hour, from the following mesured dt : - wind speed - mbient temperture T - reltie humidity h r - horizontl globl irrdition R g The next step consists in estimting the eportion rte nd rious methods re ilble to rech this gol with more or less precision. Seerl formultions he been compred [11-1]. We decided to use for our study the Penmn method which is bsed in physicl principles nd gies lues tht should sere for most project studies [15-16, 19]. The Penmn formule cn be expressed by: L = R + γ n + h c ( Ps,w ( T ) P ( T )) ( + γ ) (3) where γ is the psychometric constnt gien by [19]: ( 168. P ) L w, γ = 6 (33) is the slope of pour pressure ersus temperture cure gien by [15-16]: ( P ( T ) P ( T )) ( T T ) = (34) s,w s,w From hourly meteorologicl dt of Ajccio (wind speed, solr irrdition, mbient temperture nd reltie humidity), we clculted the eportion rte. The results for one yer re presented in Fig. 11. We note tht sometimes during the night the eportion is negtie i.e. the wter surfce temperture is below the dew point temperture nd condenstion occurs. At lst, we estimted the monthly erge lues of the dily eportion rte with the meteorologicl dt for two sttions Ajccio (Frnce) nd m. Bote (Bulgri) with colder climte (Fig. 1). We note tht the eportion phenomenon is not negligible nd influences the energy blnce of the system. 6. CONCLUSION The pumping storge is the most widely implemented lrge-scle electricl energy storge. PHS is mture technology with lrge olume, long storge period, high efficiency nd reltiely low cpitl cost per unit of energy. We estimted the stored energy nd the pump nd pek powers ccording to the chrcteristic of the wter storge. Then, the influence of the eportion phenomenon t the wter surfce hs been estimted. This work gies n ide for some procedures in the pumping storge nlysis before its integrtion in electricl power system with renewble energy sources. 71

9 1 1 Hourly solr irridtion, portion rte Wind Speed (m/s) Reltie Humidity (%) Hourly Solr Irrdition (Wh/m²) Ambient Temperture ( C) Reltie humidity, wind speed, Ambient temperture Hour 1,4 1, portion Rte (mm/h) portion rte (mm/hour) 1,8,6,4, , Hour Fig. 11. Meteorologicl prmeters nd estimted eportion rte for Ajccio Dily Solr Irrdition, Dily eportion, Reltie humidity Dily Solr Irrdition (kwh/m²) Wind Speed (m/s) Ambient Temperture ( C) reltie humidity (x1%) 5 Dily portion Rte (mm/dy) Wind speed, Ambient temperture JAN FB MAR APR MAY JUN JUL AUG SP OCT NOV DC Dily Solr Irrdition, Dily eportion, Reltie humidity Dily Solr Irrdition (kwh/m²) 1 Wind Speed (m/s) Ambient Temperture ( C) reltie humidity (x1%) Dily portion Rte (mm/dy) JAN FB MAR APR MAY JUN JUL AUG SP OCT NOV DC Fig. 1. Monthly erge lues of the dily eportion rte for Ajccio nd Bote Wind speed, Ambient temperture 7

10 RFRNCS Pirker, O. (8) Role of storge nd pump storge hydro power plnts for grid regultion. nd urelectric Smll Islnds Seminr, April 8, Ajccio, Frnce. 3. lectricity Storge Assocition. (9) Kldellis, J.K., Zfirkis, D., Kdis, K.. (9). Techno-economic comprison of energy storge systems for islnd utonomous electricl networks. Renewble nd Sustinble nergy Reie, 13, Chen, H., Ngoc Cong, T., Yng, W., Tn, C., Li, Y., Ding, Y. (9). Progress in electricl energy storge system: criticl reiew. Progress in Nturl Science, 19, Aherne, J. (4). Storge of electricl energy. Report on reserch nd deelopment of energy technologies. Interntionl Union of Pure nd Applied Physics, 4, p ilble t 7. Linden, S. (3). The commercil world of energy storge: reiew of operting fcilities (under construction or plnned). Proceeding of 1 st nnul conference of the energy storge council, Houston, Texs, Mrch, 3, D. Ppntonis. (1995). Hydrodynmic mchines: pump-hydro s. nd d., Symeon: Athens. (in Greek). 9. Swmee, P.K.; Jin, A.K. (1976). "xplicit equtions for pipe-flow problems". Journl of the Hydrulics Diision (ASC) 1 (5): Glsnoic, Z., Mrgetr, J. (9). The feture of sustinble solr hydroelectric power plnt. Renewble nergy, 34, Srtori. () A criticl reiew on equtions employed for the clcultion of the eportion rte from free wter surfces, Solr nergy, 68-1, pp ,. 1. McJnnet D.L., Webster I.T., Stenson M.P., Shermn B.S. (8) stimting open wter eportion for the Murry-Drling Bsin : A report to the Austrlin Goernment from the CSIRO Murry-Drling Bsin Sustinble Yields Project. CSIRO, Austrli. 5 pp. Noember Wtmuff J.H., Chrters W.W.S., Proctor D. (1977) Solr wind induced externl coefficients for solr collectors. COMPLS,, pp Frid M. () Miscoseptions in the nlysis of het nd mss trnsfer in drying nd frying. Drying -Proceedings of the 13 th Interntionl Drying Symposium (IDS ) Beijing, Chin, 7-3 August, ol. A, pp Belrbi N, Sïghi M. (7). tude des trnsferts de chleur et de msse entre une surfce d eu d un brrge et l tmosphère. Appliction ux régions rides et semi rides en Algérie. 18 ème Congrès Frnçis de Mécnique, Grenoble, Frnce, 7-31 Août Belrbi N, Sïghi M. (7). tude comprtie des méthodes d élution du tux d éportion à prtir d une surfce d eu libre. Appliction ux régions rides et semi rides en Algérie. 13 ème Journées Interntionles de Thermique, Albi, Frnce, 8-3 Août Yu T.W. (1977) Prmetriztion of surfce eportion rte for use numericl modelling. J. Applied Meteorology, 16, Duffie, J. A. nd Beckmn, W. A. (1991). Solr engineering of therml processes. nd d., John Wiley nd sons, New York. 994 p. Hrrison, L. P Fundmentls concepts nd definitions relting to humidity. In Wexler, A (ditor) Humidity nd moisture Vol 3, Reinhold Publishing Co., N.Y. 19. Terzi O; Keskin. (5) portion estimtion using gene expression progrmming. J. Applied Sciences, 5(3), Infrred Serices Incorportion. (9). missiity lues for common mterils. Ailble t com/mteril-1.htm. 73

11 АНАЛИЗ НА ПОМПЕНО АКУМУЛИРАЩА ВОДНОЕЛЕКТРИЧЕСКА ЦЕНТРАЛА КЪМ СИСТЕМА С ВЕТРОГЕНЕРАТОР И ФОТОВОЛТАИК ЗА ВНЕДРЯВАНЕ В ЕЛЕКТРИЧЕСКАТА МРЕЖА Ж. Нотон, В. Лазаров, Л. Стоянов Резюме: Тази статия има за цел анализ на помпено акумулираща водноелектрическа централа за запазване на енергия. Водният резервоар има за цел съхраняването на енергия за дни или сезони като по този начин се улесни разпространението на възобновяеми източници на енергия. Електричеството произведено в повече от хибридната система с възобновяеми източници на енергия се превръща в потенциална енергия посредством изпомпване на вода в горния резервоар, където тя може да се съхранява безкрайно много и в последствие да премине през хидравличната турбина произвеждайки електроенергия. Оценката на запасената енергия, на номиналните мощности на хидроенергийния възел и на загубите при изпаряване от водния резервоар са представени. Ключови думи: слънчева енергия, вятърна енергия, хибридни системи, запазване на енергия чрез изпомпване доц. Жил Нотон Изследователски център Виньола Корсикански Университет ЮМР ЦНРС 6134 рут де Сангинерс Ф Аячо, Франция Тел: Факс: имайл: доц. Владимир Лазаров Лаборатория по ВЕИ Електротехнически Факултет Технически Университет София бул. Кл. Охридски 8 1 София, България Тел: Факс: имейл: инж. Людмил Стоянов, докторант Лаборатория по ВЕИ Електротехнически Факултет Технически Университет София Изследователски център Виньола Корсикански Университет бул. Кл. Охридски 8 1 София, България Тел: Факс: имейл: Assoc Prof. Gilles Notton Reserch Centre of Vignol Uniersity of Corsic UMR CNRS 6134 route des Snguinires F Ajccio, Frnce Tel: Fx: mil: Assoc Prof. Vldimir Lzro Lbortory on Renewble nergy Sources Fculty of lectricl ngineering Technicl Uniersity of Sofi 8, Kl. Ohridski Bld. 1 Sofi, Bulgri Tel: Fx: mil: ng. Ludmil Stoyno, PhD student Lbortory on Renewble nergy Sources Fculty of lectricl ngineering Technicl Uniersity of Sofi Reserch Centre of Vignol Uniersity of Corsic 8, Kl. Ohridski Bld. 1 Sofi, Bulgri Tel: Fx: mil: 74

The International Association for the Properties of Water and Steam. Release on the Ionization Constant of H 2 O

The International Association for the Properties of Water and Steam. Release on the Ionization Constant of H 2 O IAPWS R-7 The Interntionl Assocition for the Properties of Wter nd Stem Lucerne, Sitzerlnd August 7 Relese on the Ioniztion Constnt of H O 7 The Interntionl Assocition for the Properties of Wter nd Stem

More information

THERMAL EXPANSION COEFFICIENT OF WATER FOR VOLUMETRIC CALIBRATION

THERMAL EXPANSION COEFFICIENT OF WATER FOR VOLUMETRIC CALIBRATION XX IMEKO World Congress Metrology for Green Growth September 9,, Busn, Republic of Kore THERMAL EXPANSION COEFFICIENT OF WATER FOR OLUMETRIC CALIBRATION Nieves Medin Hed of Mss Division, CEM, Spin, mnmedin@mityc.es

More information

Simulations of the irradiation and temperature dependence of the efficiency of tandem photoelectrochemical water-splitting systems

Simulations of the irradiation and temperature dependence of the efficiency of tandem photoelectrochemical water-splitting systems SUPPORTING INFORMATION FOR Simultions of the irrdition nd temperture dependence of the efficiency of tndem photoelectrochemicl wter-splitting systems Sophi Hussener, Shu Hu, Chengxing Xing, Adm Z. Weber,

More information

NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.

NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by. NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with

More information

Factors affecting the phonation threshold pressure and frequency

Factors affecting the phonation threshold pressure and frequency 3SC Fctors ffecting the phontion threshold pressure nd frequency Zhoyn Zhng School of Medicine, University of Cliforni Los Angeles, CA, USA My, 9 57 th ASA Meeting, Portlnd, Oregon Acknowledgment: Reserch

More information

A027 Uncertainties in Local Anisotropy Estimation from Multi-offset VSP Data

A027 Uncertainties in Local Anisotropy Estimation from Multi-offset VSP Data A07 Uncertinties in Locl Anisotropy Estimtion from Multi-offset VSP Dt M. Asghrzdeh* (Curtin University), A. Bon (Curtin University), R. Pevzner (Curtin University), M. Urosevic (Curtin University) & B.

More information

Measuring Electron Work Function in Metal

Measuring Electron Work Function in Metal n experiment of the Electron topic Mesuring Electron Work Function in Metl Instructor: 梁生 Office: 7-318 Emil: shling@bjtu.edu.cn Purposes 1. To understnd the concept of electron work function in metl nd

More information

Wind-induced vibration: a serviceability study

Wind-induced vibration: a serviceability study Proceedings of the 9th Interntionl Conference on Structurl Dynmics, EURODYN 214 Porto, Portugl, 3 June - 2 July 214 A. Cunh, E. Cetno, P. Ribeiro, G. Müller (eds.) ISSN: 2311-92; ISBN: 978-972-752-165-4

More information

FBR Neutronics: Breeding potential, Breeding Ratio, Breeding Gain and Doubling time

FBR Neutronics: Breeding potential, Breeding Ratio, Breeding Gain and Doubling time FBR eutronics: Breeding potentil, Breeding Rtio, Breeding Gin nd Doubling time K.S. Rjn Proessor, School o Chemicl & Biotechnology SASTRA University Joint Inititive o IITs nd IISc Funded by MHRD Pge 1

More information

Fundamentals of Analytical Chemistry

Fundamentals of Analytical Chemistry Homework Fundmentls of nlyticl hemistry hpter 9 0, 1, 5, 7, 9 cids, Bses, nd hpter 9(b) Definitions cid Releses H ions in wter (rrhenius) Proton donor (Bronsted( Lowry) Electron-pir cceptor (Lewis) hrcteristic

More information

Acceptance Sampling by Attributes

Acceptance Sampling by Attributes Introduction Acceptnce Smpling by Attributes Acceptnce smpling is concerned with inspection nd decision mking regrding products. Three spects of smpling re importnt: o Involves rndom smpling of n entire

More information

Freely propagating jet

Freely propagating jet Freely propgting jet Introduction Gseous rectnts re frequently introduced into combustion chmbers s jets. Chemicl, therml nd flow processes tht re tking plce in the jets re so complex tht nlyticl description

More information

Synoptic Meteorology I: Finite Differences September Partial Derivatives (or, Why Do We Care About Finite Differences?

Synoptic Meteorology I: Finite Differences September Partial Derivatives (or, Why Do We Care About Finite Differences? Synoptic Meteorology I: Finite Differences 16-18 September 2014 Prtil Derivtives (or, Why Do We Cre About Finite Differences?) With the exception of the idel gs lw, the equtions tht govern the evolution

More information

Intro to Nuclear and Particle Physics (5110)

Intro to Nuclear and Particle Physics (5110) Intro to Nucler nd Prticle Physics (5110) Feb, 009 The Nucler Mss Spectrum The Liquid Drop Model //009 1 E(MeV) n n(n-1)/ E/[ n(n-1)/] (MeV/pir) 1 C 16 O 0 Ne 4 Mg 7.7 14.44 19.17 8.48 4 5 6 6 10 15.4.41

More information

Department of Mechanical Engineering ME 322 Mechanical Engineering Thermodynamics. Lecture 33. Psychrometric Properties of Moist Air

Department of Mechanical Engineering ME 322 Mechanical Engineering Thermodynamics. Lecture 33. Psychrometric Properties of Moist Air Deprtment of Mechnicl Engineering ME 3 Mechnicl Engineering hermodynmics Lecture 33 sychrometric roperties of Moist Air Air-Wter Vpor Mixtures Atmospheric ir A binry mixture of dry ir () + ter vpor ()

More information

Tests for the Ratio of Two Poisson Rates

Tests for the Ratio of Two Poisson Rates Chpter 437 Tests for the Rtio of Two Poisson Rtes Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson

More information

Math 8 Winter 2015 Applications of Integration

Math 8 Winter 2015 Applications of Integration Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

More information

CML: Series Low Resistance Chip Resistor Specification

CML: Series Low Resistance Chip Resistor Specification CML: Series Low Resistnce Chip Resistor Specifiction 1. Scope This specifiction pplies to CML**** Series Low Resistnce Chip Resistor for use in electric equipment. *****METAL FOIL***** 2. Prt number CML

More information

The Thermodynamics of Aqueous Electrolyte Solutions

The Thermodynamics of Aqueous Electrolyte Solutions 18 The Thermodynmics of Aqueous Electrolyte Solutions As discussed in Chpter 10, when slt is dissolved in wter or in other pproprite solvent, the molecules dissocite into ions. In queous solutions, strong

More information

APPLICATION OF WATER EQUIVALENTS METHOD TO CALCULATION OF CERAMIC HEAT EXCHANGERS

APPLICATION OF WATER EQUIVALENTS METHOD TO CALCULATION OF CERAMIC HEAT EXCHANGERS METALLURGY AND FOUNDRY ENGINEERING Vol. 35, 2009, No. 2 Krzysztof Krczewski* APPLICATION OF WATER EQUIVALENTS METHOD TO CALCULATION OF CERAMIC HEAT EXCHANGERS Nottion: c het cpcity of ir, kj/(m 3 K), c

More information

The Wave Equation I. MA 436 Kurt Bryan

The Wave Equation I. MA 436 Kurt Bryan 1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string

More information

Thermal Diffusivity. Paul Hughes. Department of Physics and Astronomy The University of Manchester Manchester M13 9PL. Second Year Laboratory Report

Thermal Diffusivity. Paul Hughes. Department of Physics and Astronomy The University of Manchester Manchester M13 9PL. Second Year Laboratory Report Therml iffusivity Pul Hughes eprtment of Physics nd Astronomy The University of nchester nchester 3 9PL Second Yer Lbortory Report Nov 4 Abstrct We investigted the therml diffusivity of cylindricl block

More information

DIFA, University of Basilicata, Italy

DIFA, University of Basilicata, Italy Guido Msiello nd Crmine Serio DIFA, Uniersity of Bsilict, Itly 1 Bsic methodologicl steps to retriee surfce emissiity Step 1: Represent emissiity with Fourier cosine truncted series to lower it dimensionlity

More information

POLYPHASE CIRCUITS. Introduction:

POLYPHASE CIRCUITS. Introduction: POLYPHASE CIRCUITS Introduction: Three-phse systems re commonly used in genertion, trnsmission nd distribution of electric power. Power in three-phse system is constnt rther thn pulsting nd three-phse

More information

Section 4.8. D v(t j 1 ) t. (4.8.1) j=1

Section 4.8. D v(t j 1 ) t. (4.8.1) j=1 Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions

More information

Simulation of Eclipsing Binary Star Systems. Abstract

Simulation of Eclipsing Binary Star Systems. Abstract Simultion of Eclipsing Binry Str Systems Boris Yim 1, Kenny Chn 1, Rphel Hui 1 Wh Yn College Kowloon Diocesn Boys School Abstrct This report briefly introduces the informtion on eclipsing binry str systems.

More information

Determination of the activation energy of silicone rubbers using different kinetic analysis methods

Determination of the activation energy of silicone rubbers using different kinetic analysis methods Determintion of the ctivtion energy of silicone rubbers using different kinetic nlysis methods OU Huibin SAHLI ohmed BAIEE Thierry nd GELIN Jen-Clude FETO-ST Institute / Applied echnics Deprtment, 2 rue

More information

#6A&B Magnetic Field Mapping

#6A&B Magnetic Field Mapping #6A& Mgnetic Field Mpping Gol y performing this lb experiment, you will: 1. use mgnetic field mesurement technique bsed on Frdy s Lw (see the previous experiment),. study the mgnetic fields generted by

More information

Overview. Before beginning this module, you should be able to: After completing this module, you should be able to:

Overview. Before beginning this module, you should be able to: After completing this module, you should be able to: Module.: Differentil Equtions for First Order Electricl Circuits evision: My 26, 2007 Produced in coopertion with www.digilentinc.com Overview This module provides brief review of time domin nlysis of

More information

Minimum Energy State of Plasmas with an Internal Transport Barrier

Minimum Energy State of Plasmas with an Internal Transport Barrier Minimum Energy Stte of Plsms with n Internl Trnsport Brrier T. Tmno ), I. Ktnum ), Y. Skmoto ) ) Formerly, Plsm Reserch Center, University of Tsukub, Tsukub, Ibrki, Jpn ) Plsm Reserch Center, University

More information

NUMERICAL SIMULATION OF FRONTAL MIXED CLOUD SYSTEMS AND CLOUD MICROSTRUCTURE EFFECT ON SATELLITE SIGNAL

NUMERICAL SIMULATION OF FRONTAL MIXED CLOUD SYSTEMS AND CLOUD MICROSTRUCTURE EFFECT ON SATELLITE SIGNAL NUMERICAL SIMULATION OF FRONTAL MIXED CLOUD SYSTEMS AND CLOUD MICROSTRUCTURE EFFECT ON SATELLITE SIGNAL V. Bkhnov, O. Kryvook, B. Dormn Ukrinin Hydrometeorologicl Reserch Institute, Avenue of Science 37,

More information

PHYSICS 211 MIDTERM I 21 April 2004

PHYSICS 211 MIDTERM I 21 April 2004 PHYSICS MIDERM I April 004 Exm is closed book, closed notes. Use only your formul sheet. Write ll work nd nswers in exm booklets. he bcks of pges will not be grded unless you so request on the front of

More information

HT Module 2 Paper solution. Module 2. Q6.Discuss Electrical analogy of combined heat conduction and convection in a composite wall.

HT Module 2 Paper solution. Module 2. Q6.Discuss Electrical analogy of combined heat conduction and convection in a composite wall. HT Module 2 Pper solution Qulity Solutions wwwqulitytutorilin Module 2 Q6Discuss Electricl nlogy of combined het conduction nd convection in composite wll M-16-Q1(c)-5m Ans: It is frequently convient to

More information

COMMISSIONING THE NEW PITCH DAMPING RIG AT THE CSIR WIND TUNNEL FACILITIES

COMMISSIONING THE NEW PITCH DAMPING RIG AT THE CSIR WIND TUNNEL FACILITIES 3 RD INTERNATIONAL SYMPOSIUM ON BALLISTICS TARRAGONA, SPAIN 6-0 APRIL 007 COMMISSIONING THE NEW PITCH DAMPING RIG AT THE CSIR WIND TUNNEL FACILITIES F. Dionisio, L. Vn Zyl nd A. Stockenström 3 CSIR DPSS,

More information

Continuous Random Variables

Continuous Random Variables STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht

More information

Flow in porous media

Flow in porous media Red: Ch 2. nd 2.2 PART 4 Flow in porous medi Drcy s lw Imgine point (A) in column of wter (figure below); the point hs following chrcteristics: () elevtion z (2) pressure p (3) velocity v (4) density ρ

More information

MAC-solutions of the nonexistent solutions of mathematical physics

MAC-solutions of the nonexistent solutions of mathematical physics Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements MAC-solutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE

More information

Line Integrals. Partitioning the Curve. Estimating the Mass

Line Integrals. Partitioning the Curve. Estimating the Mass Line Integrls Suppose we hve curve in the xy plne nd ssocite density δ(p ) = δ(x, y) t ech point on the curve. urves, of course, do not hve density or mss, but it my sometimes be convenient or useful to

More information

Design Data 1M. Highway Live Loads on Concrete Pipe

Design Data 1M. Highway Live Loads on Concrete Pipe Design Dt 1M Highwy Live Lods on Concrete Pipe Foreword Thick, high-strength pvements designed for hevy truck trffic substntilly reduce the pressure trnsmitted through wheel to the subgrde nd consequently,

More information

Design and Testing of Coils for Pulsed Electromagnetic Forming

Design and Testing of Coils for Pulsed Electromagnetic Forming Design nd Testing of Coils for Pulsed Electromgnetic Forming. Golovshchenko, N. Bessonov, R. Dvies Ford Reserch & Advnced Engineering, Derborn, UA University of Michign-Derborn, Derborn, UA Pcific Northwest

More information

Seasonal Variation of Internal Parameters of an Amorphous Silicon (a-si) Thin Film Photovoltaic Module

Seasonal Variation of Internal Parameters of an Amorphous Silicon (a-si) Thin Film Photovoltaic Module NTENATONAL JOUNAL of ENEWABLE ENEGY EEACH ustu Eke et l., ol., No.4, 0 esonl rition of nternl Prmeters of n Amorphous ilicon (-i) Thin Film Photovoltic Module ustu Eke*, **, ener Oktik*** * Mugl ıtkı Kocmn

More information

Strategy: Use the Gibbs phase rule (Equation 5.3). How many components are present?

Strategy: Use the Gibbs phase rule (Equation 5.3). How many components are present? University Chemistry Quiz 4 2014/12/11 1. (5%) Wht is the dimensionlity of the three-phse coexistence region in mixture of Al, Ni, nd Cu? Wht type of geometricl region dose this define? Strtegy: Use the

More information

Chapter H1: Introduction, Heat Equation

Chapter H1: Introduction, Heat Equation Nme Due Dte: Problems re collected on Wednesdy. Mth 3150 Problems Hbermn Chpter H1 Submitted work. Plese submit one stpled pckge per problem set. Lbel ech problem with its corresponding problem number,

More information

The Basic Functional 2 1

The Basic Functional 2 1 2 The Bsic Functionl 2 1 Chpter 2: THE BASIC FUNCTIONAL TABLE OF CONTENTS Pge 2.1 Introduction..................... 2 3 2.2 The First Vrition.................. 2 3 2.3 The Euler Eqution..................

More information

Precalculus Spring 2017

Precalculus Spring 2017 Preclculus Spring 2017 Exm 3 Summry (Section 4.1 through 5.2, nd 9.4) Section P.5 Find domins of lgebric expressions Simplify rtionl expressions Add, subtrct, multiply, & divide rtionl expressions Simplify

More information

200 points 5 Problems on 4 Pages and 20 Multiple Choice/Short Answer Questions on 5 pages 1 hour, 48 minutes

200 points 5 Problems on 4 Pages and 20 Multiple Choice/Short Answer Questions on 5 pages 1 hour, 48 minutes PHYSICS 132 Smple Finl 200 points 5 Problems on 4 Pges nd 20 Multiple Choice/Short Answer Questions on 5 pges 1 hour, 48 minutes Student Nme: Recittion Instructor (circle one): nme1 nme2 nme3 nme4 Write

More information

Math 0230 Calculus 2 Lectures

Math 0230 Calculus 2 Lectures Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two

More information

An Application of Phase Change Technology in a Greenhouse 1

An Application of Phase Change Technology in a Greenhouse 1 An Appliction of Phse Chnge Technology in Greenhouse 1 Yuning Liu Cho Chen Hifeng Guo Hilin Yue grdute student professor grdute student grdute student Deprtment of Building Environment nd Fcilities Engineering,

More information

Computation and Modeling of the Air-Sea Heat and Momentum Fluxes. By Dr. Nasser H. Zaker

Computation and Modeling of the Air-Sea Heat and Momentum Fluxes. By Dr. Nasser H. Zaker Computtion nd Modeling of the Air-e et nd Momentum Fluxes By Dr. Nsser. Zker ICTP June 00 1 Introduction The ocen receives energy through the ir-se interfce by exchnge of momentum nd het. The turbulent

More information

Lecture 10: PN Junction & MOS Capacitors

Lecture 10: PN Junction & MOS Capacitors Lecture 10: P Junction & MOS Cpcitors Prof. iknej eprtment of EECS Lecture Outline Review: P Junctions Therml Equilibrium P Junctions with Reverse Bis (3.3-3.6 MOS Cpcitors (3.7-3.9: Accumultion, epletion,

More information

Data Assimilation. Alan O Neill Data Assimilation Research Centre University of Reading

Data Assimilation. Alan O Neill Data Assimilation Research Centre University of Reading Dt Assimiltion Aln O Neill Dt Assimiltion Reserch Centre University of Reding Contents Motivtion Univrite sclr dt ssimiltion Multivrite vector dt ssimiltion Optiml Interpoltion BLUE 3d-Vritionl Method

More information

7.6 The Use of Definite Integrals in Physics and Engineering

7.6 The Use of Definite Integrals in Physics and Engineering Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 7.6 The Use of Definite Integrls in Physics nd Engineering It hs been shown how clculus cn be pplied to find solutions to geometric problems

More information

Numerical Integration

Numerical Integration Chpter 1 Numericl Integrtion Numericl differentition methods compute pproximtions to the derivtive of function from known vlues of the function. Numericl integrtion uses the sme informtion to compute numericl

More information

ENERGY-BASED METHOD FOR GAS TURBINE ENGINE DISK BURST SPEED CALCULATION

ENERGY-BASED METHOD FOR GAS TURBINE ENGINE DISK BURST SPEED CALCULATION 28 TH INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES ENERGY-BASED METHOD FOR GAS TURBINE ENGINE DISK BURST SPEED CALCULATION Anton N. Servetnik Centrl Institute of Avition Motors, Moscow, Russi servetnik@cim.ru

More information

FIRE PROTECTION HYDRAULICS

FIRE PROTECTION HYDRAULICS GAP.1.0.1 A Puliction of Glol Asset Protection Services LLC FIRE PROTECTION HYDRAULICS INTRODUCTION Fluid mechnics is concerned with the forces nd motions ssocited with gses nd liquids. Hydrulics, distinct

More information

Optimum Design of a Horizontal Condenser

Optimum Design of a Horizontal Condenser Fourth Interntionl Wter echnology Conference IWC 99, Alexndri, Egypt 9 Abstrct Optimum esign of Horizontl Condenser A.M. Abdllh nd Aly Krmeldin Atomic Energy Authority-Nucler eserch Center ectors eprtment

More information

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1 3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =

More information

Introduction to Mechanics Practice using the Kinematics Equations

Introduction to Mechanics Practice using the Kinematics Equations Introduction to Mechnics Prctice using the Kinemtics Equtions Ln Sheridn De Anz College Jn 24, 2018 Lst time finished deriing the kinemtics equtions some problem soling prctice Oeriew using kinemtics equtions

More information

l 2 p2 n 4n 2, the total surface area of the

l 2 p2 n 4n 2, the total surface area of the Week 6 Lectures Sections 7.5, 7.6 Section 7.5: Surfce re of Revolution Surfce re of Cone: Let C be circle of rdius r. Let P n be n n-sided regulr polygon of perimeter p n with vertices on C. Form cone

More information

ESCI 343 Atmospheric Dynamics II Lesson 14 Inertial/slantwise Instability

ESCI 343 Atmospheric Dynamics II Lesson 14 Inertial/slantwise Instability ESCI 343 Atmospheric Dynmics II Lesson 14 Inertil/slntwise Instbility Reference: An Introduction to Dynmic Meteorology (3 rd edition), J.R. Holton Atmosphere-Ocen Dynmics, A.E. Gill Mesoscle Meteorology

More information

MATHEMATICAL SIMULATION OF GEOTHERMAL HEAT TRANSFER IN HOT DRY ROCK UNDERGROUND HEAT EXCHANGERS

MATHEMATICAL SIMULATION OF GEOTHERMAL HEAT TRANSFER IN HOT DRY ROCK UNDERGROUND HEAT EXCHANGERS SCIENIFIC PROCEEDINGS 009, Fculty of Mechnicl Engineering, SU in Brtislv MAHEMAICAL SIMULAION OF GEOHERMAL HEA RANSFER IN HO DRY ROCK UNDERGROUND HEA EXCHANGERS Kristín KÁZMÉROVÁ, Michl MASARYK Slovk University

More information

A Convergence Theorem for the Improper Riemann Integral of Banach Space-valued Functions

A Convergence Theorem for the Improper Riemann Integral of Banach Space-valued Functions Interntionl Journl of Mthemticl Anlysis Vol. 8, 2014, no. 50, 2451-2460 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/10.12988/ijm.2014.49294 A Convergence Theorem for the Improper Riemnn Integrl of Bnch

More information

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz

More information

Mapping the delta function and other Radon measures

Mapping the delta function and other Radon measures Mpping the delt function nd other Rdon mesures Notes for Mth583A, Fll 2008 November 25, 2008 Rdon mesures Consider continuous function f on the rel line with sclr vlues. It is sid to hve bounded support

More information

A. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.

A. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C. A. Limits - L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c

More information

Deteriorating Inventory Model for Waiting. Time Partial Backlogging

Deteriorating Inventory Model for Waiting. Time Partial Backlogging Applied Mthemticl Sciences, Vol. 3, 2009, no. 9, 42-428 Deteriorting Inventory Model for Witing Time Prtil Bcklogging Nit H. Shh nd 2 Kunl T. Shukl Deprtment of Mthemtics, Gujrt university, Ahmedbd. 2

More information

Non-Linear & Logistic Regression

Non-Linear & Logistic Regression Non-Liner & Logistic Regression If the sttistics re boring, then you've got the wrong numbers. Edwrd R. Tufte (Sttistics Professor, Yle University) Regression Anlyses When do we use these? PART 1: find

More information

Z0103/07/09 series. Blocking voltage to 800 V (NA and NN types)

Z0103/07/09 series. Blocking voltage to 800 V (NA and NN types) Rev. September Product dt. Product profile. Description Pssivted trics in conventionl nd surfce mounting pckges. Intended for use in pplictions requiring high bidirectionl trnsient nd blocking voltge cpbility.

More information

Section 14.3 Arc Length and Curvature

Section 14.3 Arc Length and Curvature Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in

More information

Thermodynamic analysis of geothermal heat pump during the cold season

Thermodynamic analysis of geothermal heat pump during the cold season IOP onference eries: Mterils cience nd ngineering PAPR OPN A hermodynmic nlysis of geotherml het pump during the cold seson o cite this rticle: G Dumitrcu et l 2016 IOP onf. er.: Mter. ci. ng. 147 012137

More information

Construction and Selection of Single Sampling Quick Switching Variables System for given Control Limits Involving Minimum Sum of Risks

Construction and Selection of Single Sampling Quick Switching Variables System for given Control Limits Involving Minimum Sum of Risks Construction nd Selection of Single Smpling Quick Switching Vribles System for given Control Limits Involving Minimum Sum of Risks Dr. D. SENHILKUMAR *1 R. GANESAN B. ESHA RAFFIE 1 Associte Professor,

More information

Solution of Tutorial 2 Converter driven DC motor drive

Solution of Tutorial 2 Converter driven DC motor drive chool of Electricl Engineering & Telecommunictions, UNW olution of Tutoril Converter driven DC motor drive Question 1. T V s D V I L E V 50 V,.5, I 0 A rted rted f 400 Hz, 0 rev/ min s rted (i) 0 6.8 rd

More information

ECO 317 Economics of Uncertainty Fall Term 2007 Notes for lectures 4. Stochastic Dominance

ECO 317 Economics of Uncertainty Fall Term 2007 Notes for lectures 4. Stochastic Dominance Generl structure ECO 37 Economics of Uncertinty Fll Term 007 Notes for lectures 4. Stochstic Dominnce Here we suppose tht the consequences re welth mounts denoted by W, which cn tke on ny vlue between

More information

3.4 Numerical integration

3.4 Numerical integration 3.4. Numericl integrtion 63 3.4 Numericl integrtion In mny economic pplictions it is necessry to compute the definite integrl of relvlued function f with respect to "weight" function w over n intervl [,

More information

Key comparison BIPM.RI(I)-K7 of the air-kerma standards of the ENEA-INMRI, Italy and the BIPM in mammography x-rays

Key comparison BIPM.RI(I)-K7 of the air-kerma standards of the ENEA-INMRI, Italy and the BIPM in mammography x-rays Key comprison BIPM.RI(I)-K7 of the ir-kerm stndrds of the ENEA-INMRI, Itly nd the BIPM in mmmogrphy x-rys C Kessler, D T Burns, P Roger, M P Toni*, M Pinto*, M Bovi*, G Cppdozzi*, C Silvestri* Bureu Interntionl

More information

MATH STUDENT BOOK. 10th Grade Unit 5

MATH STUDENT BOOK. 10th Grade Unit 5 MATH STUDENT BOOK 10th Grde Unit 5 Unit 5 Similr Polygons MATH 1005 Similr Polygons INTRODUCTION 3 1. PRINCIPLES OF ALGEBRA 5 RATIOS AND PROPORTIONS 5 PROPERTIES OF PROPORTIONS 11 SELF TEST 1 16 2. SIMILARITY

More information

EMF Notes 9; Electromagnetic Induction ELECTROMAGNETIC INDUCTION

EMF Notes 9; Electromagnetic Induction ELECTROMAGNETIC INDUCTION EMF Notes 9; Electromgnetic nduction EECTOMAGNETC NDUCTON (Y&F Chpters 3, 3; Ohnin Chpter 3) These notes cover: Motionl emf nd the electric genertor Electromgnetic nduction nd Frdy s w enz s w nduced electric

More information

Using QM for Windows. Using QM for Windows. Using QM for Windows LEARNING OBJECTIVES. Solving Flair Furniture s LP Problem

Using QM for Windows. Using QM for Windows. Using QM for Windows LEARNING OBJECTIVES. Solving Flair Furniture s LP Problem LEARNING OBJECTIVES Vlu%on nd pricing (November 5, 2013) Lecture 11 Liner Progrmming (prt 2) 10/8/16, 2:46 AM Olivier J. de Jong, LL.M., MM., MBA, CFD, CFFA, AA www.olivierdejong.com Solving Flir Furniture

More information

High speed machines using advanced magnetic materials analyzed by appropriate finite element models

High speed machines using advanced magnetic materials analyzed by appropriate finite element models High speed mchines using dvnced mgnetic mterils nlyzed y pproprite finite element models G. D. KALOKIRIS, A. G. KLADAS Lortory of Electricl Mchines nd Power Electronics, Electric Power Division Fculty

More information

A Signal-Level Fusion Model for Image-Based Change Detection in DARPA's Dynamic Database System

A Signal-Level Fusion Model for Image-Based Change Detection in DARPA's Dynamic Database System SPIE Aerosense 001 Conference on Signl Processing, Sensor Fusion, nd Trget Recognition X, April 16-0, Orlndo FL. (Minor errors in published version corrected.) A Signl-Level Fusion Model for Imge-Bsed

More information

Impact of Device Scaling on Deep Sub-Micron Transistor Reliability - A Study of Reliability Trends Using SRAM

Impact of Device Scaling on Deep Sub-Micron Transistor Reliability - A Study of Reliability Trends Using SRAM Impct of Device Scling on Deep Sub-Micron rnsistor Relibility - A Study of Relibility rends Using SRAM Mrk White,, Bing Hung, Jin Qin, Zvi Gur, Michel lmor,, Yun Chen, Json Heidecker, Duc Nguyen, Joseph

More information

On the Pairs of Orthogonal Ruled Surfaces

On the Pairs of Orthogonal Ruled Surfaces EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5 No. 2 2012 205-210 ISSN 1307-5543 www.ejpm.com On the Pirs of Orthogonl Ruled Surfces Filiz KANBAY Deprtment of Mthemtics Fculty of Arts nd Science

More information

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0. STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t

More information

CALCULATED POWDER X-RAY DIFFRACTION LINE PROFILES VIA ABSORPTION

CALCULATED POWDER X-RAY DIFFRACTION LINE PROFILES VIA ABSORPTION 16 17 CALCULATED POWDER X-RAY DFFRACTON LNE PROFLES VA ABSORPTON Keji Liu nd Heifen Chen School of Mteril Science nd Engineering, Shnghi nstitute of Technology, Shnghi, Chin 2233 ABSTRACT We hve clculted

More information

x = a To determine the volume of the solid, we use a definite integral to sum the volumes of the slices as we let!x " 0 :

x = a To determine the volume of the solid, we use a definite integral to sum the volumes of the slices as we let!x  0 : Clculus II MAT 146 Integrtion Applictions: Volumes of 3D Solids Our gol is to determine volumes of vrious shpes. Some of the shpes re the result of rotting curve out n xis nd other shpes re simply given

More information

Questions on Ship and Offshore Hydromechanics

Questions on Ship and Offshore Hydromechanics Questions on Ship nd Offshore Hydromechnics (in progress of formtion) JMJ Journée Contents: Question 1 Stnding Wves Question Regulr Wve Observtion Question 3 Axes Systems in Ship Motion Clcultions Question

More information

15. Quantisation Noise and Nonuniform Quantisation

15. Quantisation Noise and Nonuniform Quantisation 5. Quntistion Noise nd Nonuniform Quntistion In PCM, n nlogue signl is smpled, quntised, nd coded into sequence of digits. Once we hve quntised the smpled signls, the exct vlues of the smpled signls cn

More information

Lecture 1. Functional series. Pointwise and uniform convergence.

Lecture 1. Functional series. Pointwise and uniform convergence. 1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is

More information

Physics Honors. Final Exam Review Free Response Problems

Physics Honors. Final Exam Review Free Response Problems Physics Honors inl Exm Review ree Response Problems m t m h 1. A 40 kg mss is pulled cross frictionless tble by string which goes over the pulley nd is connected to 20 kg mss.. Drw free body digrm, indicting

More information

Lecture 14: Quadrature

Lecture 14: Quadrature Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl

More information

Numerical integration

Numerical integration 2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter

More information

Experiment 9: DETERMINATION OF WEAK ACID IONIZATION CONSTANT & PROPERTIES OF A BUFFERED SOLUTION

Experiment 9: DETERMINATION OF WEAK ACID IONIZATION CONSTANT & PROPERTIES OF A BUFFERED SOLUTION Experiment 9: DETERMINATION OF WEAK ACID IONIZATION CONSTANT & PROPERTIES OF A BUFFERED SOLUTION Purpose: Prt I: The cid ioniztion constnt of wek cid is to be determined, nd the cid is identified ccordingly.

More information

1 Online Learning and Regret Minimization

1 Online Learning and Regret Minimization 2.997 Decision-Mking in Lrge-Scle Systems My 10 MIT, Spring 2004 Hndout #29 Lecture Note 24 1 Online Lerning nd Regret Minimiztion In this lecture, we consider the problem of sequentil decision mking in

More information

Your Thoughts. Mechanics Lecture 16, Slide 1

Your Thoughts. Mechanics Lecture 16, Slide 1 Your Thoughts I get dizzy with ll the equtions being shifted, spun nd switched so much in the pre-lectures. If the pre-lectures for, 3 nd 4 re like tht, I m pretty worried. Are we going to be rcing spheres,

More information

Solutions to Supplementary Problems

Solutions to Supplementary Problems Solutions to Supplementry Problems Chpter 8 Solution 8.1 Step 1: Clculte the line of ction ( x ) of the totl weight ( W ).67 m W = 5 kn W 1 = 16 kn 3.5 m m W 3 = 144 kn Q 4m Figure 8.10 Tking moments bout

More information

11.1 Exponential Functions

11.1 Exponential Functions . Eponentil Functions In this chpter we wnt to look t specific type of function tht hs mny very useful pplictions, the eponentil function. Definition: Eponentil Function An eponentil function is function

More information

u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph.

u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph. nlyzing Dmped Oscilltions Prolem (Medor, exmple 2-18, pp 44-48): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + $

More information

MATHS NOTES. SUBJECT: Maths LEVEL: Higher TEACHER: Aidan Roantree. The Institute of Education Topics Covered: Powers and Logs

MATHS NOTES. SUBJECT: Maths LEVEL: Higher TEACHER: Aidan Roantree. The Institute of Education Topics Covered: Powers and Logs MATHS NOTES The Institute of Eduction 06 SUBJECT: Mths LEVEL: Higher TEACHER: Aidn Rontree Topics Covered: Powers nd Logs About Aidn: Aidn is our senior Mths techer t the Institute, where he hs been teching

More information

Czechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction

Czechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction Czechoslovk Mthemticl Journl, 55 (130) (2005), 933 940 ESTIMATES OF THE REMAINDER IN TAYLOR S THEOREM USING THE HENSTOCK-KURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When rel-vlued

More information

Basic model for traffic interweave

Basic model for traffic interweave Journl of Physics: Conference Series PAPER OPEN ACCESS Bsic model for trffic interweve To cite this rticle: Ding-wei Hung 25 J. Phys.: Conf. Ser. 633 227 Relted content - Bsic sciences gonize in Turkey!

More information