The Pelton Wheel

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Transcript of The Pelton Wheel

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    Joseph GAZALET Monday, November 9th, 2015H !"# H001$%&00'En( Me)han*)a+ En(*neer*n(, ear -

    '59E! . Me)han*)a+ En(*neer*n( /)*en)e 9

    The e+ton hee+/perv*sor# "r M Na3ar*n*a

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    3

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    !ntrod)t*on

    Des%ite the 2ide variety of fluid "achinery availale, fluid "achines can e classified into t2o roadcategories ased on their o%erating %rinci%les  positive-displacement   "achines and rotodynamic "achines. 1he o%eration of %ositive-dis%lace"ent "achines is ased on volu"e variations of the fluid2ithin a control s%ace such as a %iston or %ortion of conduit 2ith a fle&ile oundary. With the %ossilee&ce%tion of relatively ne2 and li"ited use oceanic energy harnessing technology, the "a7ority of %ositive-dis%lace"ent "achines are pumps, 2hich are used to convey 8inetic or %ressure energy to a

    fluid. 'n the other hand, energy transfers et2een fluids and rotodyna"ic "achines occur at thecontact et2een the fluid and a rotating %art called a rotor  2hich consists of a set of radially sy""etriclades or vanes. De%ending on the orientation of the rotor relative to the direction of flo2 across it,rotodyna"ic "achines can further e differentiated into axial-flow  "achines, radial-flow  "achines, andmixed-flow  "achines. 1he ter" 9a&ial-flo2: descries "achines for 2hich the rotation a&is of the rotor is %arallel to the direction of flo2, 2hereas "achines for 2hich the rotating "otion of the rotor occurs2ithin the %lane of fluid flo2 are classified as radial-flo2 "achines. ;f the orientation of the "achineE5 of land, air, and "arine vehicles de"onstrate another, so"e2hat si"ilar a%%lication of turine technology, 2herey the flo2 of gases fro" an ;>EE

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    Apparats

    1he e&%eri"ental a%%aratus used for the %ur%ose of this e&%eri"ent is sho2n in figure +. @igure + a5 ista8en fro" the Heriot-Watt University C Duai >a"%us laoratory sheet titled The Pelton Wheel , andfigures + 5, + c5, and + d5 are %rovided y 1aaish S;DD;U;, 2ho too8 a%%aratus %hotogra%hs for grou% of the Mechanical Engineering Science "odule of the year (*+.

    Water is %u"%ed fro" a reservoir at the otto" of the hydraulics ench y the %u"% sho2n in figure +c5, 2hich %rovides enough head to the 2ater flo2 in order to ensure the %ro%er functioning of the

    a%%aratus. !t its o%ti"al level, the %u"% dra2s0.37 kW 

     of %o2er. 1he flo2 rate can e controlled

    y a valve %laced in the %i%e syste", and is left in the fully o%en throughout the e&%eri"ent. ! no00leensures the %ro%er for"ation of an unconfined 7et at at"os%heric te"%erature. 1he no00le is e4ui%%ed2ith a s%ecial valve called a spear valve 2hich controls the flo2 rate y ad7usting the area at theno00le e&it in order to 8ee% the 7et velocity constant. 1his is achieved y "oving a s%ear sha%edco"%onent in the valve hence the na"e of the valve5 in2ards and out2ards along the direction of flo2 using a "anually o%erated s%indle on the no00le. !s the s%ear is "oved for2ard along the flo2direction, it induces a reduction in annular area availale at the no00le e&it, 2hich reduces flo2 rateand increases %ressure at the no00le. 1he o%%osite occurs 2hen the s%ear is retracted. 1he s%ear isso sha%ed that it ensures the annular flo2 it %roduces coalesces into a circular flo2 al"osti""ediately. ! %ressure gauge "ounted on the no00le %rovides an indication of its %ressure in ars

    1 ¿̄105 Pa 5. 1he 7et i"%acts the Pelton 2heel uc8ets sho2n in figure in the ne&t section5 at a

    5

    Figure 1 – Experimental Apparatus

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    %itch radius ofr p=80mm  and causes the 2heel to rotate. ! "echanical ra8e, 2hich co"%rises

    t2o s%ring alances attached to a fra"e and connected y a stra%, is used to a%%ly a loading tor4ue

    on the turine dyna"o"eter 2ith a "easured circu"ferenceldynamo=34.5 cm , as sho2n in figure

    + 5. 1he s%ring alance designated as :s%ring alance +: dis%lays the tensile load to 2hich it is

    su7ected in grams of force    1gf =9.81×10−3

     N  5, 2hile the s%ring alance deno"inated 9s%ring

    alance (: dis%lays its tensile load in a scale of kilograms of force   1kgf =9.81 N  5. ! reflector is

    %laced on the dyna"o"eter and %er"its the use of a %ortale laser tacho"eter for the "easuring of dyna"o"eter rotational s%eed 2hich is the sa"e as that of the turine5. 1he availale tacho"eter 

    "odel dis%lays rotational s%eed in rotations per minute orrpm

       1rpm=

    2 π 

    60rad / s

    5. ! valve at

    the otto" of the "easuring tan8 can e shut, 2hich initiates the filling of the tan8 as 2ater isdischarged fro" the turine. 1he flo2 rate can then e deter"ined y "easuring the ti"e re4uired tofill an aritrary tan8 volu"e using a sto%2atch 2hich dis%lays ti"e in seconds. 1he filled volu"e is

    dis%layed y the indicator sho2n in figure + d5 in liters    1l=1dm3=0.001m3 5.

    Theory

    When dealing 2ith fluids, the conce%t of  pressure  is funda"ental. Pressure can only %ro%erly e

    defined y considering an infinitely s"all outer surfacedA

     of a rando" volu"e 2ith a total outer 

    surface A

     such as the one illustrated in figure (.

    Pressure is defined at a %oint and is the result of the total force a%%lied at that %oint in the directionnor"al i.e. %er%endicular5 to the infinitely s"all surface around it eing distriuted unifor"ly over thatele"entary surface. ;f a non-nor"al force is a%%lied at a %oint, only the nor"al co"%onent of that force

    2ill result in %ressure at that %oint. ;f a nor"al forceδF 

     is a%%lied to a %oint 2ithin the ele"entary

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    Figure 2 – Random Volume under Pressure

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    surface dA , the %ressure   p  e&erted at that %oint on the ele"entary surface is  p=δF 

    dA . ;f 

    several nor"al forcesδF i   are a%%lied at n  different %oints across the outer surface of the

    considered control volu"e, the total nor"al force a%%lied to the surface is  F =∑i

    n

    δF i . 1his total

    force is different fro" the net force 2hich ta8es into account the directions of the forces. ;f a force

    δF i   is a%%lied at every %oint i  on the volu"e outer surface, the %ressure at each %oint is

     pi=δF i

    dA . 1he total force a%%lied nor"ally to the surface is F =∑ δF i=∯ p i dA .  ;f 

    additionally the forces a%%lied at every %ointi

     across the volu"e surface are e4ual, such that

    δF i=δF  , then the %ressure  pi   at every %oint i   on the surface is the sa"e and

     pi= p=

    δF 

    dA . ;n such a case, the %ressure is said to e unifor" across the surface, and the total

    nor"al force is unifor"ly distriuted over the entire surface A

    . 1hus for a unifor" %ressure field

    over a surface A

    , the total nor"al force a%%lied on that surface can e 2ritten as follo2s

     F =∯  p dA= pA yields→

     p= F 

     A

     

     !ny surface su"erged 2ithin a fluid volu"e e&%eriences %ressure due to the 2eight of the fluid

    directly aove it. 1his %ressure is called hydrostatic pressure. !s its na"e i"%lies, this %ressure is aninherent conse4uence of the si"%le e&istence of the fluid volu"e 2hich has a "ass5 2ithin agravitational field, and is %resent in all fluid volu"es, even at rest. ;n order to 4uantify this %ressure,

    the density ρ

     of the fluid, 2hich is the "ass of this fluid %er unit volu"e, "ust e 8no2n. 1he

    density of an infinitely s"all volu"edV 

     of "atter 2ith a corres%onding "assdm

     is e&%ressed

    as follo2s

     ρ=dm

    dV 

    ;f this ratio is the sa"e for all the infinitely s"all volu"es that constitute a volu"e V   of "atter, the

    "ass M   of that "atter is given y the follo2ing e&%ression

     M =∰ ρ dV = ρ∰dV = ρV yields→

     ρ= M 

    ;n such a case, the volu"e is said to e of unifor" or constant5 density. @luids for 2hich the densityfluctuates significantly 2ith changes in a%%lied %ressure are characteri0ed as compressible.>onversely, fluids for 2hich significant increases in density only occur at %ressures several orders of 

    7

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    "agnitude greater than the a%%lied %ressures are considered to e incompressible. 1he density of 

    2ater at an at"os%heric %ressure Patm ≈1.013 ¿̄   and an a"ient te"%erature of 20°     is

     ρ!ater ≈1000 kg /m3

    . Within a ho"ogenous volu"e of fluid of unifor" density, an infinitely thin

    layer of that fluid is su7ected to the 2eight of the fluid colu"n aove it. ;f the considered layer has an

    area A

    , and is located at a distance "

      elo2 the fluid surface, the colu"n volu"e is

    V = "A . 1herefore, the 2eight of that colu"n is  F = ρg"A , 2here g=9.81 N /kg   is the

    acceleration due to gravity on the earth

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    $eferring to figure 3, if a s"all ele"ent located et2een t2o flo2 cross-sections se%arated y a

    distanceδs

     is considered, the force acting on the ele"ent surface u%strea" of the flo2 is pA

    and the force acting on the ele"ent surface do2nstrea" of the flo2 is( p +δp )( A+δA)

    , 2here

    δp   is the %ressure variation et2een the t2o cross-sections, and δA   is the variation in area

    et2een the t2o cross sections. !dditionally, the %ressure at the outer oundary of the flo2 varies

    across the length of the ele"ent, ut can e assu"ed as e4uivalent to a "ean %ressure  p+kδp ,

    unifor" across the entire oundary, 2herek 

     is a %ositive fraction less than unity. ;f the flo2 is

    radially sy""etrical, this %ressure does not result in a net force co"%onent a%%lied to the ele"ent inthe direction nor"al to the flo2. ;t does ho2ever result in a net force co"%onent in the direction of flo2

    due to that %ressure acting u%on the %ro7ected volu"eδA

    . 1he resulting force on the ele"ent in

    the direction of flo2 is thus ( p+kδP)δA . ;t can e noted that this force is non-e&istent if there is

    no variation in cross-sectional area et2een the t2o %oints of the flo2. @inally, the flo2 ele"ent is alsosu7ected to the a&ial co"%onent of the 2eight of the fluid it contains, the radial co"%onent eingco"%ensated y the flo2 oundary. 1he a&ial co"%onent of 2eight acting on the ele"ent is

    W  cos$= ρgδV  cos$   2here  δV    is the volu"e of the ele"ent. ;f the length δs   of the

    ele"ent is infinitely s"all, it can e 2ritten asds

    , and the volu"eδV 

     of the ele"ent can e

    e&%ressed as follo2s

    limds →0

    δV =dV = A ds

    1herefore, for an ele"ent of flo2 2ith an infinitely s"all length, the net forceδF 

     acting on the

    ele"ent is

    limds →0

    δF =dF = pA+( p+kδp ) δA−( p+δp ) ( A+δA )− ρgA dscos$

    /oting thatdscos$=d"

    , 2here "

     is the elevation of a flo2 %oint aove a conveniently set

    reference, and a%%lying /e2ton

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    is initially different fro" the initial velocity of the %article yδ#s , and the fact that in the ti"e δt  ,

    the i"%osed velocity at that %oint changes yδ#t  . 1he change in a&ial velocity is then

    δ#=δ#t 

    δt   δt +

    δ#s

    δs  δs

    ;f this variation of a&ial velocity is considered over an infinitely s"all %eriod of ti"edt 

     at a given

    %oint of the flo2, then the corres%onding dis%lace"ent of a %article at that %oint isds=# dt 

    .

     !ssu"ing the variations in a&ial velocity 2ith regards to ti"e and %osition are %redictale at that %oint,

    the 4uantities

    δ#t 

    δt   and δ#sδs  are such that

    limδt→ 0

    δ#t 

    δt   =

    & #

    & t  and

    limδt →0

    δ#s

    δs  =

    & #

    & s, and the

    total variation of a&ial velocity at the considered %oint can e e&%ressed as an e&act differential asfollo2s

    d#=

    & t  dt +

    & #

    & s ds=

    & #

    &t   dt +#

     & #

    & s dt 

    @inally, an e&%ression for a&ial acceleration 2ith regards to ti"e at the considered %oint can ederived

    as=d#

    dt  =

    & t  +#

     & #

    & s

    ;f the variations in a&ial velocity are %redictale 2ith regards to ti"e and %osition at every cross-sectionand at any ti"e, this e&%ression is a%%licale to the entire flo2 for its entire duration. $ecalling that thestudied flo2 is steady, its velocity at any %oint is constant 2ith regards to ti"e, 2hich i"%lies that thevelocity of any %article in the flo2 is only de%endent on its %osition in the direction of flo2. 1hus

    & t  =0

     and the a&ial acceleration of any %article at a given cross-section eco"es

    as=# & #

    & s=#

     d#

    ds

    ;n7ecting this result in the e4uation for net force a%%lied to the ele"ent considered %reviously yields

     pA+( p+kδp ) δA− ( p+ δp ) ( A+δA )− ρgA d"= ρA# d#

    ds ds

    /eglecting s"all 4uantities of the second order si"%lifies the e&%ression. @urther"ore, if the variationin %ressure along the length of the ele"ent is %redictale, it can e e&%ressed as an e&act differential

    dp.

    10

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    − A dp

    ds− ρgA

     d"

    ds= ρA#

     d#

    ds  yields

    1

     ρg

    dp

    ds +

    #

    g

    d#

    ds +

    d"

    ds=0

    Since the considered fluid is inco"%ressile, its density is constant and the aove e&%ression can eintegrated to yield

     p

     ρg + #

    2

    2g + "= ' 

    Where ' 

      is a constant. 1his result is 8no2n as Bernoullis e!uation, and is an e&%ression of 

    conservation of flo2 energy along a flo2 u%on 2hich no e&terior forces act. 1his is not to e confused2ith the energy of a volu"e of fluid "oving along the flo2, 2hich is variale 2ith regards to %osition inthe flo2, and thus de%endent on ti"e. ;n the for" %resented aove, this e4uation %rovides the totalhead  of the flo2. Head is a 4uantity 2hich is ho"ogenous 2ith a length, and re%resents the theoreticalheight that a static colu"n of the sa"e fluid as the actual fluid 2ould need to have in order to %rovidea hydrostatic %ressure at the otto" of the colu"n e4uivalent to the total %ressure at the considered%oint of flo2. $eferring to the ter" on the left side of the e4uation, the 4uantity on the left is the

     pressure head , the 4uantity in the "iddle is the dynamic head , and the 4uantity on the right is theelevation head . 1he %ressure head %ossessed y the flo2 at a given %oint is the head %rovided y the%ressure at that %oint. 1he Dyna"ic head at a given %oint is due to the "otion of the flo2 at that %oint.1he elevation of a %oint of the flo2 deter"ines the elevation head availale to the flo2 at that %oint. ;tessentially re%resents the head that has the %otential to e converted into %ressure head or dyna"ichead. @or the ideal flo2 considered, this velocity is unifor" across any cross sectional area of the flo2.;n reality ho2ever, actual /e2tonian fluids %ossess a certain viscosity 2hich induces a differentdistriution of fluid velocity across the flo2 cross-section, 2ith the "a&i"u" velocity occurring at thecentral strea"line of the flo2, and a gradual decrease of strea"line velocity occurring to2ards theoundaries of the flo2. 1he rate at 2hich the velocity decreases de%ends on the %ro%erties of the/e2tonian fluid in the flo2, as 2ell as the %ro%erties of the flo2. 1his is generally circu"vented yconsidering that the flo2 velocity at any %oint of a cross-section is the "ean velocity of the flo2 at thatcross-section. ! further conse4uence of fluid viscosity is the e&istence of friction at the flo2 oundary

    and 2ithin the fluid itself. Because the oundary friction forces are e&terior forces acting on the flo2 atits oundary 2ith the outside environ"ent, they initiate a transfer of energy et2een the t2oafore"entioned syste"s. 1he detailed analysis of fluid friction is eyond the sco%e of this %articular e&%eri"ent, and for the sa8e of revity, it 2ill e assu"ed that the losses incurred can e e&%ressedusing the follo2ing "teady "tate #nergy #!uation $""##% for t2o se%arate %oints 2ithin the flo2,2hich is an ada%tation of Bernoulli

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    Wheren

     is the total nu"er of flo2 %ortions 2ith unifor" velocities throughout the" et2een

    %oints + and (, and ' l )i  is a constant for a %ortion i  of unifor" velocity

    #i , de%endent on

    the conduit in 2hich that section of flo2 ta8es %lace, as 2ell as the flo2 %ara"eters at that section andthe inherent %ro%erties of the fluid involved. /o further detail shall e %rovided regarding fluid friction in

    this section, as it is a vast to%ic. ;f a %u"% is %laced et2een %oints + and (, the head *  p  %rovided

    y the %u"% is accounted for in the SSEE as follo2s

     p1

     ρg+

    #12

    2g+ "

    1+ *  p=

     p 2

     ρg+

    #22

    2g+ "

    2+(l

    Si"ilarly, insertion of a turine 2hich e&tracts a head * t    et2een the t2o %oints "odifies the

    e&%ression as follo2s

     p1

     ρg

    +#1

    2

    2g

    + "1+ *  p=

     p 2

     ρg

    +#2

    2

    2g

    + "2+(l + * t 

     !ttention 2ill no2 e directed to2ards the e&%eri"ental a%%aratus. @igures , and ) are ta8en fro"the Heriot-Watt University C Duai >a"%us laoratory sheet titled The Pelton Wheel   and slightly"odified to ensure no"enclature continuity.

    12

    Figure 4 – Pelton (Impulse) ur!ine "onstru#tion (Arrows indi#ate $uid $ow dire#tion)

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    1hese figures illustrate the general o%erating %rinci%les of a Pelton 2heel, as 2ell as the action of thefluid 7et on the turine vanes, 2hich 2ill henceforth e referred to as buckets. ;t is 2orth noting thatfigures and ) re%resent sections of the 7et and uc8et ta8en such that the sectioning %lane is %arallelto the Pelton 2heel a&is and contains the a&is of the 7et 2hich is assu"ed to e circular in sha%e5.Since the 7et is se%arated evenly and sy""etrically u%on reaching the uc8et, the results %roduced ythe analysis of one half of the uc8et can e a%%lied to the other y sy""etry. 1hroughout thefollo2ing analysis, it 2ill e assu"ed that the Pelton 2heel has already een accelerated to an angular 

    velocityѡ 

     2hich re"ains constant. ;f the %itch radius of the Pelton 2heel the radius at 2hich the

     7et acts %er%endicularly to the uc8et5 isr p , the "agnitude +  of the tangential velocity ⃗+  of 

    at any %oint on the %itch radius is+ =r p ѡ  . 1his velocity is called the velocity of whirl , and it acts in

    the direction called the direction of whirl . 1he "agnitude+

     is oviously constant 2ith regards to

    ti"e since the angular velocity and %itch radius are oth constant. While the 7et is acting on a uc8et,the Pelton 2heel is rotating and, as a conse4uence, so are all the uc8ets. Ho2ever, the resultingeffects are assu"ed to e relatively negligile, and it is considered that the tangential velocity of theuc8et is al2ays in the sa"e direction as the 7et. ;nherently due to the %revious assu"%tions, the fluid

    13

    Figure % – 'emati# o t'e Pelton 'eel *u#+et

    Figure , – *u#+et Velo#it- .iagram

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    is also discharged fro" the uc8et at the %itch radius. !nother assu"%tion is that the 7et

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    Whereδm

     is the "ass of fluid 2ithin the considered %ortion of 7et flo2. Since the 7et is steady

    2ithin the uc8et

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    unity5, 2hich e4uates to a180°   deflection of the 7et at the uc8et. 1his is unfortunately

    unachievale in %ractice, as it 2ould result in the deflected 7et i"%acting the ac8 of the succeeding

    uc8et, lo2ering overall %erfor"ance. ;n %ractice, the deflection is confined to around165°

    . 1he

    variations of efficiency 2ith regards to variations of the s%eed ratio can easily e studied y derivingthe e&%ression for efficiency 2ith regards to the s%eed ratio as follo2s

    d2d  r

    =2 (1−k lcos$ ) (1−2 s )

    1he efficiency is "a&i"al 2hen the derivative aove is null, 2hich occurs 2hen s=

    1

    2 . 1he

    asolute "a&i"u" efficiency theoretically %ossile is2=1

    , and occurs 2hen the 7et is deflected a

    full180°

      2ith no friction occurring along the uc8et and the s%eed ratio is s=

    1

    2 . 'n the

    other hand, the efficiency is null 2hen the velocity ratio is either null or e4ual to unity, 2hich occurs2hen the velocity of 2hirl is null or 2hen it is e4ual to the 7et velocity and eyond5. ;ndeed, if the2heel is not rotating, the 7et si"%ly s"ashes against the uc8et 2ithout causing any "ove"ent of the2heel and all its energy is dissi%ated y other "eans heat, %otential defor"ation of the uc8et etc.5.'n the other hand, if the uc8et is "oving at the sa"e velocity as the 7et or at a greater velocity, the 7etis no longer ale to estalish contact 2ith the uc8et and no %o2er can e transferred. ;t can also e

    noted that the turine driving %o2er Pt#r.ine   follo2s a si"ilar trend, 2ith the "a&i"u", asolute

    "a&i"u" 2hich is e4ual to the 8inetic %o2er of the 7et5 and oth null values occurring at the sa"evalues of s%eed ratio and angular deflection as for the efficiency. ;n %ractice, the s%eed ratio conducive

    to o%ti"al efficiency is slightly lo2er than0.5

    , and its value generally falls in the vicinity of0.46

    . 1his is due to the fact that the frictional forces at the earings connecting the 2heel to the turineshaft, as 2ell as the frictional forces ter"ed windage5 of the surrounding at"os%heric fluid, increasesignificantly 2ith relatively s"all increases of 2hirl velocity. @ro" the %receding analysis, it is a%%arentthat the s%eed ratio "ust e carefully controlled and "aintained to ensure o%ti"al o%eration of theturine. ;n a %ractical conte&t, the 7et s%eed is generally fi&ed y the head availale at the no00le,2hich is itself fi&ed y the environ"ent in 2hich the turine is installed. 1he velocity of the 7et can edeter"ined y a%%lying the SSEE et2een the entrance and e&it of the no00le. @or si"%licity, the headloss ter" is o"itted, and losses of energy 2hich occur at the no00le shall e accounted for later in theanalysis. 1he SSEE a%%lied to the no00le is

     pno""le)a

     ρg  +

    #no""le2

    2g  + "no""le=

     p /et ) a

     ρg  +

     c12

    2g+ " /et 

    1he no00le and 7et are at the sa"e elevation. 1he e&%ression for 7et velocity is thus

    c1=√

    2 ( pno""le )a− p /et ) a ) ρ

      +#no""le2

    ;n order to account for friction losses 2ithin the no00le, a constant n  2hich is less than unity

    usually around0.98

     or0.99

    5 can e i"%le"ented such that

    18

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    c1= n√

     2 ( pno""le )a− p /et ) a ) ρ

      +#no""le2

     !%%lying the SSEE e4uation et2een so"e 8no2n reference %oint u%strea" of the flo2 for 2hich the%ara"eters are designated y the suscri%t +5 and the no00le, the head availale at the no00le can e

    deter"ined as follo2s

     p1

     ρg+

    #12

    2g+ "

    1=

     p no""le) a

     ρg  +

    #no""le2

    2g  + "no""le+(l

    Which yields

    2 ( pno""le )a− p1 ) ρ

      +#no""le2=#

    1

    2+2g ( "1− "no""le )−2 g (l

    /oting that the %ressure  p1  can e 2ritten as   p1=( p1− p /et ) a )+ p /et ) a , the head availale atthe no00le can e deter"ined

    2 ( pno""le )a− p /et ) a ) ρ

      +#no""le2=

    2 ( p1− p /et ) a ) ρ

      +#1

    2+2g ( "1− "no""le )−2g ( l

    Hence, the 7et velocity can also e e&%ressed as follo2s

    c1= 

    n

     2 ( p1− p /et ) a )

     ρ  +#

    1

    2+2g

    ( "

    1− "

    no""le )−2g (

    l

    =enerally, it is not unreasonale to o"it the effects of flo2 s%eed on the availale head since they areusually negligile co"%ared to the total head. 1his results in a slightly si"%ler e&%ression

    c1≈  n√

    2 ( p1− p /et ) a ) ρ

      +2g ( "1− "no""le )−2g (l

    1he Pelton 2heel can then e designed to ensure an o%ti"al s%eed ratio for the desired turine shaft

    s%eed

    ѡ 

    . ;ndeed, since the o%ti"al velocity of 2hirl is

    + =r p ѡ =0.5c1, the o%ti"al %itch radius

    of the 2heel is

    r p=0.5c

    1

    ѡ 

    1he factor0.5

     "ay e re%laced y another slightly lo2er value otained y e"%irical "eans if it is

    dee"ed "ore suitale usually around *.)5. 1urines in real 2orld a%%lications are inevitalysu7ected to inconsistencies in the loads a%%lied to the", due to fluctuations in %o2er de"and.Ho2ever changes in the rotational s%eed of a turine are usually not %er"issile, as such

    19

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    "odifications 2ould alter the fre4uency of the electrical signal %roduced 2ithin the generator. @or agiven turine, 2hich has a set %itch dia"eter and is to o%erate o%ti"ally, this constraint transfers over to the velocity of 2hirl, 2hich in turn i"%oses a constancy restriction on the 7et velocity y 2ay of theo%ti"al s%eed ratio. ;t has already een estalished that the %o2er driving the 2heel is

     Pt#r.ine=ḿ (c1−r p ѡ ) (1−k l cos$ ) r p ѡ  . !ny fluctuation of load %o2er "ust e "et 2ith acorres%onding change in the afore"entioned driving %o2er. Since constancy restrictions are i"%osed

    on oth velocities, and that the angle of deflection and loss factor

    k l are turine de%endent and

    unchanging, the only variale re"aining is the "ass flo2 rate. Since the fluid is consideredinco"%ressile, its density is constant and therefore only the volu"etric flo2 rate "ay e ad7usted.1his volu"etric flo2 rate can e e&%ressed as follo2s y considering an infinitely s"all volu"e

    dV =ds A  /et   of 7et fluid eing dis%laced y a distance e4uivalent to its length

    0=dV 

    dt  = A / et 

    ds

    dt  =c

    1 A /et 

    1he 7et velocity eing constant, only the area of the 7et "ay e "odified. Such an ad7ust"ent is carried

    out at the no00le entrance y 2ay of the s%ear valve. 1he effect of changes in flo2 rate on the%ressure differential at the no00le eco"e a%%arent fro" the %reviously derived e&%ression for 7ets%eed

    c1= n√

     2 ( pno""le )a− p /et ) a ) ρ

      +#no""le2 yields

    c1= n√

    2 ( pno""le)a− p /et ) a ) ρ

      +(   0 Ano""le )2

    Where A no""le   is the cross-sectional area at the no00le. $earranging to e&%ress flo2 rate 2ith

    regards to no00le %ressure differential, the resulting e&%ression is as follo2s

    0= A no""le

     n   √c12−

    2 n2 ( pno""le) a− p /et )a )

     ρ

    @ro" the aove e&%ression, it can e seen that a reduction of flo2 rate is acco"%anied y acorres%onding rise in the %ressure differential at the no00le. >onversely, a flo2 rate increase isacco"%anied y a dro% in no00le differential %ressure.

    E6per*menta+ Method

    1he ste%s used to conduct the e&%eri"ent are as follo2s

    +. >learance of the turine dyna"o"eter is chec8ed to ensure no har" or da"age or occurs to%eo%le or e4ui%"ent.

    (. ;f necessary, the ad7usting scre2s on the s%ring alances are loosened to ensure theconnecting stra% is clear of the dyna"o"eter.

    3. 1he %u"% is %o2ered y using the %u"% %o2er s2itch on the %u"% control %anel.. 1he valve handle is rotated until the valve is fully o%en.

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    . 1he s%indle on the no00le is used to ad7ust its %ressure until its gauge %ressure pno""le) g

    reaches a value of pno""le) g=1.0 ¿̄ . 1his %ressure is to e "aintained constant until stated

    other2ise in the follo2ing ste%s.). ;n order to "easure the flo2 rate, the valve at the otto" of the tan8 is shut, and the ti"e

    ta8en for an aritrary volu"e of the "easuring tan8 to e filled is "onitored using a sto% 2atchand noted do2n. 1he filled volu"e of the tan8 is "onitored using the volu"e indicator on the

    side of the tan8. >are "ust e ta8en as a s"all volu"e of the tan8 is initially filled, even 2iththe valve fully o%en. 1his volu"e "ust e sutracted to the total filled volu"e for an accurate"easure"ent. 'nce the flo2 rate "easure"ent is co"%lete, the valve at the otto" of thetan8 is fully o%ened.

    F. Ensuring once "ore that the dyna"o"eter is initially unloaded, the tacho"eter ea" isdirected at the reflector %laced on the dyna"o"eter, and the dis%layed rotational s%eed isnoted do2n once it eco"es so"e2hat stale.

    A. !fter the first unloaded "easure"ent, the ad7usting scre2s on the s%ring alances aretightened until the connecting stra% egins to act as a "echanical ra8e on the dyna"o"eter.>are is ta8en to ensure that the oth s%ring alances are ad7usted to avoid the dis%lace"entof the connecting stra% over to one side. 1he tension readings on oth s%ring alances arenoted, and a ne2 reading of rotational s%eed is ta8en using the tacho"eter.

    . 1he load on the dyna"o"eter is increased y tightening the ad7usting scre2s further, and the

    "easure"ents of s%ring alance load and rotational s%eed are re%eated as %reviously"entioned.

    +*. Ste% is re%eated until a total of eight sets of "easure"ents for increasing loads on thedyna"o"eter are otained, including the "easure"ents otained for the unloadeddyna"o"eter.

    ++. 'nce eight sets of "easure"ents are otained for a constant reading of no00le gauge%ressure, the ad7usting scre2s are loosened until the dyna"o"eter is unloaded once "ore.

    +(. Ste%s -+* are re%eated for no00le gauge %ressures of1.2 ¿̄

     and1.5 ¿̄

    .

    +3. 'nce all "easure"ents have een carried out, the valve is shut, the %o2er to the %u"% cut,and all used e4ui%"ent %laced ac8 in its original %osition.

    21

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    8es+ts

    TestNo

    No33+e(a(e

    ressrebar:

    ;+o< 8atem- =s:

    "ynamometer 8es+ts

    ;or)e/pee

    drpm:

    Tor>eNm:

    !npto

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    17 A (** +F* +.(()( +**.*)F33+

    +.*3+3)) +*.(A))A+

    19 +F )** +3 F* .()F3 +F* *.(33+3) 3.(+) 3(.(*())

    20 (* ** (+* AF* ).F) +)**.3+*(

    )+.*FA .FF*+F(

    21 33* +(** (* ++F* A.)3(A +**.F*+3F

    +F3.)++ .(3A3)

    22 3* +)** 3* +F* ++.FA +(A *.)3)*3 A3.F333 )+.*)*(A

    2- )(* ((** A* (+F* +.F F3* *.A))) ).F(+A3 A.*+(3

    2& F(* ()** )A* (F* +A.* 3+.*+A*(+

    A3).FA*+* ().F(3F3

    23

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    "*s)ss*on o4 8es+ts

    @igure F sho2s the evolution of tor4ue %roduced y the turine 2ith regards to turine rotational s%eed for the threeno00le %ressures that 2ere "aintained during "easure"ent.

    200 400 600 800 1000 1200 1400 1600 1800 2000 2200

    0

    0.2

    0.4

    0.6

    0.8

    1

    1.2

    1.0 bar 1.0 bar 1.2 bar 1.2 bar 1.5 bar 1.5 bar

    &peed (rpm)

    or/ue (0m)

    @ro" the analysis done in the theory section, the tor4ue is e&%ected to vary linearly 2ith the rotational s%eed.

    ;ndeed, if it is recalled that the force a%%lied on the turine y the 7et is F = ḿ ( c1−+ ) (k lcos$−1 ) , an

    e&%ression for the tor4ue3 

     a%%lied to the 2heel at the %itch radius can easily e deter"ined as follo2s

    3 =r p ḿ ( c1−+ ) (k l cos$−1)=r p2

    ḿ( c1r p−ѡ )( k l cos$−1 )

    Ho2ever, the evolution of the "easured tor4ue at the turine out%ut follo2s no such trend. 1his discre%ancyet2een the tor4ue "easured at the out%ut, and the theoretical driving tor4ue of the 2heel "ay %erha%s ee&%lained y frictional forces at the earings. Evidently, these frictional forces increase significantly and non-linearly2ith the rotational s%eed of the turine, and cause a non linear dro% in tor4ue as the 2heel rotates "ore ra%idly.

    1he variations of volu"etric flo2 rate 2ith no00le %ressure are sho2n in figure A.

    24

    Figure – Variations o or/ue wit' Rotational &peed

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    0.9 1 1.1 1.2 1.3 1.4 1.5 1.6

    0

    0

    0

    0

    0

    0

    0

    0ole Pressure (!ar)

    Flow Rate (m3s)

     !s e&%ected, flo2 rate decreases 2ith increases in no00le gauge %ressure. Ho2ever, the variations of out%ut %o2er and efficiency do not occur as %redicted in the theoretical analysis. ;ndeed, it is e&%ected that oth efficiency and%o2er out%ut vary si"ilarly 2ith regards to rotational s%eed, attaining "a&i"u" and "ini"u" levels for the sa"evalues of rotational s%eed. ;t is also e&%ected that, regardless of no00le %ressure and flo2 rate5, these notalerotational s%eeds 2hich corres%ond to "a&i"a and "ini"a of efficiency and %o2er out%ut5 re"ain constant. 'nefinal e&%ectation "ay e dra2n fro" the e&%ression of turine driving %o2er

     Pt#r.ine= ḿ (c1−r p ѡ ) (1−k l cos$ ) r p ѡ 

    .@ro" the aove e&%ression, it "ay e concluded that, should the 7et and rotational s%eeds of the 2heel re"ainconstant, the driving %o2er of the turine is "a&i"al 2hen the "ass flo2 rate and thus the volu"etric flo2 rate5 is"a&i"al. !s illustrated y figures and +*, although for a given no00le %ressure the e&istence of an o%ti"alrotational s%eed 2hich correlates turine "a&i"u" %o2er out%ut and "a&i"u" efficiency is evident, this o%ti"als%eed is de%endent on no00le %ressure, 2hich goes against the e&%ectation resulting fro" the theoretical analysis.@urther"ore, for the sa"e rotational s%eed, the %o2er out%ut of the turine increases if the no00le %ressureincreases, 2hich once again does not correlate 2ith e&%ectations. 1he "a&i"u" out%ut %o2er otained is

    a%%ro&i"ately83.7W 

    , for a no00le %ressure of1.5 ¿̄

     and a rotational s%eed of1258 rpm

    .

    25

    Figure 5 – Variations o Volumetri# Flow Rate wit' 0ole 6auge Pressure

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    200 400 600 800 1000 1200 1400 1600 1800 2000 2200

    0

    10

    20

    30

    40

    50

    60

    70

    80

    90

    f(x) = - 0x^2 + 0.16x

    f(x) = - 0x^2 + 0.15x

    f(x) = - 0x^2 + 0.14x

    1.0 bar 1.0 bar 1.2 bar 1.2 bar 1.5 bar 1.5 bar

    &peed (rpm)

    7utput Power ()

    200 400 600 800 1000 1200 1400 1600 1800 2000 2200

    0

    10

    20

    30

    40

    50

    60

    70

    f(x) = - 0x^2 + 0.11xf(x) = - 0x^2 + 0.13xf(x) = - 0x^2 + 0.13x

    1.0 bar 1.0 bar 1.2 bar 1.2 bar 1.5 bar 1.5 bar

    &peed (rpm)

    E8#ien#- (9)

    1he change in o%ti"al rotational s%eed et2een different no00le %ressures "ay e due to the fact that the headavailale at the no00le is %rovided y a %u"%. ;ndeed, the head %rovided y a %u"% is highly de%endent on theflo2 rate at 2hich the %u"% o%erates. Since the s%ear valve is used to control the flo2 rate, it li8ely also induces achange in the head %rovided y the %u"%, 2hich in turn "odifies the 7et velocity. ;n general, a %u"% %rovideshigher heads as the flo2 rate at 2hich it o%erates decreases, 2hich is the case in this e&%eri"ent since the

    26

    Figure : ; Power 7utput Variations wit' &peed

    Figure 1< – ur!ine E8#ien#- Variations wit' &peed

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    %ressure at the no00le increases. ;n the %resent case, such a head increase induces an increase in 7et s%eed,2hich is conducive to higher %o2er out%ut, and re4uires a %ro%ortionally higher rotational s%eed in order to satisfythe o%ti"al s%eed ratio. Ho2ever, as %reviously stated and de"onstrated in the theoretical analysis, the 7et velocityis de%endent on head availale on the no00le, 2hich is itself usually fi&ed y the to%ogra%hy of the area or thelayout of the %o2er %lant. 1herefore, the only %ractical "eans of increasing turine %o2er in%ut is y increasing the"a&i"u" %ossile flo2 rate. 1his "ay e done y increasing the "a&i"u" area of the 7et allo2ed y the s%ear valve, 2hich "ust e "et 2ith a corres%onding increase in uc8et si0e and 2heel dia"eter. !nother alternative,2hich is generally %referred, is the inclusion of "ulti%le no00les, 2hich "ulti%lies the flo2 rate y the nu"er of 7ets

    %roduced 2ithout re4uiring an increase in 7et dia"eter. !s al2ays, 2or8ing to reduce frictional losses also %lays asignificant %art in increasing %o2er out%ut.

    on)+s*ons

    1he analysis of the otained results confir"s the e&istence of an o%ti"al 2heel rotational s%eed for a given 7etvelocity, 2hich correlates "a&i"u" %o2er out%ut 2ith "a&i"u" efficiency. @urther"ore, the results see" toindicate that higher 7et velocities are conducive to higher "a&i"u" out%ut %o2er, 2hich is e&%ected considering thetheoretical analysis of the turine. ;t "ay also e concluded that, in order to ensure a stale 7et velocity andconse4uently a stale o%ti"al 2heel rotational velocity, the head availale at the no00le should not e de%endenton the flo2 rate, since the latter is the vector y 2hich the s%ear valve e&erts control over the in%ut %o2er. 1heseresults "ay e ta8en into account 2hen selecting an a%%ro%riate location for the installation of a hydro-%o2er %lant,

    the ty%e of turine used in the afore"entioned %o2er %lant, and the turine di"ensions, in order to ensure thatavailale resources are e&%loited to their full %otential and at a "a&i"u" efficiency.

    27

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    @igure !+ sho2s a sche"atic of the loaded dyna"o"eter.

    1he s%inning dyna"o"eter generates friction forces on a %ortion of the connecting stra%, 2hich result in a total

    forcef f  . Since the stra% is inca%ale of sustaining any load other than a tensile load, any force a%%lied to it

    "anifests itself in the for" of tension along the stra%. Since the stra% is stationary, the net force acting on it is nulland the follo2ing e&%ression "ay e estalished

     F 2− F f − F 1=0 yields

     F f = F 2− F 1

    Where F 

    1   and F 

    2  are the tensions on the stra% due to s%ring alances + and ( res%ectively. 1he total

    friction force on the stra% is also the o%%osite of the ra8ing force on the dyna"o"eter, y a%%lication of /e2ton

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    @or test nu"er (, the "easuring tan8 volu"e increased y %V =36 l

      in a duration %t =33.5 s

    . 1he

    "easured flo2 rate is thus0=

     % V 

     % t  =

      36

    33.5l / s ≈1.075×10−3 m3/ s

    . Both s%ring alances need to e

    calirated, since they dis%lay a load even 2hen the dyna"o"eter is unloaded. S%ring alance + dis%lays a load of 

    40 gf   2hile the dyna"o"eter is unloaded, 2hich "ust e sutracted to any suse4uent "easure"ent.

    Si"ilarly, s%ring alance ( dis%lays a load of 30gf   2hen the dyna"o"eter is unloaded. 1herefore, after 

    caliration, the load on s%ring alance + has a value of F 

    1=130−40=90 gf 

    , and li8e2ise, the load on s%ring

    alance ( is F 

    2=400−30=370 gf 

    . 1he net force on the dyna"o"eter is thus

     F dynamo=370−90=280gf ≈2.75 N  . 1his e4uates to a tor4ue eing a%%lied on the dyna"o"eter of 

    3 dynamo=rdynamo F dynamo . 1he radius of the dyna"o"eter rdynamo  can e deter"ined fro" the "easured

    circu"ference of the dyna"o"eterld=34.5 cm . ;ndeed, rdynamo=

      ld

    2π 

     ≈ 5.49 cm ≈0.0549m . 1herefore, the

    tor4ue a%%lied on the dyna"o"eter is3 dynamo ≈0.0549×2.75≈0.15 N - m . 1he in%ut %o2er has already een

    sho2n to e Pimp#t =( pno""le) a− p /et )a ) 0 . Since the 7et is at at"os%heric %ressure, the %ressure differential at the

    no00le is e4ual to the no00le gauge %ressure  pno""le) g=1.0 ¿̄105 Pa . 1he in%ut %o2er is then

     Pimp#t ≈105

    ×1.075×10−3

    ≈107.5W  . 1he out%ut %o2er is

     Po#tp#t =3 dynamo ѡ ≈0.15× 2 π ×1580

    60≈24.8W 

    . @inally, the efficiency can e calculated as follo2s

    2= Po#tp#t 

     P inp#t ≈  24.8

    107.5≈ 0.23

    .

    29

    Appendix * – &ample "al#ulations

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    8e4eren)es

    Massey B., Mechanics of @luids, Ath Edition, (**), 1aylor G @rancis

    The Pelton Wheel , #aoratory E&%eri"ent Handout, Heriot-Watt University Duai >a"%us