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Dimensional Analysis for Determining OptimalDischarge and Optimal Penstock Diameter inImpulse and Reaction Water Turbines
1
Arturo Leon1and Ling Zhu
1
School of Civil and Construction Engineering, Oregon State University2Department of Civil and Environmental Engineering, Louisiana State University
X Congreso Latinoamericano de Estudiantes de Ingenieria Civil - XXII Congreso
Nacional de Estudiantes de Ingenieria Civil del Per, August 4-8, !"4#
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Ackno!ledgements"
#inancial support"
$onneville Po%er Administration o& the'#(# )e*artment o& Energ+ )E. undera%ard num/er 0IP128#
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Presentation outline
$ac3ground5otivation
)imensional anal+sis A**lication Conclusions
$
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%ydropo!er systems
&arkamis hydroelectric po!er plant' Turkey
Three (orges hydropo!er
plant' )hina
%%%#%ater-technolog+#net
%oo*er Dam' +,
%%%#hdrinc#com
%%%#*o%er-technolog+#com
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-ackground./oti*ation
6"7 o& glo/al electricit+ *roduction came &romh+dro*o%er in !!9
6)evelo*ment o& all the remaining h+droelectric*otential could not ho*e to cover total &uture %orld
demand
6:+dro*o%er due to its associated reservoirstorage, can *rovide i/ilit+ and relia/ilit+ &or
energ+ *roduction in integrated s+stems# therdi&&use and varia/le rene%a/le energ+ sources
%ind, %ave, solar. can *la+ a larger role in
*roviding electrical *o%er o& commercial
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-ackground./oti*ation 0)ont2
6Im*lementation o& remaining h+droelectric *otentialcan ma3e a vast contri/ution to im*roving livingstandards in the develo*ing %orld (outh America, Asia
and A&rica., %here the greatest *otential still e;ists#
65inimi=ing %ater consum*tion &or *roducingh+dro*o%er is critical given that overuse o& &lo%s &or
energ+ *roduction ma+ result in a shortage o& &lo%s &or
other *ur*oses such as irrigation or navigation
60heoretical &rame%or3 &or determining o*timal design&lo% and o*timal *enstoc3 diameter &or im*ulse and
reaction tur/ines is N0availa/lein literature
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3lectrical po!er 0P2 of impulse andreaction turbines 0)ont2
4
( )g LP Q H h=
= t g = const
Hg : gross head
hL : total head losses
: specific weight of water = g( )
: water densityg : acceleration of gravity
Q : flow discharge
t : turbine efficiency
g : generator efficiency
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3lectrical po!er 0P2 of impulse andreaction turbines 0)ont2
5
P =Q Hg hL( )h
L =
Q2
2gA22 C
L
2
21 2
2
2
21 2
2
for an impulse turbine
for a reaction turbine
N
N
L
d
ALf k k
D AC
ALf k
D A
+ + =
+ +
L : length
D2 : diameter of penstock
A2 : cross - section area of penstock
AN : nozzle area at exit
f : friction factor
1-2
N
2
: draft tube cross-sectional area at outlet
k : sum of local losses in penstock
due to entrance, bends, penstock fittings and gates
k : nozzle head loss coefficient
11, : 0.98 0.99
d
N V
V
A
k CC
=
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A dimensionless relationship bet!eenpo!er and flo! discharge 0)ont2
6
6Choose re&erence *o%er Pr. and re&erence discharge Qr.
6Choose Pras the ma;imum *o%er can /e generated %ith
&i;ed CL and o& "!!
2
2
22
g LQP Q H CgA
=
dP dQ = 0
Qr= 2A3
1
3gH
g , Pr =4
3H
gA3
1
3gH
g
A3 = A
N for an impulse turbine
Ad
for a reaction turbine
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A dimensionless relationship bet!eenpo!er and flo! discharge 0)ont2
17
2
2
2
2g L
QP Q H C
gA
=
P+ = P Pr, Q+ = Q Qr
with
= CLA3
A2
2
= f L, D2, f, k , A3 A2(
!!
2
P Q Q + + + =
[ ]
[ ]
21
0.01, 1.0 for impulse turbines"n applications,
10, 1000 for reaction turbines
D
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11
Im*ulse tur/ines > !#8.
Dimensionless discharge *ersus Dimensionless Po!er
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1
?eaction tur/ines > !#8.
Dimensionless discharge *ersus Dimensionless Po!er
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Determine optimal 8 and optimal D 0)ont2
1$
2! !! 0,2 2
dP QdQ
+ ++
=
@P @Q> !achieve ma;imum Pho%ever, not o*timalB
P+( )max =1
2
, Q+( )max =1
2
5inimi=ing %ater
consum*tion
Achieving large @P
%ith small @
5a;imi=e @P @
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Determine optimal 8 and optimal D 0)ont2
19
2! !! 0,2 2
dPQ
dQ + +
+
=
5inimi=ing %ater
consum*tion
Achieving large @P
%ith small @
5a;imi=e @P @
@P @Q> D
Ideal condition to
save %ater /ut *o%er
*roduced is small
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1:
Im*ulse tur/ines > !#8.;P
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1?
Im*ulse tur/ines > !#8.;P
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14
Determine optimal 8 and optimal D 0)ont2
o*timal head loss hL > hL/ Hg. "2#7
Qopt upper =2
3A2
7
10
gHg
CL
Popt upper =
76
135 HgA27
10
gHg
CL
dP+dQ+
opt
= 0.8
Q+( )opt upper =7
30
P+( )opt upper =19
15
7
30
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15
Influence of changing penstock
diameter on po!er *ariation
P+
= Q
+
3
!!
2P Q Q + + +
=
!
!!
2
P QP
Q Q
+ +
++ +
=
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Im*ulse tur/ines Q+> D.
;P
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Qopt upper =2
3A2
7
10
gHg
CL
Popt upper =76
135HgA2
7
10
gHg
CL
Qopt = 4538
PHg
P s*eci&ied
Popt =38
45
HgQ
CL( )opt
A22
14
45
gHg
Q2
Q s*eci&ied
Turbine design !ith P or Qspecified
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3@ample" Impulse Turbine design
( )
( )
( )
( )( )
2
1 2
# 2
$onditions:%ross head 200 m
&enstock length '00 m
1#
Nozzle (elocit) coefficient 0.98'
1.'
*oughness height of penstock material 0.0+' mm
inematic (iscosit) 10 m -s
urbine efficien
g
N
V
H
L
A
A
C
k
=
=
=
=
=
==
( )
( )
c) 82/
%enerator efficienc) 90/
t
g
=
=
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Impulse Turbine design 0)ont2
)ase A1F design an im*ulse tur/ine %ith Q> !#7 mD
s
( )( )
( )
( ) ( )
opt -+
2 2
2
2
10 0.9
2
2
1+1#9!.8 m 1
+'
1 1 0.0!0
0.2' 2
'.+log!. *e
'001.' 1# !
L g
N
V
L N
C gH
A Q
kC
f
D
fC k
D
= =
= =
=
+
= + +
D2 = 0.3968 m
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Impulse Turbine design0)ont2
Case A"F design an im*ulse tur/ine %ith Q> !#7 mD
sssuming that a schedule 80 steel pipe is reuired due to structural
considerations, a 18 in outside diameter pipe 3ould be selected.
3all thickness 4 0.9!8 in
internal diameter 4 1#.12+ in 5+09.' mm6
LC
( )
2
272
2
2
2'.!'
dimensionless head loss 0.1!+ 51!.+/6 1'.#/2
electrical po3er 0.82 0.90 1000 9.8 0.#
0.#200 2'.!' 4'1+21 4'1.+ k
2 9.8 0.1!1
L L
g
Qh C
gH A
P
=
= =
=
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/ATLA- toolbo@ for turbine designhtt*F%e/#engr#oregonstate#eduGleonCodes#html
http://web.engr.oregonstate.edu/~leon/Codes.htmlhttp://web.engr.oregonstate.edu/~leon/Codes.html8/10/2019 Hydropower Optimal Power Flow
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)O)L+,IO, 0he *resent anal+sis resulted in various
dimensionless relationshi*s /et%een P, Qand hL&or
determining o*timal design &lo% and *enstoc3
diameter &or designing im*ulse and reaction tur/ines
0he derived relationshi*s %ere used to %ithdra%general insights on h+dro*o%er o*timi=ation# Hor
instance, it %as &ound that &or minimi=ing %ater
consum*tion, the ratio o& head loss to gross head
hL:g. should not e;ceed "2# 0o &acilitate the calculations, a 5A0LA$ h+dro*o%er
tool/o; %as develo*ed#
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/any thanks for your attentionB/any thanks for your attentionB
)ontact")ontact"Arturo LeonArturo Leon(chool o& Civil and Construction Engineering,(chool o& Civil and Construction Engineering,regon (tate 'niversit+regon (tate 'niversit+
3Cmail"3Cmail"arturo#leonoregonstate#eduarturo#leonoregonstate#eduResearch Web page"htt*F%e/#engr#oregonstate#eduGleon
?
mailto:[email protected]:[email protected]://web.engr.oregonstate.edu/~leonhttp://web.engr.oregonstate.edu/~leonmailto:[email protected]