Variable speed chax:acteristics of Francis turbine F3. (N = 1.43, fixed .... less
costly than Kaplan turbines, their potential cost benefits should be examined in ...
UNIVERSITY OF MINNESOTA
... I
ST. ANTHONY FALLS HYDRAULIC LABORATORY I
, :
Project Report No. 225
HYDROMECHANICS OF VARIABLE SPEED TURBINES .' by , f·
.
Cesar Farell, Javier Arroyave, Nicholas Cruz, and.John S. Gulliver
..
,
Prepared for STATE OF MINNESOTA DEPARTMENT OFNAtVRAL RESOURCES St. Paul, Minnes.ota
. August, 1983 Minneapolis, Minnesota
I'·t
University of Minnesota st. Anthony Falls Hydraulic Laboratory
Project Report No. 225 e-
HYDROMECHANICS OF VARIABLE SPEED TURBINES
by Cesar F'arell, Javier Arroyave, Nicholas Cruz, and John S. Gulliver
Prepared for STATE OF MINNESOTA DEPARTMENT OF NATURAL RESOURCES st. Paul, Minnesota
August, 1983 .j..
ACKNOWLEDGEMENTS
This study was performed under the general supervision of Pro Roger E~A. Arndt, Direotor of the St. Anthony Falls Hydraulio Laboratory, University of Minnesota.
The University of Minnesota is committed to the policy that all persons shall have equal access to its programs, facilities, and employment without regard to race, creed, color, sex, national origin, or handicap.
i
TABLE OF CONTENTS
Page No. Acknowledgemen ts ....... ., .......
!II" '" . . . . . . " "
" .... '" • • "
. . . fI . . . . . . . . . . . . . '"
List of Figures •••••••••••••••••••••••••••••••••••••••••• I.
INTROOUCT ION ".....
II . . . . . . . _ .... "
•
,. . . . . . "
.. ,. "
•
,.
~
.... ,
•
"
,. . . . . . . "
....
tI
•
•
.. ..
i
iii 1
II.
POSSIBLE APPLICATIONS OF THE VARIABLE-SPEED TURBINE
3
III.
SIMILARITY CONDITIONS AND OPERATING CHARACTERISTICS
6
111.1.
Specific Speed ••••••.••••••.•••.••••••••••••••••
111.2.
Performance Curves and Characteristic Diagr.ams " •. ""." ..... ,.
7
"II'" " ... " ..... " ....
10
IV.
VARIAB:J:,.E-SPEED TURBINE PERFORMANCE AT CONSTANT HEAD .,.......
12
V.
SOME THEORETICAL CALCULATIONS ••••••••••••••••••••••••••••••
26
V.I.
Axial-Flow Turbines .... '" ...... ~.. .. . . .. .. .. . ... . .... .. . .. ........... .. ..
28
V.2.
Estimation of the Lattice Parameter
.'...........
30
V.3.
Application to A&ial-F1ow Turbine A2 ..............
31
... " " ..... " ... " .. " . "
35
VI. VI I •
"'''II' .... " .. " .... " ....
SUMMARY AND CONCLUSIONS .. ,. ,. ........................... " REFERENCES ... " '" • ,. .......
it ..
"
............
"
ii
"
....
"
.....
"
40
0
DISCHARGE
0
z 30 w
w 64 . z·
..... 48 1-1 40
r
.4
->-
-110
./ . I
(2 . 8 . . :. HEA D BET WEE N INL ET AND TAIL WAT ER (FT )
Fig. 3.
10 I6
flow turbin e. Typic al head- disch arge- effici ency curve s for an axial
lL LL
w
IV.
VARIABLE-SPEED TURBINE PERFORMANCE AT CONSTANT HEAD
Va~iable-speed
performance diagrams at constant head for two small
axial-flow turbines with fixed vanes ana blades,
Al
and
A2 , an axial pump used as a turbine, P l , three Francis turbines, F l , F 2 ,and F 3 , a pum~ turbine, Tl , and a Kaplan turbine, Kl , were obtained from information suppliea by manufacture:rs or available in the lite:rature.
The
o:riginal information was in different form for the various cases investigated, and the processing of the dat .. differed for each turbine, as outlined below.
The following table gives the specific speeds of the turbines
analyzed: N
s
n s (kW ,m, . rpm)
n (hp,ft,rpm) s .
(,
Al
1.61
267
70
A2
2.65
440
115
PI
2.68
444
116
Fl.
0.51
84
22
F2
0.81
F3
1.43
237
62
Tl
0.76
127
33
Kl
3.26
542
142 .
134
35
On the H-Q plane, if 'the fixed-gate character istic curve of a turbine is known for a given speea, , the similarity laws c,an be used as explained in the preceding section to obtain the characteristic curves at 'other speeds. , Generally, the form of this H-Q curve is as shown in 11'ig. 4.
I f the tur-
bine fixed-gate I H-Q· curve at constant speed is steeper than the similarity parabola linking equal-efficiency points, then it is seen that for
12
4.5
...--------,,~---,----_r_---__r____,_-____,
FIXED BLADES
n/~=1.5~1Io..J
4.0
3.5
SIMILARITY PARABOLA ('1]/1\ =CONSTANT) '1]
3.0
I
/ 2.5
/
/
H/A
H
I
2.0 n/~=I.O -~
/
/ .1.5
/ /
./ 1.0
/ \ 5 1 MIL ARIIY
PARABOLA 0 ..5
'c:'
O~~~--L-------~------~------~------~
o
0.5
1.0
1.5
2.0
2.5
a/A
a
Fig. 4.
Sketch of fixed-gate A characteristic curves on H-Q plane for - "'''' different speeds. (H, Q, n: best-effic1ency values) 13 i
constant
H,
Q will be an increasing function of
n.
The opposite is
true if the similarity parabola is steeper than the H-Q curve.
.Both cases
actually can occur, as shown later on. A computer program was prepared to carry out numerically the similarity calculations indicated in the preceding paragraph.
Original values
from the known performance curves of the turbine at oonstantspeed were stored in disorete ;form, for small head increments
(::.H,
column matrices QREF(I), HREF(I), PREF(I), and EFF(I).
in reference The values genera-
ted using the similarity relationships for different turbine speeds were stored .in matrices QFLOW 0: ,J), HEAD (I ,J), POWER (t ,J), EFF (I), and RPM (J) • inde~
Here the the
inde~
J
I
denotes a point in the original performance curve, and
identifies the speed.
FOr constant head operation with
variable speeds, the elements of the matrix HEAD(I;J) were searched to identify the intervals
{::.H. where the constant-head value fell, and linear
interpolation was then used to calculate the other operational parameters. These oalculations were performed for the axial turbines
Al
and
A2 , the axial pump PI' and the Francis turbine F3' at fixed gate. Similar oalculations for operational conditions other than constant head H could also be performed by introducing the appropriate constraint in the matrices.
Only constant head results are presented in this report, however ~
Figure 5 shows the variable-speed characteristics of the axial-flow' Al (N s =1.61), with fixed gate and blade~at constant head. It is seen that the discharge Q increases only slightly with the rotational
turbine speed,
n.
This means that operating the turbine at variable speed will
not greatly facilitate flow control. ponding characteristics for turbine
Figures 6 and 7 present the corresA2
and the' axial~f16w pump
PI
used
A2 , a peak efficiency value.fj = 0.9 was assume.d in the calculations.) In both cases, there is a steeper rise of
as turbine.
(For turbine
the curve of discharge versus rotatiorial speed I which wili be discu.ssed fUrther in section V.
Figure S gives a comparison of variable-speed off-
design performance curves for turbines' AI' A2 ,and Pl' The corresponding ourve for a full Ka)?lan,constant-speed turbine is also included. The coordinates are percent of peak efficiency and percent of peak-efficiency discharge to facilitate the comparison.
14
The larger variation in discharge
c
"
CONSTANT HEAD OPERATION
100
l" __--------"----
u
Q
JBO ~ 75 f()
1LL
70
w
(.!)
65 a:: c:{
80
J: U
}-
z
w
.... tJ1
80 }-
u a:: w
a..
(f)
o 60
>-
:::>
a..
u
0
w u
~ 60 }}c:{
z
40
LL LL
;: 40
0
W
7J
20
-..J ~
20
p
o 150 Fig. 5.
175
200
225 275 250 TURBINE SPEED, RPM
Variable speed characteristics for axial-flow turbine Al eNs
=
300
325
1.61, fixed gate and blades.
~
l
O
a
.
7J
80
I16.0
~
I-
:::>
z w t.) a::
900
a..
~ 800 I-'
Il
a..
./
.. >-
700
,;'
(J) If)
13.0 :E
..
"" " "
l1.. l1...
w .........
/
()
W
t.)
,;'
w 40 I-
15.0 I 4.0
,;'
Z
3: 600
I
,;'
t.)
0
--' -~ 500
,;'
w 60
0 CI'I
./Q
20
w
t 2. a
C,!)
11.0
a:::
(J)
10.0
o ~,----~--~~----~--~~----~--~--------~~--------~----~ 9.0 150
200
250
300
350
400
T U RBI N E SP E ED, RPM Fig. 6.
Variable speed characteristics foraxia.l-flow turbine A2 (N s gate and blades).
2.65 1 fixed
0
-.
CONSTANT HEAD OPERATION
35
~
0.. ~
30
100
f()
I
0
25 X w I-
z
20
80
:r: u en
a:: w 80
0..
.. r
I-
u
0..
w
I-
::> 0
60
0
z
::>
60
u 40 LL LL
lI-
w
«
~40 0
~.
20
1
".."
--.I ~
p
20
0::
«
w u
I-' -..J
(!)
o ~'----~--~------~--------~------~--------~------~------~ 350
400
450
500
550
600
650
700
TURB I NE SPEED, RPM ·F~g.
7.
Variable speed characteristics for axial-flow'pump used as turbine (Ns fixed gate and bl.ades) •
2.68,
100
/
~I(~
..
>u
z
AXIAL-FLOW I\~ TURBINE ~ (N =2.65)
s
FULL KAPLAN (FI XED SPEED)
80
w u I.J.. I.J.. I-'
co
w
60
~
«
w a..
I.J..
40
0
.-z
w
u a:: w a..
20~ 00
)\ AXIAL-FLOW TURBINE (N s =L61)
20
40
PERCENT
60 OF
80
100
LOW HEAD PUMP OPERATED AS TU RBI N E ( N s :: 2.68 )
120
140
160
BEST EFF1CIENCY DISCHARGE, ~ Q
Fig. 8.
Off-design variable-speed performance of fixed-gate, fixed-blade propeller turbines Al., A2' . and Pl' Constant-speed Kaplan curve also shown for comparison.
obtained with turbines for which figure. and
Pl'
A2
and
PI
in telation to that for turbine
varies only slightly with
Q
n)
Al
is cleatly exhibited in this
This may be related to the higher specific ,speed of turbines
A2
FiguteS indicates that the performance information supplied by
the manufacturers, related to fixed-speegoperation, is not sUfficiently comprehensive to ,compute the full range of vadable speed performance. best, t\lrbines
A2
and
P1
At
indicate a ±20 percent range in turbine dis-
charge with acceptable, performance / Which is much less than that which is achieved by fixed vane, variable .... pitch blade turbines operating at constant rotational speed.
This operational range, however, m?iY be greater
in higher specific speed turbines. NoW let us consider variable speed operation with a variable-gate
ll\,
capability, such as turbines
F 2' F3' and Kl • Poin tslying along the constant efficiency contours sketched in Fig. !) represent different com-
binations of ,openings.
H,Q
values for constant speed,
Given any such contour, say
the smallest discharge,
n = nA,
no'
and different gate
it is possible to obtain
such that for constant head
H' B
,l
equal to the best-efficienoy head in Fig. 9), the effioienoy
(taken
,n,A
can be
Q < Q'B • This is simply done, by following the similarity parabola (passing through the origin) tangent to
achieved for the contour
Q = QB
,
n2= nA
but not for any
,from
H
,A
to
HB"
The speed"
n B,
is given' by
nB = nA(HB/H,A) • The discharge Q13 may be read from Fig. 9 or determined from the eguation Qs = QA n13/nA (see Eq. 7). In practice, this was done by applying the simil-arity laws to all points on the curve n = nA and finding the homologous valueS for the minimum value of
H = HB • Point Q ,computed from the contour
a n
then corresponds to =
nA.'
The pedormance 'curves for the Francis turbines analyzed herein were obtained from Bureau of Reclamation Monograph 20 (1976).
For turbine
F 3'
Fig. 33 of this reference shows typical head-discharge hill curves for a range of n
N values (n=48 to 75 in English units)" An average valUe of s s is assumed herein. Figure 10, shows the results for a fixed gate
=62 s opening egual to 0.82 using the same procedure that was applied to the
data fo):' turbines function of
It is seen that
Al , A2 , and P l "
n"
19
Q
is n.ow a decreasing
O·~~-L~~~~~~~~L-~~~~--~~~~~
o
0.2
0.4
0.6
0:8
1.0
1.2
1.4
1.6
1.8
Q/A Q A
Fig. 9.
A
Variable-gate characteristic curves on H/H - Q/Q best efficiency values). . 20
A
A
plane (H, Q:
e'
'")'
CONSTANT HEAD OPERATION I
I u
1,000 W
en
...... ~
}- 100
}-
Z W,
1..1-
900
0
a::
8,000 r N I-'
z
0-
6,000
u
0::
"
I
60
:c
-1800
"
«
~ 4,000 ....I
2,oool
w
40
"',p
I
20 150
175
200
225
250
275
300
325
TURB I NE SPEED, RPM Fig. 10.
Variable speed characteristics of Francis turbine
F3
.~ 0
1..1,1..1-
}-
}-
:l :::> 0
..
t-
80
>-
}-
}-
W 0-
w..
(Ns = 1.43 r fixed gate).
This result. agrees with the calculations of Sheldon (1983), who found also a decrease in
Q
with increasing
n
for three Francis turbines with
ns
values
(in English units)o;f 24.5, 44.7, and 56.1.
Actually, Sheldon's resUlts show a
trend toward smaller variations in
n
bine with
ns = 56.1
Q
with
with increasing
exhibited practically no change in
n:
s
the tUr-
Q with an inorease in exhibited
rotational speed from 1.00 rpm t.o 160 rpm, while the unit with
a change in unit discharge from 0.85 cfs to about 0.55 cfsfor a change in speed from 60 rpm to 130 rpm. For turbines with variable gate, the prooedure outlined in the preoeding paragraph is of more interest, since one would want to use the variable gateoapability along with variable speed.
The results obtained for the three Francis turbines
F 1 , F 2 , and F 3 , and for the Kaplan turbine, Kl , are summarized in Table 1. No CUrves are presented because the differences are small, as oan be seen from the Table.
This indicates that the variable speed capability would not significantly
improve the performanoe of these turbines at off-design flow and constant head.
TABLE 1.
Turbine At
g=
Comparison of Turbine Efficiencies for Fixed and Variable Rotational Speeds
F2
Fl
0.2
F3
Kl
nvar speed/nfixed speed
Q
= 0.4
74.0/74.0
75.1/75.0
0.6
85.3/85.0
83.7/83.2
82.0/81.3
90.5/90.0
= 0.8 = 1.0
91.2/91.0
89.5/89.2
89.3/88.9
91.5/91.0
92.2/92.2
92.0/92.0
92.0/92.0
92.0/92.0
== 1.2
86.0/86.0
90.1/90.1
89.8/89.8
91.5/91.5
::::
83.8/83.5
Finally, the same variable'"'gate procedure was applied to a Bureau of Reclamation pump turbine (1977), 12.
It is seen that
Q
Tl •
The results are shown in Figs. 11 and
is now an inoreasing funotion ofn
turbining and pumping modes.
in b.oth the
There is significant improvement in turbine
22
""
"
100
VARIABLE GATE
CONSTANT HEAD OPERATION
TURBINE EFFICIENCY 90 u
80
w
CJ)
I-
"-
z
rt)
W
IlL. N
w
N I
..
o x
70
()
0::
w
a..
60
..
50~ ~
w 40
50 w 40
(!)
()
TURB l NE 0 I SCHARGE ./ ",-
0::
~
u Cf) 0
30[
~.
w
30
20
20
1:[
10
/
PUMP 01
0 60
80
100
120
140
Ji"BE P
£':::::J
160
180
TURB I NE SPEED N, RPM Fig. 11.
Variable-speed characteristics of USBR pump turbine Tl (Ns
= 0.76).
'""".
o
r"
VARIABLE GATE ,CONSTANT HEAD OPERATION
100
I:2
/~
80
W U
a:: w
a. 60 N
"'"
>-
I
U
Z
w 40 u
/
/
/
/"
I ----VARIABLE SPEED OPERATION FIXED SPEED OPERATION
lL. lL.
W.
20
o
40
20
60
80
100
140
120
160
PERCENT OF BEST EFFICIENCY DISCHARGE, Q
Q Fig. 12.
Variable and fixed-speed characteristics of USBR pump turbine Tl (Ns
0.76).
efficiency at off-design flow and constant head, even though the specific speed of the unit is similar to turbine efficiencY' improvement.
F 2'
The authors are not certain as to the reason for this
difference in results between units 1) (:
which experienced a very limited
F2
and
'1'1'
Two possible reasons are:
That geometric shape of a pump-turbine runner is much closer to that of a pump than a turbine.
units
F2
and
T1
are therefore
likely to be asimilar geometries. 2)
The pump-turbine unit may not have been designed for off-design discharge at constant head, which would result in a very poor constant speed efficiency curve.
25
V.
SOMETHEORETlCAL CALCULATIONS
c' The calculations in Section IV indicate that for constant
H,
the
discharge (0) appears to be a decreasing function of rotational speed (n) at low specific speed (N s ) (Francis turbines), while for larger Ns (~ial-flow turbines) 0 increases wtih increasing n. Within each tUrbine type this behavior depends on the characteristics of the individual machines, so it may be possible to have different
Q
versus
n
relation-
ships for the l3ame value of
N, depending on the specific design. In s other words, one may be able to influence the speed dependence by l3uitable
design modifications.
In this section, we examine briefly this problem in
the case of axial-flow turbines, for which a relationship
O(n)
can be
derived theoretically on the basis of certain parameters characterizing Q
the blade lattice geometry. To introduce the calculations, we look first at the case of a reaction turbine (not necessarily of the axial-flow type) with a sUfficient number of blades so that the direction of the fluid flow at the exit from the runner is tangential to the exit blade angle.
We write Euler's equation in the
form (8)
where
H
is the turbine net head,
g
is the acceleration of gravity,
llh
is the turbine hydraulic effioiency, the subscripts 1 and 2 deno.te inlet to exit fram the runner, respectively, magnitude,
a
V is
the fluid velocity and
is the angle that the velocity vector
rotational velocity
-+
u,
and
u
is the magnitude of
V +
u.
V
its
makes with the In the space
between distributor .exit and runner inlet, one has, wi th sufficient approximation,
26
where
r
is radial distance and the subscript
distributor.
(I
where
0
denotes exit from the
We have, furthermore,
b o is the ohannel width.
~hus,
Q 1 21tb or l tan ao
~hen
from the exit velocity triang1e, one obtains V2 cos a 2
where
Sz
is the angle that the runner blades make with the negative
direot,ion, and
A2
is the exit flow cross-sectional area.
+
u2
Combining Eg. (8)
and the last two equations, one gets,
(9)
where
rG ~his
Q on nh
n
== rotational speed (rad/sec)
= u/r.
equation shows that under the condi tiens stated, the dependence of An + BTlh/n,
is of the form
were constant when
n
where
A
and
Bare oonstants.
varies (which it is not), the
be of the form of the curve labeled "Eq. (.14,)
II
in Fig. 13.
Q(n)
If
graph would
The minimum of the,
curve oocur,s in this case where (r 2rG • ) 2 = nh(gH}, that is m /0, 2 m1n' min peak) '= 2 nh(9H)/(r z 0, k2), with rG, k best-efficiency or c1esign speed. Depending , pea 'pea 2 ' on the value of the parameter gH/(r 2 rG peak 2) (that is, of the parameter 0:;:;
,
2 2
gH/n P , with
n = npeak )' the design point could be to the right or left of the minimum, thus implying an ascending or descending Q(n) curve. In
practice,
nh
varies with
n,
and the actual
Q(n)
curve will depend on
the type of turbine and the characteristics of the specific design.
27
V.1.
Axial-F.low Turbines The above analysis is not applicable to axial-flow turbines because
the
e~it
fluid angle from the runner is not generally equal to
In the
~2'
calculation that follows, this angle is first determined on the basis of the blade lattice parameters.
The presentation is kept brief and only the major
resulting equations are given here. Consider a cylindrical section of the turbine of diameter D is the runner diameter (a < 1). the blade spaoing in the lattice,
d
= aD, where
For this cylindrioal section, let 5/,
t
be
the blade ohord length,
V the relative velocity and v its mOdulus, (3 the angle that the relative velocity vector makes with the negative 0 direction, ~oo = (~l + ~2)/2, and va = v a = V00 the axial or meridional flow velocity. The subscripts 1 and 2 again repres~nt the inlet and outlet of the runner, respectively. irrotational flow, the magnitude of the lift force
FL
If we assume
on each blade is given
by (10 )
(I
where
r
is the circulation around each blade in the lattice, given by
Using superposition, one can show that zero-lift-angle,
(30'
r can be e:xpressed in terms of the
for the lattice, and a dimensionless parameter
8,
in
the form:
( 1.1)
Theparameter
