hydromechanics of variable speed turbines

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