Blade calculations for water turbines of the Banki type

By P. Verhaart

March 1983

Department of Mechanical Engineering -.­• = Eindhoven University of Technology

Report WPS3-83.03.R351

BANKI TURBINE BLADE CALCULATIONS

THE BLADE STRENGTH PRODUCT

By P.Verhaart Department of Mechanical Engineering

Eindhoven University of Technology 9 March 1982

SUMMARY

During the years 1978 and 1979 in Indonesia a number of previously installed Banki type water turbines developed cracks at the blade ends near the supports leading to breakage. In order to prevent recurrence of blade failure a calculation procedure was developed which can be programmed on a pocket calculator (HP-67). Blades calculated according to this procedure have so far (early 1982) behaved well. In this report the method is explained, hoping thereby to swell the so-far meagre stock of practical design information on this useful small water turbine.

Introduction

During the years 1977 and 1978 a number of small (8 to 30 kW) Banki turbines was designed and built in Indonesia under the responsibility of personel of the Eindhoven University of Technology at the time employed in a cooperation project with the Bandung Institute of Technology. Not long after installation cracks developed in the runner blades starting from the periphery and near the supporting flanges. Further use of the turbine resulted in breaking of the affected blades. A little later similar troubles occurred in Nepal where a Swiss team had been developing the same type of water turbine. In response to this problem a calculating procedure was developed which is quite amenable to further refinement but which has, so far as we know, resulted in longer blade life of existing machines and a better design for newly produced machines. In this paper, details about the calculation of the fatigue strength of blades are given.

2 Geometry of the Banki Runner

The runner geometry is best explained with the aid of Fig. 1. From Banki's article [1] we derive the following ratios:

o = 2r = 6,1236g 1

r 2 = 2,0177g

R = 2,2516g

830321 Banki Turbine Blade Calculations 2

-------

rM -< -< L .J

I

I I i ...:I i

!

! ... ,

L.J

~-A

FIGURE 1. BANKI RUNNER DIMENSIONS.

The following symbols occur in the figure:

c D L I n R r

velocity of the water from the nozzle runner outer diameter total effective blade length length of a blade section between supports number of blades pitch circle radius outer runner radius

r2

inner runner radius, locus of the ends of

the skeleton lines of the blades

t u w

a

width of channel between two succesive

blades normal to the flow direction thickness of the blades peripheral velocity of the runner relative velocity of the water in a channel between two succesive blades angle between absolute water velocity and peripheral velocity of runner angle between relative velocity wand peripheral velocity u angle of pipe section forming a blade curvature radius of blade inside surface pitch of blades on pitch circle

2.1 Constructive Dimensions

mls m m m

m m

m

m m mls

mls

rad

rad rad m m

8\

Some of the dimensions defined above need further clarification. The total effective blade length L properly is the length of the runner minus the total thickness of the supporting flanges. It is the length of blade that actually deflects the water jet and as such it cannot be represented properly in the drawing. The blade length between supports 1 is self explanatory. It plays a major role in this report. The number of supporting disks can be chosen such

2

830321 Banki Turbine Blade Calculations 3 that the bending stress due to hydraulic forces on the blade can be kept at a safe value. The number of blades n is also a compromise. From a fluid dynamics point of view a large number of blades is desirable but in practice only a limited number can be accomodated due to manufacturing constraints. The pitch circle radius R is a purely manufacturing parameter. It needs to be known in order to be able to scribe the slots for the blades on the supporting disks. The inner radius r

2 is another constructive dimension. It fixes the

position of the inner ends of the blades on the supporting disks. It is good practise to drill holes of a diameter equal to the blade thickness t to obtain well defined end positions for the blade slots in the supporting disks. The width of the channel s1 formed by two successive blades is a very

important dimension determining the distributed load on the blades. The blade thickness t not only has its influence on the bending strength of the blade, it also decreases the theoretical channel width between blades. In order to admit a certain volume flow of water the real runner has a slightly greater length than the theoretical one where no allowance was made for blade thickness. The radius of curvature of the inside of the blade Q is needed to scribe the blade slot outlines on the supporting disks. The pitch [ is used to set out the n blade slots on the pitch circle.

2.2 Fluid Dynamical Dimensions The absolute water velocity of the jet c is usually taken to be the

velocity attained by the complete conversion of the head into kinetic energy using a nozzle efficiency ~ of:

~ = 0,96 ... 0,98

The peripheral velocity of the runner u. With the values of the angles of entry and relative velocity as set out in Banki's article [1] the ratio:

u/c = 0,484

The relative velocity w of the water in a blade channel at the entry of the runner normally satisfies the condition that the ratio:

w/c = 0,5546

In the theoretical analysis the relative velocity is not constant throughout the blade channel.

The angle between the absolute and peripheral velocity a is taken as:

a = 0,2792 rad or 16 0

The angle between relative velocity and peripheral velocity ~ is taken as:

~ = 0,5236 rad or 30 0

The angle of pipe section forming the blade profile 6 is taken as:

6 = 1,2828 rad o or 73,5

3

830321 Banki Turbine Blade Calculations 3 Outline of the Procedure

After a brief description of the quantities that have been taken into account the relations that are assumed to exist between them are defined. In the succeeding section the final expression is derived in detail. The calculator program is described in a separate section.

3.1 Factors of Influence The components of the load on the blades that were taken into

account in the analysis are the following:

4

a) The hydraulic force, periodic in nature, resulting from the water jet entering the runner.

b) The centrifugal force, constant at constant angular runner velocity.

The components of the load not taken into account are the following:

c) Torsion resulting from transmission of torque to the output shaft end.

d) Shear stress resulting from transmission of torque to output shaft end.

e) Bending moment from the entry of the water jet into the runner.

3.2 Discussion Component b) in most cases turns out to be insignificant compared

to component a) but was easy to include into the calculator program. Component c) is of the same magnitude as in the shaft of the runner. The latter has been designed to withstand the combined stress resulting from torsion as well as from bending. Component d) is constant at constant power output and angular velocity. It is the result of the parallel displacement of the blade supports when the latter undergo angular displacement as the result of torque transmission to the shaft. This component was dismissed without thorough examination. It would add a constant amount of tensional stress near some of the supporting flanges.

------- -- - - ----

- compression

~E:::·=-~r-·--- ---------r-' ~.

- tension

FIGURE 2. FORCES ON A RUNNER.

Component e) results in a periodic tension and compression imposed on the blades. This, however, is not superimposed upon the hydraulic induced stress as it occurs a quarter revolution before and after the latter as can be seen in Fig. 2.

4

830321 Banki Turbine Blade Calculations 5

4 Developing the Strength Product Concept.

The strength product concept enables one to do the fatigue strength calculation for the blade beforehand as it were. Usually the blades for a Banki type turbine are made from steel pipe which is sectioned lengthwise. When the outer diameter and the wall thickness of the pipe with which one intends to build a Banki runner is known, the strength product can be calculated. This enables the turbine manufacturer to specify the maximum head of water for which his turbine is suited. Conversely it gives him the safe length of blades between supports if the head of water under which the turbine is going to be used is specified. As steel pipes come in standardised sizes it is also possible to give a turbine builder a list of pipe sizes with the associated strength product for each pipe size.

4.1 Assumptions

a) The blades are treated as uniform beams of constant cross section, fixed at both ends.

b) The hydraulic force of the water jet is treated as a uniformly distributed centric load along the length of the blades.

4.2 Functional Relations The well known relation between the uniformly distributed load and

the maximum bending moment states that it is directly proportional to the magnitude of the distributed load and to the square of the length of the beam.

M Wl

2 = 12 ( 1 )

where:

M is the bending moment Nm W ~s the distributed load N/m 1 is the length of the beam between supports m

A further relation exists between the bending momemt and the flexural stress:

where:

a =

a e

I

Me I

is the flexural stress is the maximum fibre distance from the neutral plane in the blade cross section is the area moment of inertia of the blade

cross section

(2)

m

The distributed load W from (1) with fixed runner geometry is equal to the force on the concave side of a blade divided by its length. The force on the concave side of the blade is a result of the change in moment of momentum of the part of the water jet that enters a blade channel. As the moment of momentum can be expressed in the square of the relative velocity of the water on entering the blade channel the distributed load can be

'. ,

5

830321 Banki Turbine Blade Calculations written in the following form:

where:

2 W = Aw

W A

w B

g Z

2 A(B/(2gZ»

is the distributed force along the blade constant dependent on the geometry and number of blades relative velocity in blade channel geometry dependent proportionality constant between absolute and relative velocity

acceleration of gravity head of water

(3 )

N/m

m/s

2 m/s m

6

Combining (1) and (3) we get a relation between the bending moment and the head where the bending moment M is directly proportional to the product of the head Z and the section length 1 to the second power.

(4 )

where:

C is a constant of proportionality

Combining (4) with (2) we obtain an expression for the bending stress a in terms of the blade geometry quantities (I, e and 1) and the head Z e.g.

a = 2

CZl e I

The expression can be rearranged as follows:

(5 )

(6)

When from literature or experiment a safe value for the bending stress a can be found and substituted in the expression (6), it will produce a maximum value for the head times the square of the blade section length when the blade quantities e and I are substituted. In this way the safe section length for given blade dimensions and a given head can be calculated. The term between brackets on the left of the = sign we call "Strength product".

4.3 Derivation of the Expressions

4.3.1 Massflow Through a Blade Channel From inspection of Fig. 1 the mass flow per blade channel ~s seen

to be:

Qmc = QwlS 1W

where:

Qmc ~s

Qw is

the mass flow

the density of water

(7 )

kg/s

kg/m3

6

830321 Banki Turbine Blade Calculations 7

In order to find an expression in known quantities for s1 we look again at

Fig. 1. We see that the channel width s1 is the projection of the chord

of the arc between two successive intersections of blade skeleton lines with the runner perimeter. The arc length 1 is equal to the circumference divided by the number of

a blades.

where:

n is the number of blades

Now in practice the number of blades is always above 20. When the chord length is substituted for the arc length a very small error is made. The angle between the tangent to the concave blade surface at its intersection with the runner periphery and the tangent to the runner periphery at that same spot is p. From the figure we see that:

This is without taking into account the blade thickness t. When blade thickness is taken into account the expression changes into:

lTDsinp = ------ - t

n (8)

The channel width s1 is only one of the two dimension of the cross section

of the blade channel. The other is the length of the blade section 1. The cross sectional area is the product:

A = s 1 1

(9 )

The relative velocity can be found from the velocity triangle, see Fig. 3.

FIGURE 3. VELOCITY TRIANGLE AT RUNNER ENTRANCE.

From the figure and using the cosine rule we obtain:

222 w = u t C - 2uccoscr

Under design conditions:

u = O,484c

7

830321 Banki Turbine Blade Calculations 8

a = 0,2792 rad

So that, substituting these values in the cosine expression and working it out, we obtain:

2 2 w = O,3038c

or:

w = O,5511c

Expressing the relative velocity w in the head we obtain:

w = O,7794f(gZ) ( 10)

Combining equations (8), (9) and (10) we get an expression for the volume flow Q through a blade channel of width s1 and length 1

vc

where:

lfOsin~ = O,7794( ------ - t)lf(gZ)

n

is the volume flow per blade channel 3

m /s

To obtain the massflow Q per blade channel the volume flow is multiplied mc

by the density of water Q . Expressing the massflow per channel section in w

known quantities and ratios we obtain:

Q 2439 94(~L§!~~~ - t)lfZ mc = , n ( 11)

4.3.2 Torque Transmitted by a Blade Channel The torque transmitted by a blade channel is the change in moment

of momentum of the massflow passing through a blade channel. At the entrance the circumferential component c of the absolute velocity is:

u

c = ccosa u

While the distance from the centre of rotation is r 1 (= 0/2)

At the exit of the blade channel the relative velocity has a radial direction and thus the circumferential component is equal to the local runner velocity u2:

u = 2

And the radius is r 2 .

Expressing u2 in c we get:

8

830321 Banki Turbine Blade Calculations The moment transmitted by a blade channel then becomes:

T = Q c(r 1cosa-c mc

2 0,484r 2 -------)

r 1

substituting (11) for Q and the current values for the proportions, mc

expressing the radii in Q we get:

9,619Q Tc = 24842,1(--n--- - t)QZI (12)

4.3.3 Point of Application of the Hydraulic Forces In Fig. 4 we see that the tangential component of the hydraulic

force is assumed to have its point of application at the centre of the blade arc.

F---_-.-...t.

FIGURE 4. POINT OF APPLICATION OF HYDRAULIC FORCES.

From the figure and with the cosine rule we get:

where:

y equals 2Qsin(6/4) m

substituting the current values in the expression gives the result:

r/Q = 2,6236 ( 13)

The position of the blade in relation to the tangential force F can be seen from Fig. 5. In the triangle formed by r, Rand Q, the centric force Fc works along Q

while the hydraulic force F is perpendicular to r. Thus the angle ~ is the complement of the included angle ~ between rand Q.

9

9

830321 Banki Turbine Blade Calculations --------------- ---------

F----~_,_~

FIGURE 5. FORCES ON A BLADE.

Using the cosine rule:

or:

222 r + Q - R ~ = arccos(---2rg--- )

Using the current values we obtain:

~ = 1,0049

The angle 1 is then:

1 = U/2 - 1,0049 = 0,5659

o or 32,4

4.3.4 Distributed Load on a Blade

rad

rad

In Fig. 5 the relation between F and F is clearly seen. As the c

10

blade can only transmit forces resulting from pressure (and thus normal to the blade surface) the centric force must apply at the centre of the arc. If the pressure is not constant along the arc a torsional moment is superimposed. This latter is not here taken into account. The force F is therefore the projection of the centric force F :

c

or:

F cos1 = F c

F 6~8443 = 1,1847F ( 14)

10

· ------y

x

830321 Banki Turbine Blade Calculations The force F can be found from (12) dividing the torque T by r:

c

F = T

c r

Using the current values we obtain:

9,619Q F = 9468,7(------ - t)Zl

n

And for the centric force normal to the blade F : c

9,619Q Fc - 11217,4(--n--- - t)Zl N

Finally, dividing F by the section length 1 we obtain the distributed c

load W:

( 15)

where:

W is the distributed load N/m

11

In (15) we have an expression for the distributed load on a blade expressed in properties of the blade e.g. thickness and inside radius of curvature, in a property of the runner e.g. the number of blades and in a "condition of employment" e.g. the head.

4.3.5 Area Moment of Inertia of the Blade In Fig. 6 the simplified shape of the blade cross-section is

shown.

y

------,- --- -----

\ \

FIGURE 6. BLADE CROSS SECTION.

--------

What we need to know is the area moment of inertia in respect to the neutral plane X-X. As a direct evaluation of the area moment of inertia with respect to the X-axis appears to be a rather complicated operation, the indirect method

'1

830321 Banki Turbine Blade Calculations is employed using Steiner's theorem.

I x

where:

2 I - Ay Y

I is the area moment of inertia with respect x

I Y

A

y

to the X-axis is the area moment of inertia with respect

to the Y-axis 1S the cross sectional area of the

blade profile is the distance of the centre of gravity of the section from the Y-axis

From Fig. 6 the terms in (16) can be derived:

6t(2g + t) A = ----------2

y = ~il~_±_~l:_:_~=!~~~l~L~l 6t(2g + t)

I = ilQ_±_~l~:_e~!l~_±_~!~~l y 8

( 16)

4 m

4 m

2 m

m

The extreme fibre distance from the neutral plane e is found to be:

e = y - gcos(6/2)

Substituting the current values into the above equations we obtain the following:

A = 0,6414(2g + t)t

y = QL§~!~ile_±_tl=:_e=l (2g + t)t

4 4 I = 0,2231{(g+t) - g } y

The extreme fibre distance e is:

e =

The area moment of inertia can be written out as follows:

4.3.6 Bending Stress on Blade Section In this analysis the blade is treated as a beam of constant cross

section, rigidly fixed at both ends (see Fig. 7).

12

12

830321 Banki Turbine Blade Calculations

FIGURE 7. BLADE SECTION AS A RIGIDLY FIXED BEAM.

Adapting the well known equation from applied mechanics for this case we can write for the bending moment:

2 W{6X(1 - X) - 1 }

M = -----------------12

where X varies from zero to 1. The maximum values are found for X = 0 and for X = 1 when:

M =

Nm

( 17)

Which means that the highest values for the bending moment occur near the ends of the blade sections. The negative sign means that the topside of the beam near the supports (in Fig. 7) is subjected to tensional stress. The tensional stress that occurs in the beam can be expressed in known quantities:

13

substituting W from (15) and rearranging the following expression emerges:

(J = ( 18)

4.3.7 Strength Product The tensional stress (J occurs in each runner blade once per

revolution. In order to make a safe runner construction the value for (J

has to be chosen according to fatigue strength criteria. As the alternating load takes place in the presence of water a low fatigue strength results. Consulting various sources (2], [3], (4], a value:

a =( 22 2 N/mm

for mild steel seems in the correct order of magnitude. When this value is substituted for a in (18), the expression can be rewritten as follows:

Z12 = 23534,951 /{(9,619Q/n - t)e} x ( 19)

13

830321 Banki Turbine Blade Calculations 14

This expression gives the safe value of the product of the net head Z (in m) and the square of the length 1 (in m) of a runner blade section

between supporting discs (if the fatigue stress of 22 N/mm2 is considered safe) . In practice runner dimensions are given in mm and the head is given in m. How does (19) change in that case? Expressing 1 in mm requires the part to the right of the = sign to be

multiplied by 106

. . . 4 -12

Expresslng I ln mm produces a factor 10 x

Expressing Q, t and e in mm gives a factor 10-6in the denominator and thus

106

in the numerator. Adding these powers of 10 together we get exactly zero which means that

4 (19) is also valid for Z in mi 1, Q, t and e in mm and I in mm. This

x gives manageable figures. In (18), on the other hand, all powers of 10 cancel out when the above mentioned quantities (except Z) are expressed in mm and thus 0 remains ln

2 N/m .

NOT E The expression (19) is not dimensionallY homogenous. In the

numerical coefficients the numerical values of the acceleration of gravity and the density of water have been used. In the unlikely case of anyone wishing to rewrite (19) in other units conversion has to start from equation (10) ..

5 Working with the Strength Product

5.1 Summary of Formulae

1) Strength Product:

Z12 = 23534,95Ix/{(9,619Q/n - t)e} ( 19)

2) Area Moment of Inertia:

3) Maximum Fibre Distance:

e =

With the aid of the above collection of formulae it should not be too difficult to calculate the strength product, especially if a modern programmable pocket calculator is used.

5.2 Using the Strength Product The expression (19) gives the highest value of the product of the

head Z and the square of the section blade length 1 which will keep the

alternating tensional stress in the blade below the value of 22 N/mm2 (which is considered safe for mild steel).

14

830321 Banki Turbine Blade Calculations 15 If the head under which the turbine has to work is known, the safe section length is easily calculated. If the head is not known, as in the case of a series production, it is easy to indicate the maximum permissible head.

6 HP-67 Calculator Program As well as the strength product, a number of other useful

calculations was programmed in order to make the fullest use of the programming memory space available in the HP-67 pocket calculator. In order to be able to instruct small metal workshops in urban and rural areas in developing countries, it seemed useful to include the calculations for all the main dimensions of a Banki runner. Under the assumption that mild steel pipe is the raw material for the blades, the program starts after the pipe's wall thickness t (mm) and its outer diameter d (mm) are entered. From these data all the runner dimensions as well as the strength product are generated. For the number of blades a simple rule of thumb has been evolved. A second part of the program calculates dimensions and the strength product for other numbers of blades. A third part of the program calculates the centrifugally induced tensional stress S in the blades.

c For a given head Z and blade section length 1 the tensional bending stress is calculated.

6.1 Number of Blades The number of blades must be such that the runner can be

fabricated. If the runner has more than two supporting discs, welding will also have to be done between the blades. It was felt there should be some direct relation to the diameter of the runner which resulted in the following relation:

or, expressed in g:

Thus the program can run after the dimensions of the pipe, the blades are going to be made from, are entered.

6.2 Program Details The program is available from:

HP User's Program Library Europe

under number:

60954 D, Banki Crossflow Turbine Runner Design.

The addres is:

HEWLET-PACKARD SA. User's Program Library Europe 7 Rue du Bois-du-Lan P.O.Box CH-1217 Meyrin 2 GENEVA-SWITSERLAND

15

830321 Banki Turbine Blade Calculations 16 When ordered the program description comes complete with two magnetic cards, one containing the 220 program steps, the other containing a number of constants to be entered into the memories. As the program, at the time it was composed, used a different method to calculate the distributed load W, the resulting strength product is some 58 % higher than with the calculation method just described. To remedy this it is sufficient to change the value of the constant that goes into memory A to:

23534,95 (was 37037,8)

The values that have to be entered into memory registers if no data card is available are the following:

REG.NR VALUE COMMENT

0 0,6414085 standard value of 6/2 4 0,1633 = Q/D 5 0,3295

A 23534,95 constant for calculating Zl2

I 6,541666667*10- 10

6.2.1 Description The program consists of 4 parts under the labels A through D.

Part A:

Calculates runner dimensions and Strength Product after entering wall thickness t and outer diameter d (in that order) of the pipe that will be used to make the blades.

Part B:

Calculates the new pitch [ and Strength Product when a new value for the number of blades n is entered.

Part C:

Calculates the bending stress a at the edge of the blades near the supports resulting from the hydraulic load when the head Z and the blade section length I are entered.

Part D:

Calculates the bending stress S in the blades resulting from centrifugal c

forces acting on the blades.

16

830321 Banki Turbine Blade Calculations 6.2.2 User Instructions

STEP INSTRUCTION

Load program card, both sides Load data card, 1 side only

VARIABLE KEY

1 2 3 4

Enter wall thickness t t (mm) ENTER Input outer diameter of pipe d in mm d (mm)

5 6 7 8

Initialize A A Display runner diameter D R/S Display number of blades n R/S Display radius of inner circle r

2 9 Display radius of pitch circle

R 10 Display pitch r 11 Display strength Product

Z12

For other wall thickness and/or

R/S

R/S R/S

R/S

other pipe diameter repeat steps 3 ... 11

12 Key in desired number of blades n

1 13 Initialize B 14 Display Strength Product

Zl2

B

R/S

For other numbers of blades repeat steps 12 ... 14. Steps 12 ... 14, can be run immediately after step 5.

15 16

17

Enter head Z Input blade section length 1

Initialize C

Steps 15 ... 17 can be run immediately after step 5.

18 Initialize D 19 Enter rotational frequency

f (Hz) 20 Input blade section length

1 (mm) 21 Restart program

Z (m) ENTER 1 (mm)

C

D

f (Hz) ENTER

1 (mm) R/S

Steps 18 ... 21 can be run immediately after step 5.

RESULT

t (mm)

d (mm) Q (mm) D (mm) n (-)

R (mm) r (mm)

2 2 Zl (m*mm)

n1

(-)

r (mm)

Z (m) 1 (mm)

r (N/mm2)

54435

f (Hz)

1 (mm) S (N/mm) c

17

17

830321 Banki Turbine Blade Calculations 6.2.3 Example

In this example all the possible calculations will be done. This enables the user to check the program.

The following data are from one of the turbines that broke down. Head: 5,3 m ; Blades made from pipe of 174,8 mm outside diameter and 4,4 rom wall thickness. The section length was 640 mm.

STEP

1 2 3 4

5 6 7 8

9

10 11

12

13 14

15 16

17

18 19

20

21

INSTRUCTION VARIABLE KEY

Load program card, both sides Load data card, 1 side only Enter wall thickness t in rom 4.4 Input outer diameter of pipe d in mm 174.8 Initialize A to display Q (mm) Display runner diameter D (mm) Display number of blades n Display radius of inner circle r

2 (mm)

Display radius of pitch circle R in mm Display pitch [ (mm) Display strength Product

2 2 Zl (m*mm)

Key in desired number of blades n1 22

Initialize B for pitch [ (mm) Display Strength Product

2 2 Zl (m*mm)

Enter head Z (m) 5.3 Input blade section length I (mm) 640

Initialize C to calculate a (N/mm2)

Initialize D Enter rotational frequency f (Hz) Input blade section length I (mm) Restart program to calculate

S (N/mm2) c

2.97

640

ENTER

A RIS RIS

RIS

RIS RIS

RIS

B

RIS

ENTER

C

D

ENTER

RIS

RESULT

4.4

174.8 83. 508.3 24.

167.5

186.9 48.8

819815.

22.

53.2

742071.

5.

640.

64.

54435.

3.

640.

5.

18

18

0321

STEP

001 002 003 004 005 006 007 008 009 010 011 012 013 014 015 016 017 018 019 020 021 022 023 024 025 026 027 028 029 030 031 032 033 034 035 036 037 038 039 040 041 042 043 044 045 046 047 048 049 050 051 052 053 054 055

Banki Turbine Blade Calculations 19

6.2.4 Program Listing --.~---.

KEY KEY COMMENT ENTRY CODE

f LBL A 31 25 .11 h RAD 35 42 2 02 · 81 · 5TO 2 33 02 h x<=>Y 35 52 STO 1 33 01 - 51 STO 3 33 03 RCL 4 34 04 · 81 · STO 6 33 06 RCL 5 34 05 * 71 STO C 33 13 9 x"2 32 54 RCL 3 34 03 -- .. ---.-

9 x"2 32 .54 • + 61 c-.:J

f Vx 31 54 STO D 33 14 "l:)

c: RCL 6 34 06 cu

h PI 35 73 ar:

* 71 , N

RCL 6 34 06 ~ L

83 • , · • c: 1 01 • ... , 2 02 c

en

* 71 0. ._, LLlQ.

RCL 1 34 01 I-en

+ 61 .... · · 81 0

83 c: · 0 5 05 .~ .... + 61 cu

~

h CF 1 35 61 01 :::J u

f LBL 0 31 25 00 ~

cu f INT 31 83 u

STO 7 33 07 h 1/x 35 62 h PI 35 73 * 71 f sin 31 62 RCL D 34 14 * 71 2 02 * 71 STO E 33 15 h F? 1 35 71 01 GTO 2 22 02 RCL 2 34 02 4 04 h y"x 35 63 RCL 3 34 03 4 04

,

- ------,-..--....... ---..... ---..... ---_ ..... _. ~.. --_ ..... _-_ .. _-.

STEP KEY ENTRY

056 h y"x 057 -058 RCL 0 059 2 060 * 061 ENT 062 f sin 063 + 064 * 065 8 066 : 067 STO 8 068 RCL 2 069 3 070 h y"x 071 RCL 3 072 3 073 h y"x 074 -075 ENT 076 9 x"2 077 RCL 2 078 9 x"2 079 RCL 3 080 9 x"2 081 -082 ENT 083 h R" 084 h x<=>y 085 : 086 2 087 * 088 3 089 : 090 RCL 0 091 : 092 RCL 0 093 f sin 094 * 095 h Rv 096 : 097 RCL 0 098 f sin 099 9 x"2 100 * 101 4 102 * 103 9 104 : 105 RCL 0 106 : 107 STO - 8 108 h R" 109 RCL 0 110 f cos

KEY CODE

35 63 51 34 00 02 71 41 31 62 61 71 08 81 33 08 34 02 03 35 63 34 03 03 35 63 51 41 32 54 34 02 32 54 34 03 32 54 51 41 35 54 35 52 81 02 71 03 81 34 00 81 34 00 31 62 71 35 53 81 34 00 31 62 32 54 71 04 71 09 81 34 00 81 33 51 08 35 54 34 00 31 63

COMMENT

---.--

• ., .... o (II .... L

/'-CU 00. ...

• "l:) • c: • cu 0-

-4")( ..... en 0. .... LLlO I­enc: o

.~ .... cu ~

:J u ~

cu u

~.- -----· . i -4" ., : ... !.-'+-, · • 0 , . , • c: '00 0

!~~ I en ~ i 0. :::J · LLI u· I- ~. en cu

u

19

830321 Banki Turbine Blade Calculations 20

.-~ .-_' .. _._--- ---<-- ----- . "- --". -, -- ---' ... -- - _.---._-

STEP KEY KEY COIIIfIIIENT STEP KEY KEY CO .... ENT ENTRY CODE ENTRY CODe

111 RCL 3 34 03 '" -- --- - _ . 166 * 71 112 71 •

* N 167 h RTN 35 22 113 51 c - N -' 168 f LBL D 31 25 14 114 STO 9 33 09 fO"I -II,

.- N 169 RCL 0 34 00 115 f LBL 2 31 25 02 • · .... 170 f cos 31 63 116 RCL 9 34 09 • 0 11\ 171 RCL 3 34 03 117 h 1/x 35 62 .- c

.- 0 172 * 71 118 RCL 8 34 08

.~

(I) .. 173 RCL 9 34 09 119 * 71 CL. 1'0

W -' 174 + 61 120 RCL A 34 11 .... ::J

! (I) U 175 ENT 41 121 71 -'

* ; 1'0 176 ENT 41 122 RCL 6 34 06 u 177 RCL 0 34 00 123 h PI 35 73 178 f sin 31 62 ._.- -

124 * 71 • 179 * 71 125 2 02 1oo,J' 180 RCL C 34 13 126 · 81 I · 181 + 61 -_ .. ~---

127 RCL 7 34 07 1J

128 81 c · 1'0 · 182 h x<=>y 35 52 183 RCL 0 34 00

129 RCL 1 34 01 fO"Ia::

130 51 11\ - .- , 131 81 • N · • L ·

184 f cos 31 63 185 * 71 186 CHS 42

132 STO B 33 12 • fO"I ,

133 h F? 1 35 71 01 fO"I c .-134 GTO 3 22 03

, (l)Q

135 RCL 3 34 03 CL. w , 136 DSP 1 23 01 .... Ci

(I)

137 RIS 84 138 RCL 6 34 06 >-

1'0

139 RIS -' 84 Q.

140 DSP 0 23 00 (I) .~

141 RCL 7 34 07 Q

......

187 RCL 3 34 03 188 + 61 •

U

189 9 ->P 32 72 0(1) N

190 h x<=>y 35 52 N .... • 0

191 CHS 42 • 192 h PI 35 73 aO~

-0 .~

193 + 61 .- .. 1'0

194 RCL 0 34 00 (I) -'

CL. ::J 195 - 51 wu .... -' 196 f cos 31 63 (I) 1'0

u 142 RIS 84 • c 143 DSP 1 23 01 .. 144 RCL C 34 13 c

41 145 RIS 84 L

41 146 RCL D 34 14 .... .... 147 RIS 84 .~

C01J 148 f LBL 3 31 25 03 11\

.- L 149 RCL E 34 15 • 0 · .... 150 RIS 84 •

04'N 151 RCL B 34 12 11\ C .- -' 152 DSP 0 23 00 -II

(l)N

153 h RTN 35 22 CL. w .... 154 f LBL B 31 25 12 .... 0

(I)

155 DSP 1 23 01 c 156 h SF 1 35 51 01

0 ---. ~ • .. ,.... t) 157 GTO 0 22 00 1'0 -' -0

158 h RTN 35 22 ::J .... U · .... 159 f LBL C 31 25 13 -' • 0 1'0 • 160 9 xA2 32 54 u coc

11\0 161 * 71 .- .~ .. 162 RCL B 34 12 (I) 1'0

CL.-' 163 · 81 W::J · .... U 164 2 02 (I) -'

1'0 165 2 02 u

197 * 71 198 RCL 2 34 02 199 9 )("2 32 54 200 RCL 3 34 03 201 9 xA2 32 54 202 - 51 203 RCL 0 34 00 204 * 71 205 * 71 206 RIS 84 207 * 71 208 2 02 209 * 71 210 h PI 35 73 211 * 71 212 9 x"2 32 54 213 * 71 214 RCL 9 34 09 215 * 71 216 RCL 8 34 08 217 . 81 . 218 h RC 1 35 34 219 * 71 220 h RTN 35 22

830321 Banki Turbine Blade Calculations 6.3 Bibliography

1. Banki, D. (1918 ) Neue Wasserturbine. Zeitschrift fuer das Gesamte Turbinenwesen. Vol. 15 Nr 21 (30 July 1918) R.Oldenbourg Verlag. Berlin, Munich.

2. Rolfe, S.T., Barson, J.M. (1977) Fracture and fatigue control in structures Prentice Hall.

3.Dubbel's Taschenbuch fuer die Maschinenbau (1974) Springer.

4. Overbeeke, J.L. (1982) Eindhoven University of Technology Personal communication.

21

21

830321 Banki Turbine Blade Calculations 6.4 List of Symbols and Units

Symbol

A

A B

C D I

I x

I Y

L M

Qmc

Qvc R W W Z c e

t u w

y

[

a

Meaning

constant dependent on the geometry and number of blades

cross sectional area geometry dependent proportionality constant between absolute and relative velocity constant of proportionality runner outer diameter area moment of inertia of the blade

cross section area moment of inertia with respect

to the X-axis area moment of inertia with respect

to the Y-axis total effective blade length bending moment mass flow

volume flow per blade channel

pitch circle radius distributed force along the blade distributed load head of water velocity of water from the nozzle maximum fibre distance from neutral plane in blade cross section

acceleration of gravity length of a blade section between supports length of the beam between supports number of blades inner runner radius, locus of the ends of

the skeleton lines of the blades width of channel between two succesive

blades normal to the flow direction thickness of the blades peripheral velocity of the runner relative velocity of the water in the channel between two succesive blades distance of the centre of gravity of the section from the Y-axis pitch of the blades on the pitch circle angle between absolute water velocity and peripheral velocity of runner angle between relative velocity wand peripheral velocity u angle of pipe section forming a blade curvature radius of blade inside surface

density of water

flexural stress

Unit

2 m

m

4 m

4 m

4 m m Nm kgls

3 m Is m N/m N/m m m/s

m 2

m/s m m

m

m m m/s

mls

m m

rad

rad rad m

kg/m3

N/m2

22

22