Hermod
Brekke
Norwegian University of Science and Technology
NTNU, Waterpower laboratory, Alfred Getz veg 4, N-7491 Trondheim, Norway
Telephone: +47 73 59 38 56, Fax: +47 73 59 38 54,
E-mail:
hermod.brekke@maskin.ntnu.no
Abstract: The paper describes the state of the art in Francis turbine design and focuses on the improvement of efficiency, cavitation performance and dynamic behaviour. Examples of theoretical analyses of the flow regime in runners are presented together with an explanation of the advantages obtained by splitter blades with an optimal shape. A discussion on the optimal use of geometry parameters by means of analysis based on classical potential theory is also included.
A brief description is given of the philosophy behind pressure-balanced blades with skewed outlets that are made to stabilize dynamic behaviour and improve efficiency during off-design operation. An explanation is given about the possibility of avoiding cavitation with this design. Because of the reduced specific weight (Tonne & MW) through the introduction of high tensile strength steel, the working stresses have increased and critical material defects have decreased. A discussion on acceptable defects, necessary inspections, safety aspects and the lifetime of turbines is also given.
During the last 50 years the demand for the improved performance of Francis turbines has been a challenge for manufacturers. In addition, there has been a demand for an increase in unit sizes and an improvement of the performance over a wide range of operation with a large variation in head without the occurrence of cavitation- and dynamic problems. The reason for this is meeting the requirements for peak load operation in electricity production all over the world. Peak load operation from hydropower production brings a reduction in greenhouse gas emissions as thermal power production can be based on running these plants at a constant optimal load.
The aim of this paper is to describe the research work on turbine design and the result achieved by classical approaches together with the CFD analysis and model tests.
The runners made by the industry are normally designed on the basis of existing geometry modified by means of a CFD analysis.
The optimizing of the geometry is often made by trial and error, but experts normally get a good idea of the geometries required in this work. However, because of differences in basic design, significant variation in runner geometry from different manufacturers can be seen. This is clearly illustrated in Fig. 1 taken from /Ref 1/. However, acceptable performance may also be obtained with differences in the geometry of the cross section shown in Fig. 1 even if there will also be a significant weakness in some details.
Fig. 1 Different cross of Francis runners sections of turbines from different manufacturers
Classical analyses using potential flow have been used at the Norwegian University of Science and Technology (NTNU) in order to create the optimal geometry of Francis runners without any existing geometry to start with.
The main procedure in the design of a new runner employing this method will be:
(1) Classical theory for shaping the geometry.
(2) CFD analysis for the fine tuning of the geometry of runners prior to model testing.
The procedures in such a runner design will now be presented.
(1) The runner geometry is optimized by the governing parameters based on potential theory. This work may be made by means of computerized calculation of classical theory instead of the graphical methods used up to the early sixties.
Fig. 2 Graphical calculation of stream surfaces in a Francis runner. (Advertisement KVAERNER)
Fig. 2 shows a copy of an advertisement of a graphical calculation of a high head Francis runner made by the turbine manufacturer Kværner Brug in the mid sixties. (Kværner is now GE Hydro, Norway).
The governing equations for the potential theory used in the design are:
(1) The equilibrium of forces (Newton's 2nd Law).
(2) The relative specific energy equation (ROTHALPY).
(3) The equation of continuity.
The first equation is the most important one and may be expressed explicitly from the pressure gradient (dh/dn) from crown to band after the assumed stream surfaces have been drawn as shown in Fig. 2. The meridian velocity cm is determined from the equation of continuity. The equation for equilibrium of forces yields, see Fig. 3b:
(1)
Note, reduced
values are used i.e. h =
h/H,
The geometric parameters are illustrated in Fig. 3b.
The starting pressure h at the
crown must be calculated by the ROTHALPY equation and used as the starting point
for the calculation of
. For the other stream surfaces, the pressure h must also be calculated
for each stream surface by the ROTHALPY equation for comparison.
The ROTHALPY equation yields:
(2)
An iteration
between Equations 1 and 2 must then be done by adjustment of the stream surfaces
in order to adjust
by means of the equation of
continuity in order to obtain agreement for the pressure variation h = f(n).
Fig. 2 shows the final stream surfaces with agreement between Equations 1 and 2.
This also enables the pressure distribution between the crown and band to be
determined.
Splitter blades have been introduced for high head turbines with long blades in order to obtain a better agreement between potential flow, and the real flow in a runner with a finite number of blades.
Compared to a runner without splitter blades a splitter blade runner will also reduce the danger of reversed flow on the pressure side of the blades near the inlet at part load. The reason for this is the reduced distance between the blades at the inlet.
On the other hand, the total blade loading can be increased at the inlet portion in a splitter blade runner by allowing for less loading towards the outlet.
It should also be noted that the blade loading on the splitter blades must gradually decrease to zero towards the outlet in order to obtain smooth running over a wide range of operation.
The described parameter study based on classical theory is also very useful in order to obtain pressure balanced blades for low head and medium head turbines.
Fig. 3 (a) shows the runner for the
GAMM workshop in Lausanne /Ref 2/ and this is compared with the geometry used
for student exercises at NTNU (b). Fig.
3b illustrates the parameters used in Equations 1 and 2. The most important
parameter is the blade lean angle
that is the blade angle normal to the flow direction, see Fig. 3(b). This angle
can be calculated from the blade angle in the horizontal plane (
and the angle between the meridional sections (
) and the n axis as shown in Fig. 3b.
(a) (b)
Fig. 3 Geometry illustrated by meridian sections and sections normal to the axis of rotation that is useful in the analytical analysis. The runner used in the GAMM workshop /Ref 2/ left (a) and a runner used for student work at NTNU right (b)
The blade shown in Fig. 3b is not fully pressure
balanced. A fully pressure balanced
blade must have increased the angles (
) near the inlet in order to obtain an increased pressure towards the band.
Equations 3 and 4 show how the blade lean angles
and
can be calculated from the angles
,
,
and
as shown in Fig. 3b.
(3)
Here
is the slope angle of the
streamlines and
is the angle between each meridian
cross section. Further the blade angle can be calculated by means of the
following equation:
(4)
The blade inlet angle is also of
importance especially for high specific speed runners. It is important to
control the inlet pressure h1 on the blades by adjusting the
reaction ratio. The inlet pressure may be found by the equation for the reaction
ratio
expressed by reduced dimensionless
variables as defined for Equations 1 and 2.
h1
= h2 + 2u1 cu1-
(5)
Equation 5 is valid for best efficiency only where cu2 = 0 for flow along a stream line. In Equation 5, we also find that the inlet pressure is increased by reducing the ratio cu1/u1.
In Fig. 4 the flow regime is shown on the pressure side of a pressure-balanced runner (X-blade runner) (right) compared with traditional non-pressure-balanced blade without an optimized design (left). The reduction of cross flow on the pressure side of the blades is clearly illustrated. The reduced cross flow will also decrease the danger of the interblade separation from the inlet of the runner at part load and thus this will lower the pressure pulsations.
Fig. 4 Flow regime on pressure side of a traditional blade (left) and pressure-balanced blade (right)
Fig. 5 The pressure difference between the pressure side and suction side of a traditional non-pressure-balanced blade (left) compared with a pressure-balanced blade (right)
The design procedure for a pressure-balanced runner ready for CFD analysis will be:
Step one. Shape the crown and band and the inlet and outlet edges of blades based on the Euler turbine equation. A smooth curvature in the band is important.
Step two. Calculate the blade inlet and outlet angles by means of the Euler turbine equation followed by shaping the blade's suction side along three stream surfaces between the crown and band. The reaction ratio should be used for the control of the inlet pressure.
Step three. Calculate the pressure gradient (dh/dn) normal to a minimum of four stream surfaces between the crown and band and adjust the streamwise blade shape.
Step four. Smoothen the blade shape prior to CFD analysis.
It is well known that improvement of
cavitation behaviour can be obtained by increasing the runner outlet diameter
and decreasing the outlet angle (
).
Formulas for determining the Thoma Cavitation number based on the outlet angles may be set up based on experience from model tests.
The following formula may be used:
(6)
Fig. 6 Thoma Cavitation number versus circumferential speed u2
It should be noted that the reduced
circumferential speed
is used where D2 is the
outlet diameter of the runner. A diagram based on Equation (6) is shown in Fig.
6.
Runners must be dimensioned for dynamic behaviour at off-best efficiency point operation and for the high frequency pressure oscillations from the blade passing frequency.
Normally Francis runners do not have fatigue problems. If fatigue fracture does occur, the fracture often resembles a shark bite and is caused by the loosening of a semicircular piece of the blade, which disappears in the draft tube.
Because blade cracking does not give fatal damage the acceptance criteria of material defects will only briefly be described in this discussion on safety.
The acceptance criteria for defects that may lead to fracture are determined by fracture mechanics theory based on Paris' Law, i.e. crack growth versus the stress intensity factor. The number of load cycles will be the blade passing frequency, i.e. the number of guide vanes multiplied by the speed multiplied by the time of operation. This means that if the threshold value of the stress intensity variation factor is lower than the threshold value, there is no crack growth. The equation for the stress intensity factor variation yields:
where
(7)
The threshold
value
is found from material tests that
gives following values found for
13 % Cr 4 % Ni steel /Ref 3/
and
We find as an
example the acceptable crack depth (a) of a semielliptic crack with length (2a)
that will not develop into a fatigue fracture during lifetime of the turbine if
we assume R = 0.5,
,
and stress variation
, Then the depth will be:
and the length: 2a = 4 mm.
This means that all linear indications
of surface defects longer than 4 mm should be removed in order to avoid cracks
if the stress amplitudes are 40 MPa from peak to peak and a
2 mm. For higher local stresses the
critical size will be smaller.
The pressure-loaded parts with the highest stresses are the stay ring and the spiral case shell.
During fabrication, the criteria for acceptable welding defects and material defects are established based on fracture mechanics theory for static or low frequency loads.
The load variation from depressurizing and pressurizing the spiral case in turbines occurs during start and stop if the valve or gate upstream of the turbine is closed during stop.
For a peak load turbine, three depressurizing-pressuring cycles may occur daily i.e. in extreme cases 1000 times each year. For such machines the acceptable defects in new turbines may grow to a size of an unstable crack that may cause unstable rupture during lifetime of the turbine. Because of this, the critical size of an unstable crack must be determined by fracture mechanics theory.
For smaller turbines, the unstable critical crack size should normally be large enough to penetrate the plates so that the crack can be detected by leakage indication before a critical unstable rupture occurs i.e. LEAKAGE BEFORE RUPTURE.
For large turbines, like the turbines for Three Gorges Power Plant, the plate thickness will be too large to fulfil this requirement.
Here, periodic inspections of the turbines should be done after f.ex. 5000 pressurizing cycles, in order to examine critical locations with high stresses where the critical cracks are smallest.
Based on the crack tip opening displacement (CTOD) test the critical crack sizes can be found for different materials. Then a design curve can be found showing a crack size with a given safety level against unstable fracture.
Larger cracks than those given by the design curve must be repaired.
This design curve is shown in
Fig.7(a), based on a work by M.G. Dawes /Ref 4/. The crack size in Fig.7(b) is a
through plate crack with length
and a surface crack found by
ultrasonic examination can be determined by using the diagram shown in Fig.7(b).
Note that the stress on the horizontal
axis is
for heat treated welds and
for not-heat treated welds. In the
diagram in Fig. 8, the yield point
, the elasticity modules E and the CTOD values
are all material constants. As an
example we may assume
,
and E = 2.1
. A typical stress level may be
and then
or
for a stress relieved weld.
(a) (b)
Fig. 7 Design curves for the determination of acceptable or non-acceptable crack sizes used as guidelines for inspection left and conversion to semi elliptic defects right.
The basic design of turbines should be based on the governing parameters from classical equations. By means of this approach, pressure-balanced runners should be shaped prior to a final CFD analysis. However, to achieve a slight improvement by means of a CFD analysis, this requires a geometric design based on existing runners. The safety of a turbine must be based on inspections guided by what are acceptable defects as calculated by fracture mechanics theory.
References
[1] Henry, Pierre: Turbomachinery hydrauliques. Presses Polytechnique et Universitaires Romandes, 1992.
[2] Sottas, G. and Ryhming, I.L.: 3D-computation of Incompressible Internal Flows.
Vol. 39, W. Langelüddecke, Brannschweig ISSN 0179-9614, 1993 Report of GAMM workshop.
[3] Grein, H. et al.: Inspection periods of Pelton Runners.
Proceedings 12th IAHR Symposium on Hydraulic Machinery, Sterling 1984.
[4] Dawes, M.G.: The CTOD Design Curve Approach Limitation, Finite Size and Application, WI report 278/1985.