On the geometry of the penstock lower bend for large flow Francis turbines.

Vertan, Gheorghe ; Dobanda, Eugen ; Manea, Adriana Sida 等

1. MATHEMATICAL ELEMENTS

From mathematical point of view, any bended sheet of metal is a deployment surface (Barglazan M., 1999).

Such a shell--considered as a ruled surface--is obtained by shifting in space a straight line. Some ruled surfaces are not deployable.

To be deployable, a surface must accomplish a certain condition (Sundar Varada Raj P., 1995, Sriro 1961).

So, if in a xOyz reference frame a deployable surface has the equation (1).

z = f(x,y), (1)

then this surface in deployable if satisfied the differential equation (2)

For the case of the spiral casing of certain hydraulic machines, this equation was considered from long ago, and was used including to patented technologies, and, also, the problem was studied (Vertan Gh. et al., Man E.T. et al., 2010).

[FIGURE 1 OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

2. THE DETERMINATION OF THE CURRENT

POINT COORDINATED ON THE SPATIAL CONTOUR OF THE SHELLS

For a current ruled surface for a penstock lower bend with oval sections [A.sub.1][A.sub.n][B.sub.n][B.sub.1] in a xOyz reference frame (figure 1 "a").

The end of the shell placed in xOy plane is defined by the known values [l.sub.1] and [R.sub.j] (figure 1 "b"). The other end is placed in [x.sub.1]Oy plane, which has the angle [delta] with xOy plane.

This second end is defined by the values [l.sub.2] and [R.sub.2] (figure 1 "c").

In the plane xOy, a current point from the contour [A.sub.i] is defined through the angle [alpha] and has the coordinates:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

At the same shell, in [x.sub.1]Oy plane, a current point [B.sub.i] from the contour is determined by the angle [beta], which, in the plane [x.sub.1]Oy has the abscissa

[X'.sub.Bi] = [l.sub.2] + [R.sub.2] x COS([beta]) (4)

and in the system xOyz has the coordinates

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

3. THE DETERMINATION OF THE CURRENT GENERATOR OF THE SURFACE

For any shell of the penstock lower bend the vital problem consist in determining the point [B.sub.i], as function of the current point [A.sub.i], in such a matter that the straight line [A.sub.i][B.sub.i] represents the current generator of the surface supported by the curves [A.sub.1][A.sub.i][A.sub.n] and [B.sub.1][B.sub.i][B.sub.n].

Essentially, the problem consist in determination the angle [beta] as function of angle [alpha] and the rest geometrical elements, such as: [l.sub.1], [R.sub.1], [delta], [l.sub.2], [R.sub.2]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

4. DETERMINATION OF THE CONTOUR OF THE SHELL AND THE BENDINGS LINES

In this purpose there is used the mathematical notion of geodetic curve, whose length is, generally, determined by an integral equation.

The shell surface is decomposed in n quadrilateral surfaces, going from an initial generator [A.sub.1][B.sub.1] to the final one [A.sub.n][B.sub.n].

The number "n" is determined to be convenable from technological point of view, to result finally "n" bending lines, uniform distributed on the whole surface of the shell.

Then, each quadrilateral is decomposed in two triangles.

Ones are the triangles [A.sub.i]-1[A.sub.i][B.sub.i]-1 and [B.sub.i]- 1[B.sub.i][A.sub.i], (figure 1 "a") in the first hypothesis.

In the second hypothesis, the considered triangles are [A.sub.i] [sub.1][B.sub.i]-[sub.1][B.sub.i] and [A.sub.i]-1[A.sub.i][B.sub.i] (figure 1 "a").

To find the length of the sides of this triangles means that the wholw surface of the shell is decomposed in triangles with known sides.

Going from the initial generator, each triangle is developed, obtaining both the developing of the shell and the corresponding positions of the bending lines.

Each of the four triangles, considered on the surface of the spatial shell (figure 1 "a"), is helded in plan and has two sides exactly calculable.

One of this is the generator [A.sub.i-1][B.sub.i-1] or [A.sub.i][B.sub.i], and the second is the arc curve [A.sub.i-1][A.sub.i] or [B.sub.i-1][B.sub.i]. And every of this four triangles has a third side ([A.sub.i-1][B.sub.i] or [B.sub.i-1][A.sub.i]), which, on the curve surface of the shell is a geodetic curve.

Considering that all the quadrilaters which compose a particular shell are decomposable in two triangles, following the procedure presented above, results that all points [A.sub.i] and [B.sub.i] (i = 2, n), exception [A.sub.1] and [B.sub.1], placed on the initial generator, has the coordinate approximate.

The approximation is higher for the points [A.sub.n] and [B.sub.n], which has maximum errors, the coordinates values being higher than the real ones.

In order to obtain acceptable results, the shell decompose in [n.sub.c] > n quadrilaters, growing up their number, [n.sub.c], until the difference between the coordinates calculated through both hypothesis are less than the accepted tolerance.

From all this points [A.sub.i] and [B.sub.i] will be withhelded a "n" number uniform distributed on the contour of the shell; the number "n" will be chosen to be convenient from the technologic point of view.

5. CONCLUSIONS

The absolute value of max([absolute value of [alpha]-[beta]]) differs form one shell to another. In industrial cases, this difference is about 1[degrees] to 5[degrees].

If is considered [alpha] = [beta], then results small differences from the correct variant, but the bending lines induces technological difficulties, the shells differing from the correct form.

The method based on the exposed brieffly calculus offer the coordinates of the points representing the contour of each particular shell and, also, the bending lines; this allowed realizing the shells at the site of the hydroelectric power plant, which reduce considerably the costs.

Determination, in the limits of accepted tolerance, of the coordinates of the points that represent the contour of each shell and, also, the ends of the bending lines, ensure the correct cutting off and the bending of each shell.

This particularity ensure the technological procces of realizing the penstocks lower bends in the site of the hydroelectric power plant.

6. ACKNOWLEDGEMENTS

This paper is supported by CNCSIS--UEFISCSU, project number 679/2009 PNTI--IDEI code 929/2008, Transmisii hidrodinamice inteligente (Intelligent Hydrodynamic Transmitions), and Grant CNMP 21047/1467/2007.

7. REFERENCES

Barglazan M. (1999). Hydraulic turbines and hydrodynamic transmitions, Ed. POLITEHNICA, ISBN: 973-9389-39-2, Timisoara

Man E. T., Vertan Gh., Balan G., (2010), The 6th Conference "Dorin Pavel", Bucuresti, 2010, May, 27 - 28, on CD. Technology implementation at the site of hydroelectric power plant for the spiral chambers and suction elbows with oval section of large hydraulic turbines, pg. 329. ISSN 2068-2778

Sundar Varada Raj P. (1995). Trans. of the ASME "Journal of Engineering for Industry", vol. 117, pag. 117 - 185 http://journalseek.net/cgi- bin/journalseek/journalsearch.cgi?field=issn&query=0022-0817. Evolution of Generic Mathematical Models and Algorithms for the Surface Development and Manufacture of Complex Ducts. 1995may,

Sriro I.I. (1961). Ghidroturbinostroienie, nr. 8, Masghiz. About deploying units welded spirals, Moskva

Vertan Gh., Secosan A., Lupsiasca C., Mihail E., Vertan M., Sepici G., Bon^ea M. Patent nr. 69500, Romania Assembly process for making bodies of the outline given in space Viro