Double-flux water turbine dynamics.

Stroita, Daniel Catalin ; Barglazan, Mircea ; Manea, Adriana Sida 等

1. INTRODUCTION

The Double Flux turbine belongs to action type water turbines and it was discovered by Antony Michell in 1903. Professor Donat Banki introduced the evelventic channel for water admission improving the running of the turbine.

[FIGURE 1 OMITTED]

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The double-flux name comes from the double passing of the water in the turbine runner like this the kinetic energy from the water is absolved by the runner twice, we can see this from the velocity triangle in the figure 1.

Further we will present the most important equations of this kind of turbine.

P = [rho][Qu.sub.1]([v.sub.1] cos [[alpha].sub.1] - [v.sub.4] cos [[alpha].sub.4]) (1)

Equation (1) represents the hydraulic power of the turbine, this equation comes from Euler equation adapted for the double-flux turbine. Turbine's efficiency is presented in equation 2, and is estimated like a ratio between runner's diameter and turbine's head.

[eta] = 0.863 - 0.264 D/H (2)

Although the efficiency is not very high the great advantage of this kind of micro-hydro turbine is the simplicity of the construction and the low cost. The double flux water turbine covers a great running domain and it can be compared with very popular water turbines like Francis, Kaplan or Pelton.

2. HYDRAULIC TURBINE DYNAMIC EQUATION APPLIED FOR DOUBLE FLUX TURBINE

We will start by presenting the general dynamic equation of a hydraulic turbine (Barglazan, 2001):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

This general equation will be particularized for the double-flux turbine and it will be:

d[omega]/dt = [DELTA][phi]/[[phi].sub.0] x 1/[T.sub.[phi]] + [DELTA][omega]/[[omega].sub.0] x 1/[T.sub.[omega]] = [phi]/[T.sub.[phi]] + [OMEGA]/[T.sub.[omega]] (4)

The general formula is simplified because for the doubleflux turbine we have only one element for adjusting the flowrate, which is the nozzle's vane and the density of the liquid [rho] and the head of the turbine H are considered constant. The characteristic times for this type of turbine are [T.sub.[phi]] and [T.sub.[omega]] the times for the variation of the flow-rate and of rotor's angular speed.

In this new method we will use the classical method (Barglazan, 1979), improved by computer programs (Stroita 2007). We consider two points on the hill chart (1, 2) corresponding to 2 working regimes of the hydraulic turbine. The method is presented step by step and is exemplified for the working regime (point 1), for the other regime (2) the results are only presented, the steps used being similar as for the first.

The formulas for [T.sub.[phi]] and [T.sub.[omega]] are given below:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where J is the inertial momentum of the hydraulic turbine, electric generator rotor and shaft and [Ma.sub.0] is the active momentum.

[M.sub.a0] = [rho] x g x [Q.sub.0] x [H.sub.0]/[[omega].sub.0] (7)

Through similitude relations we can calculate the flow-rate Q and the speed n, using the [Q.sub.11] and [n.sub.11] from the hill chart. Also from the hill chart we can obtain the partial derivatives through a grapho-analytical method.

Q = Q11 x [D.sup.2] x [square root of H] (8)

n = n11 x [square root of H]/D (9)

[omega] = [pi] x n/30 (10)

The hill chart is presented in the figure 2, with the coordinates [Q.sub.11] and [n.sub.11] and with the parameters [phi], which is nozzle's vane opening and [[eta].sub.rel] which is the relative efficiency.

In order to obtain the partial derivatives ([partial derivative]Q/[partial derivative][[phi].sub.0]) and ([partial derivative][eta]/[partial derivative][[phi].sub.0]).sub.0], we extract from the hill chart (Barglazan, 1974) the dependence Q=Q(([phi]) and [eta]=[eta]([phi]) presented in the figures 3 and 4

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

In those graphs the working regime 1 is presented with the big dot. In order to obtain the equation of the variation we added a trendline which corresponds to a polynomial with the order 2, it seems from practice that in this case the polynomial with the order 2 gives the best approximation for the variation. Now, with the equation known we used a MathCad program for obtaining the partial derivate in the specified point. Like this we have the value for the partial derivatives. The same method is used for the other two variations Q=Q([omega]) and [eta]=[eta]([omega]) presented in the figures 5 and 6.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

In the MathCad program is needed to enter the dependence along the line n11=const and along the curve [phi]=const, the parameters of the rotor, the operating regime and it will calculate automatically the characteristic times. For this operating point the results were:

T[phi]=0.0138800 and Tco=0.002421 seconds, and now we can write the dynamic equation of the double-flux water turbine as:

dx/dt = [phi]/0.0138800 + [OMEGA]/0.002421 (11)

As the same is done for the other two working regimes, the graphs and the characteristic times are presented below:

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

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At this operating point the characteristic times are [T.sub.[phi]]=0.0000831 and [T.sub.[omega]]=-0.058970 and the dynamic equation can be write

dx/dt - [phi]/0.0000831 - [OMEGA]/0.058970 (12)

3. CONCLUSIONS

* A presentation of Double Flux water turbine has been made

* It was presented the calculus of the characteristic times for two working regimes (1, 2).

* The presented method is an improved one and it can be applied to all types of water turbine.

* With the help of computer programs the method is more precise that the classic one.

* The dynamic equation can predict the running of the turbine in unsteady regimes

4. FURTHER WORK

As further work in the Phd. Thesis called "Dynamic Identification of Double Flux turbines" we will solve the derivative dynamic equation, identifying the dynamics of this kind of water turbine through experimental results.

AKNOWLEDGEMENT This paper was possible through the CNCSIS Grant IDEI cod 35/nr. 68/01.10.2007, director Prof Dr. Ing. Victor BALASOIU

5. REFERENCES

Barglazan, M. (2001) Hydraulic turbines and hydrodynamic transmissions, Ed. Politehnica, ISBN 973-9389-39-2, 482 p, Timisoara.

Barglazan, M. (1974) Automation control of hydraulic machinery, laboratory works, Institutul Politehnic "Traian Vuia" Timispara.

Barglazan, M. (1979) Automation control of hydraulic systems, Institutul Politehnic "Traian Vuia" Timisoara

Stroita D. C., Barglazan M. (2007) A Method for Calculating the Characteristic Times for Double-Flux Water Turbines, Scientific Bulletin of The "Politehnica" University of Timisoara, Transaction on Mechanic, Tom52(66), Fasc 4.