Mathematical model of hydraulic transients in HE Rijeka.

Skific, J. ; Druzeta, S. ; Crnjaric-Zic, N. 等

Abstract: Hydraulic transients can be a harrowing experience to the operator of a hydraulic system. Accordingly, the dynamic response of an automatic or regulated fluid conveyance can be the key to a successful complex engineering scheme. The classical mathematical model describing hydraulic transients is the Allievi model. Furthermore, since hydraulic transients can be described by the hyperbolic conservation laws, the implementation of high order accurate numerical scheme arises. Modeling of hydraulic transients in hydroelectric power plant Rijeka using high order finite volume WENO scheme is presented using software STRAN developed by the authors.

Key words: Mathematical model, hydraulic transients, numerical schemes for hyperbolic conservation laws, hydroelectric powerplant

1. INTRODUCTION

Hydraulic transients can cause high pressures, which can be dangerous for the piping systems, especially in hydroelectric powerplants. Since hydraulic transients can be described by the hyperbolic conservation laws, it would be desirable to apply numerical schemes that are high order accurate. The numerical schemes satisfying the stated requirements are finite volume WENO schemes. In this paper we present efficient implementation of finite volume WENO scheme along with succesfull application to the modelling of hydraulic transients in HE Rijeka.

2. MATHEMATICAL MODEL OF HYDRAULIC TRANSIENTS

Allievi's liquid pipe flow mathematical model is derived from mass and momentum conservation laws (Streeter, 1983), (Chaudhry, 1987). However, the model does not consider cavitation and liquid density variations as a consequence of pressure variations. Therefore it can be written as

[partial derivative]p / [partial derivative]t + [partial derivative] / [partial derivative]x ([c.sup.2][rho] / A Q) = 0 (1)

[partial derivative]Q / [partial derivative]t + [partial derivatibe] / [partial derivative] x (A / [rho] p) = -Ag dz/dx - [lambda]Q [absolute value of Q] / 2DA (2)

where Q is pipe volume discharge, p is pipe pressure, A is pipe cross-section area, D is pipe diameter, [rho] is liquid density, c is speed of sound, [lambda] is friction coefficient.

Speed of sound c can be determined by the relation

[c.sup.2] = K / [rho] / 1 (K / A)([DELTA]A / [DELTA]p) (3)

where [DELTA]p is pressure change, [DELTA]A cross-section area change, [DELTA][rho] fluid density change caused by hydraulic transients and K bulk modulus.

3. FINITE VOLUME WENO SCHEME

Since the considered one-dimensional conservation law described in the previous section is of the form

[partial derivative]u / [partial derivative]t = - [partial derivative]f(u, x) / [partial derivative]x + g(u, x) (4)

we look for the solution u(x,t). The general structure of corresponding conservative scheme for solving (4) is

d[[bar.u].sub.i] (t) / dt = - 1 / [DELTA][x.sub.i] ([[??].sub.i+1/2] - [[??].sub.i-1/2]) + 1 / [DELTA][x.sub.i] [G.sub.i] (5)

Here [u.sub.i](t) denotes the approximation to the average value of the solution over the cell [I.sub.i], [[??].sub.i+1] is the numerical approximation to the value f(u([x.sub.i+1],t),[x.sub.i+1]), while Gi approximates the term [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The left side of the above equation, i.e. the time part, is solved by using the TVD Runge-Kutta time integration. The approximations of the terms on the right side of (5) are based on WENO reconstruction (Shu, 1998) and on the solution on the generalized Riemann problem as described in the proceeding. Namely, since at each time step the average values [[bar.u].sub.i](t) are known, by using the WENO reconstructions, the values [u.sup.-.sub.i+1/2] and [u.sup.+.sub.i+1/2], which approximate the exact value of the solution at the considered cell boundary can be determined. Then the corresponding Riemman problem on the (i+1/2)th cell boundary should be solved. Since the exact values of such solution are not always available or inexpensive, the approximate Riemann solvers are used, i.e.

[[??].sub.i+1/2] = F([u.sup.-.sub.i+1/2], [u.sup.+.sub.i+1/2]) (6)

where F is the numerical flux function consistent with the physical flux. In this work the Roe and Lax-Friedrichs fluxes are used (Shu, 1998). Furthermore, for approximating the geometrical part of the source term integral [G.sub.i], the decomposed approach is applied, such that the exact conservation property is achieved(Bermudez & Vazquez, 1994), (Vukovi' et al. 2002), (Crnjaric-Zic et al., 2004).

4. NUMERICAL SIMULATION OF HYDRAULIC TRANSIENTS IN HE RIJEKA

The task of the research was to calibrate the parameters of the mathematical model of the hydroelectric powerplant Rijeka (Figure 1.) according to the measurements conducted by Brodarski Institut for operating regime of 0-75-0% load. Numerical simulations were conducted with [C.sub.cfl] coefficient 0.65.

[FIGURE 1 OMITTED]

The model consists of a reservoir, i.e. lake Vali'i, headrace tunnel, surge chamber, penstock and turbine. The reservoir was modelled as an inflow boundary condition of a known pressure, The turbine was modelled as an outflow boundary condition of the known discharge presented in Figure 2. Headrace tunnel is modelled as pipe element 3117 m long and 3,2 m in diameter. Surge tank was modelled according to Chaudhry, 1987, with geometrical properties described as in Figure 3. Penstock was modelled as pipe element 786 m long and 2,3-2,2 m in diameter.

[FIGURE 2 OMITTED]

In order to calibrate the model, the water elevation at the surge tank was measured. The calibration parameters were pipe roughness coefficients, and surge chamber head loss coefficient. The results presented in Figure 4. show a good agreement of the numerical results with the measured data.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

5. CONCLUSION

Implementation of finite volume WENO scheme to the hydraulic transients modeling was proven to be justified. Conducted numerical simulations of hydraulic transients in HE Rijeka have shown that the simulated results are in good agreement with measured data.

6. REFERENCES:

Bermudez, A. & Vazquez, M. E. Upwind methods for hyperbolic conservation laws with source terms, Comput. Fluids 23(8), 1049 (1994).

Chaudhry, M. H. Applied hydraulic transients, Van Nostrand Reinhold, New York, 1987.

Crnjaric-Zic, N.; Vukovic, S. & Sopta, L. Balanced finite volume and central WENO schemes for the shallow water and the open-channel flow equations, J. Comput. Phys. 200, 512 (2004).

Shu, C.-W. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, edited by B. Cockburn, C. Johnson, C.-W. Shu and E. Tadmor, Lect. Notes in Math. (Springer-Verlag, Berlin/New York) 160, 325 (1998).

Streeter, V. L. & Wylie, E. B. Fluid transients, FEB Press, Ann Arbor, Mich., 1983.

Vukovi', S. & Sopta, L. ENO and WENO schemes with the exact conservation property for one-dimensional shallow water equations, J. Comput. Phys. 179, 593 (2002).