Low-frequency oscillations in hydroturbines caused by cavitation together with the phase transitions effects

Among the unsteady phenomena taking place in hydroturbines one can separate oscillations caused by a cavitation bubble behind the runner. Usually a concept of cavitation compliance is used to model the cavity pulsations (Chen et al.). There was demonstrated the destabilizing effect of the diffuser and swirl. Kuibin et al. had found that the effect of swirl on stability is much smaller than that described by Chen et al. In the present work, the same analysis is repeated, but taking into account influence of the evaporation and condensation on dynamics of the cavity appearing in the draft tube. During analysis a simplified spherical cavity form was considered.


Introduction
The hydroturbines during their operation undergo unsteady phenomena of electrical, mechanical and hydrodynamical nature. The description of hydrodynamical phenomena is complicated by the possibility of existence of two-phase areas due to cavitation or involvement of air in the water flow. The reviews on approaches to solve the unsteady problems in hydro power stations can be found for example, in PhD thesis by Nicolet [1] or in monograph by Dörfler et al. [2]. One of interesting and simple enough method for analysis of instability generated due to the presence of the cavitation / air area behind the turbine runner was developed by Chen et al. [3]. The authors used one-dimensional model together with the concept of cavitation compliance and revealed the destabilizing effect of the draft tube diffuser and swirl. The same authors [4] considered water as compressible medium and found generation of highfrequency unstable modes in the penstock. Chen et al. [3,4] had shown that one can control the flow instability as well as its eigen frequencies by changing the parameters of the diffuser or the swirl intensity, or the volume of the cavity. Kuibin et al., [5], Kuibin and Zakharov [6] tried to develop onedimensional approach with help of more correct usage of the Bernoulli's equation and application of different vortex models for evaluation of the swirl effect. They demonstrated much weaker influence of swirl on the stability conditions. Moreover, this influence decreases with gas/vapor cavity growth.
When considering the cavitation phenomenon, a reasonable question arises, how the effects of phase transition, evaporation and condensation influence on the instability parameters. Just this question is the main goal of the present research. For analysis we take the main equations derived by Chen et al. [3] and introduce new terms responsible for the phase transition effect. For simplicity a spherical cavity form was considered.

Analytical model
Consider a system modelling a hydro power station water passage consisting of a penstock of length Li with cross-sectional area Ai, a turbine runner (TR) and a draft tube (DT) with the area of the inlet crosssection Ac and the exit cross-sections Ae (see figure 1). As a model of cavitational area we put volume Vc behind the turbine. Two main equations were derived by Chen et al. [3] for description of the flow rate and pressure. When there exist the cavitational bubble, the flow rate in the penstock Q1 differs from the flow rate in the draft tube Here  is the liquid density; Le = ∫(Ae A(s))ds is the DT effective length; D = (Ae Ac) 2 − 1 is the diffusor factor, 2 is the DT loss factor (2 is assumed to be constant). All terms of the right-hand part of equation (1) contain multiplier C = -∂Vc/∂pc, the cavitation compliance. The last term in equation (1) reflects the effect of flow swirl behind the runner. The quantity in brackets is the characteristic circumferential flow velocity at the outlet from TR U2 is the peripheral velocity at the runner exit, 2 is the angle of inclination of the blade at the exit from the TR, S is the runner exit area.  is the pressure coefficient responsible for the swirl effect, it can be represented through difference between the ambient pressure pa in the zone with cavitational bubble and pressure inside the bubble pc The second main equation from [3] links the pressure at the system inlet, pi, with the ambient pressure, pa, and with the exit pressure, pe At a constant rotation speed of runner and fixed angle of opening of the guide apparatus, the turbine can be considered as a resistance with a constant loss coefficient T, which depends on the opening of the guide device. Assume that cavitational bubble has a spherical form of radius Rc. The equation for the rate of bubble radius changing due to phase transition reads [7,8]   Here ql is the density of heat flux from the bubble into liquid, c is the vapor density,  is the latent heat of vapor generation,  is the vapor adiabatic exponent. Equation (1) was derived in [3] through the time derivative of the cavity pressure Q2 -Q1 = dVc/dt = -C dpc/dt. Thus, to take into account the phase transitions effects we find additional rate of the cavity pressure change in time and add new term to the right-hand side of equation (1) 3 In view of the expression for the bubble volume, Vc = 4  Rc 3 /3, we obtain the modified continuity equation It is necessary to determine also the density of heat flux from the bubble into liquid ql. It will follow from the heat transfer equation on the liquid temperature Tl The coefficient al is the thermal diffusivity of the liquid. It is evident that temperature in cavity equals the saturation temperature. Equations (3), (5), (6) allow to fulfill the stability analysis for Q1(t) and Q2(t). For linearization we assume This equation can be rewritten as a second-order differential equation The constants A and B should be found from the boundary condition. We obtain  R dp dV C R dp The dependence pc(Ts) is known from the Handbook of Physical Properties of Liquids and Gases [9] which usually is approximated by some analytical function, for example  The complex frequency consists of two parts:  = R + iI. The real part, R, is the frequency, and the imaginary part, I, is the damping rate (decrement) of the perturbation. When I is positive, one has stable disturbances and controversy, at I < 0 the amplitude of disturbances will grow.
Unlike the paper by Chen et al. Unfortunately, the model constructed obeys some lacks. First of all, the attention should be payed on the bubble radius (or volume) in denominators in number of coefficients. So, we have limitation on consideration of large enough size of bubbles. Nonetheless we try to analyze the effect of phase transitions on the disturbances development.

Results of calculations
Some examples demonstrating influence of the phase transition effects on the stability characteristics are presented in figure 2. For the base we took stability analysis made by Kuibin [6]), namely at vortex core size equal  = 0.5 R (R 2 = Ac), cavity radius rc = 0.5  and rc = 0.9 . As seen, at small flow rate the frequencies of unstable disturbances are close to ones found in [6]. The range of flow rates, when the oscillations are impossible becomes narrower in both cases for small cavity ([0.591 … 0.744] instead of [0.552 … 0.712]) and bigger one ([0.572 … 0.700] instead of [0.520 … 0.778]). For the first case this interval is shifted to area of higher flow rates. When the phase transition takes place the perturbances become more stable at small flow rates. At Q > 0. 4

Conclusion
In the paper a model is constructed for description of the phase transition influence on the low-frequency oscillations arising in flow in hydro turbine in presence of the cavitational bubble size behind the turbine runner. It was found that effect is strong enough on the perturbances increments or decrements. In the same time influence on the frequencies is relatively weak. The model needs in future development.