Analysis of the Swirling Flow at GAMM Francis Runner Outletfor Different Values of The

Analysis of the swirling flow at GAMM Francis runner outletfor different values of the discharge

Tiberiu CIOCAN*, Romeo F. SUSAN-RESIGA*, Sebastian MUNTEAN**

Corresponding author

*Department of Hydraulic Machinery, POLITEHNICA University of Timisoara

Bd. Mihai Viteazu 1, 300222, Timisoara, Romania

,

**Centre for Advanced Research in Engineering Sciences, Romanian Academy – Timisoara Branch

Bd. Mihai Viteazu 24, 300223, Timisoara, Romania

Abstract: Nowadays an important, and still open, issue in Francis runner design is the blade geometry at the trailing edge, wich directly determines the swirling flow at the runner outlet and further downstream the pressure recovery and hydraulic losses in the turbine draft tube. In this paper we present and validate a novel mathematical model for computing the radial profiles of both axial and circumferential velocity components, respectively, at the runner outlet for Francis hydraulic turbines within the full operating range. We apply this methodology of computing the swirling flow downstream the runner, over the GAMM Francis turbine model, where experimental data were available. We present the computation procedure for the dimensionless discharge and dimensionless flux of moment of momentum, respectively. We introduce the swirl-free velocity profile in order to expres the relationship between the axial and circumferential velocity components. This hypothetical velocity, the swirl-free velocity, is subject of optimization in order to achieve the best performance of the draft tube over the whole intended operating range.

Key Words: swirling flow, Francis turbine, flux of moment of momentum, swirl-free velocity

1. INTRODUCTION

Nowadays, modern hydraulic turbines meet new challenges associated with variable demand on the energy market and limited possibilities for energy storage, resulting in the need for flexibility in their operation. This need for flexibility in the operation of a turbine, means that there is often a tendency (required) the use of turbines for a wide range of operating regimes, far from the operating point for which have been designed. Hydraulic turbines, especially radial-axial turbines (e.g. GAMM Francis turbine [1]) having a rotor with fixed pitch blades, have a sharp drop in efficiency and severe pressure fluctuations at off-design operating regimes. When operating at part-load, in the turbine draft tube, it appears a high-level of residual swirl, as a result between the mismatch of the swirl configuration at the runner inlet imposed by the guide vanes, and the angular moment extracted by the turbine runner [2]. The helical vortex (which itself is part of the flow rotation) appears due to the decelerated flow rotation and it is the source of flow unsteadiness with associatedpressure fluctuations, that can lead in breakage of the runner blades.

Evaluation of the turbine efficiency for the full turbine operating range, depending on the discharge coefficient and energy coefficient with respect to the IEC regulations [3], is still the standard experimental investigation on model turbines in order to predict the performance of the real size machine, and the resulting efficiency hill-chart usually displays a peek efficiency, wich is called the “best efficiency point” (BEP). Due to technological advances in recent decades, numerical simulation can be made ​​on all components of a Francis turbine, which allow a high accuracy analysis of the spiral, the guide vanes, the rotor and draft tube, respectively [4].

For this reason in the last 20 years, it appeared numerous programs dedicated to investigating the hydraulic losses wichoccurs in thedraft tube, such as:

● GAMM Workshop – „3D - Computation of Incompressible Internal Flows” [5]

● Turbine – 99 Workshop on draft tube [7]

● FLINDT Project – Flow investigation in draft tube [8]

Better understanding of the particularities of the swirling flow at the runner outlet / draft tube inlet, is the key in developing new techniques for controlling the swirling flow downstream the Francis turbine runner. A very good example is the water injection method from the runner crown along the axis machine, in order to eliminate the helical vortex and the associated pressure fluctuations, developed by (Susan-Resiga, et al. 2006) [9]. An important, and still open, issue in Francis runner design is the blade geometry at the trailing edge, wich directly determines the swirling flow at the runner outlet and further downstream the pressure recovery and hydraulic losses in the turbine draft tube. As a result, is essential to correctly estimate the hydraulic losses and the pressure recovery in the turbine draft tube in the early design stages, if possible even before acctualy designing and computing the runner. This is possible only if one has the capability of predicting the swirling flow at runner outlet within a wide range of operating regimes, in order to establish the best swirling flow configuration at runner outlet at the design operating regime.

Susan-Resiga, et al. [10], developed a mathematical model that enables calculation of the swirling flow at the runner outlet for the whole operating range of a Francis turbine runner and showed that this model is able to capture the main features of the swirling flow. In this paper i will present the methodology of computing the swirling downstream the Francis turbine runner for variable operating regimes,in order to validate the theory. In this theory the dimensionless discharge φcalc, dimensionless flux of moment of momentum m2 and the swirl-free velocity vsf, respectively, is computed.

2. EXPERIMENTAL DATA

The experimental data further used in this paper corresponds to three operating regimes of the GAMM Francis turbine model [5], wich is a turbine with medium specific speed ν = 0.5 and it is characterized by the following main parameters:

• Q = 0.376 m3/s;
• H = 5.957m;
• ω = 52.36 rad/s.

The turbine geometry, in meridian plane is given in Figure 1, with the reference runner radius of R2= 200 mm.

The hill-diagram of the the turbine model is presented in figure 2, represented by the discharge coefficient φref and energy coefficient ψref, respectively, with respect to the IEC regulations [3].

From the hill diagram it can be seen that operating point 4 at minimal discharge, operating point 1 wich is the best efficiency point and operating point 5 at maximal discharge, respectively, are at constant head.

In table 1 is given the corresponding dimensionless discharge of the three points at constant head.

Table 1 – Dimensional discharge and head

The experimental available data wich i will use are in order to compute the swirling flow, are the velocity profiles downstream the runner, from the measurement section S2(see Fig. 1), wich has a measurement radius of Rw= 218.38 mm. Further i will present the necessary steps in order to compute the swirling flow at the runner outlet.

3. COMPUTATION METHOD

3.1 Dimensionless velocity profiles

The dimensional velocity profiles used are from survey section S2, for three operating points at constant head (see Fig. 2). The survey section has a corresponding dimensionless radius of rw= 1.092 and it is made dimensionless by the reference radius, as follows: rw= Rw / R2. All the measurement radius are made dimensionless by the reference radius and all the experimental velocity components are made dimensionless with respect to the reference velocity Vref= ωR2 = 10.472 m/s.

In figures 3, 4 and 5 it is represented the dimensionless velocity profiles, from the minimum to maximum discharge, respectively, at the runner outlet, for the three operating points at constant head. In figure 3 at part load operating regime, the axial velocity it is decreasing until it develops a stagnation region with helical vortex breakdown and at full

load operating point the axial velocity has an excess near to the axis, or with other words we

have an axial jet.

A very important fact is that the following analysis is performed only for the dimensionless

velocity profiles.

3.2. Computation of the discharge

The discharge can be computed by integrating numerically the axial velocity profile over the survey section. By introducing the dimensionless velocity v and dimensionless radius r, respectively, the dimensionless discharge can be computed by using the simplest integrating approach, as follows:

Trapezoidal rule:

where i = 0…Ndata ,with i =0 coresponding to the axis and i =Ndata to the wall, respectively.

The computed dimensionless discharge is given in table 2, for the three operating points at constant head, along with the errors between the dimensionless discharge from the velocity measurementsφcalcand the corresponding value from the measured overall discharge φexp.

Table 2– Computed dimensionless discharge

With the computed dimensionless discharge, it can be computed the discharge fraction (normalized streamfunction),wich is the fraction of the volumetric flowrate discharged through a disc of radius r ≤ rw. As a result, the discharge fraction can be computed for the experimental data, from equation (1), as follows:

where i = 1…Ndata.

3.3 Swirl-free velocity profiles

The swirl-free velocity vsf is used in order to describe the flow kinematics, instead of the relative flow angle β2, at runner outlet, because it remains unchanged at different operating regimes. Introducing the dimensionless radius and dimensionless velocity, it can be written in dimensionless form the swirl-free velocity, as follows [10]:

In order to have an average swirl-free velocity independent of the operating regime, we have to express the swirl-free velocity versus the discharge fraction.

From figure 6 it can be seen that the swirl-free velocity profile collapse on the same discharge fraction (with blue lines), but this was expected since the swirl-free velocity is related with the runner geometry at the trailing edge, wich is not changing with the operating point for fixed pitch blades as in Francis turbines.

In figure 7 it can be observed three different regions. The first 2 regions with blue lines, corresponds to the runner crown wake and wall boundary layer, and it is a result of the viscous effects developed by the runner crown surface and wall surface, respectively.

The data of these two regions, shown with hollow circles are discarded since the mathematical model is an inviscid flow model. The third region corresponds to the main swirling flow, shown with filled circles. The data shown with filled black circles from figure 7 are fitted with a linear fit, having the equation:

where vsf_ct is the constant value of the swirl-free velocity and δvsf is the slope of the swirl-free velocity profile. These two coefficients are going to be the optimization parameters for the swirling flow at the runner outlet in order to obtain the best performance of the draft tube within a given operating range.

It can be seen that the experimental data from the third region are reasonably clustered around the constant fit (red line).

3.4 Computation of the flux of moment of momentum

From the fundamental equation of turbomachines written for hydraulic turbines:

, we have the hydraulic power written as the product of the mass flow rate and the specific energy multiplied by the hydraulic efficiency, and in the right side of the equation we have the rate at wich the fluid does work on the runner, given by the difference between the flux of moment of momentum upstream the runner and downstream the runner, respectively. Because in this paper we are investigating the swirling flow at the runner outlet it is necessary to compute the flux of moment of momentum downstream the runner. By introducing the dimensionless radius and dimensionless velocity, respectively, the flux of moment of momentum downstream the runner can be written in dimensionless form, as follows:

The computed dimensionless flux of moment of momentum with equation (6), is given in table 3.

Table 3– Dimensionless flux of moment of moment of momentum

In figure 8 it is represented the moment of momentum, with filled circles, versus the discharge fraction, fitted with cubic fits, for all three operating regimes at constant head. If we multiply the average values of the dimensionless moment of momentum, given in figure 8, with the corresponding dimensionless discharges computed with Eq. (1) (table 2), as follows: , we obtain the dimensionless flux of moment of momentum, with the corresponding values from table 3. This is done mainly to verify if the dimensionless flux of moment of momentum obtained with equation (6) is correctly computed. If we compare the values from table 3, obtained with the two approaches, it can be seen that the values are in a very good agreement, especially at full-load.

4. COMPUTED SWIRLING FLOW AND RESULTS

With the computed integral quantities that characterized the swirling flow, the dimensionless discharge φcalc, and dimensionless flux of moment of momentum m2, respectively, and in addition the swirling flow satisfies the kinematic constraints corresponding to the runner blade at trailing edge described by the swirl-free velocity, we can compute the axial velocity by solving the variational problem[10]:

Given the swirl-free velocity, wich in this case we chose a constant swirl-free velocity of vsf_ct = 0.278, and by minimazing the functional (7), to the integral constraints (8), (9):

, we can compute the axial velocity vz at the runner outlet, and with the axial velocity computed, we can compute the circumferential velocity vθ, as follows:

The numerical results for the the three operating regimes at constant head investigated in this paper are represented in figure 9, togheter with the experimental data.

5. DISCUSSION

In the above figures it is represented the computed axial velocity and circumferential velocity, computed with φcalc, m2 at a constant swirl-free velocity of vsf_ct= 0.278.

Figure 9 shows that the mathematical model correctly captures the swirling flow evolution at different discharges. At part load P4, Q > QBEP, one can see that the velocity is gradually decreased until a stagnant region develops in the axis neighborhood and the flow co-rotates with respect to the runner. On the other hand at full load P5, Q < QBEP, we have an axial velocity excess near the axis and counter-rotating flow with respect to the runner rotation.

The agreement with the experimental data are quite good except the wake of the crown, but the lack of correlation is expected since the model does not account for the viscous effects which lead to the crown wake. One can see from figure 9a that at low discharge, the flow model correctly captures the stagnant region extent.

Figure 10 shows the maps of axial and circumferential velocity components versus the radial coordinate and discharge coefficient, at the reference section S2 (Fig. 1).

From figure 10 we have an overall perspective on the swirl evolution with the turbine discharge. In figure 10a, it can be easily observe the development of the central stagnant region (lower-left corner) when the discharge decreases.Figure 10b shows the development of a counter-rotation when the discharge is increased and at lower values of the discharge (underφcalc = 0.308), the whole flow rotates in the same direction with the runner. The boundary of this stagnat region is an unstable vortex sheet which evolves in the precessing helical vortex known as vortex rope [11].

6. CONCLUSION AND PERSPECTIVES

This paper presents in detail the necessary steps required in order to compute the swirling flow downstream the Francis turbine runner. It is shown that for computing the swirling flow at the runner outlet we need to compute the dimensionless discharge φcalc, dimensionless flux of moment of momentum m2and the swirl-free velocity profile vsf. This are the only three quantities required for computing the swirling flow (i.e. axial velocity and circumferential velocity) at the runner outlet. The computed velocity profiles are in a very good agreement with the experimental data, excepting the runner crown wake region, because the viscous effects are not captured by the inviscid flow model, this is a limitation of the mathematical model.In the near future, the next step is the computation of the swirling flow at the draft tube inlet in order to evaluate the draft tube hydrodynamics, followed by the optimization of the swirling flow at the draft tube inlet, by optimizing the swirl-free velocity in order to achieve the best draft tube performance that will improve the turbine efficiency.The final step is to actually design the turbine runner, through an inverse design method [12], such that we have at the runner outlet the prescribed swirling flow.