modelFrancis

Information

Francis Turbine Model

This is the Francis turbine model that gives possibilities for proper modelling of the Francis turbine. The mechanistic model is based on Euler equations for the Francis turbine.

Besides hydraulic input and output, there are input as the control signal for the valve opening and also output as the turbine shaft power and input as angular velocity.

Figure: Key quantities in the Francis turbine model showing inlet (1) and outlet (2) sections.

Euler Turbine Equations

The shaft power produced by the turbine is given by:

$$ \dot{W}_s = \dot{m}\omega \left(R_1\frac{\dot{V}}{A_1}\cot{\alpha_1} - R_2\left(\omega R_2 + \frac{\dot{V}}{A_2}\cot{\beta_2}\right)\right) $$

where ṁ and V̇ are mass and volumetric flow rates, ω is angular velocity, R₁ and R₂ are inlet and outlet radii, A₁ and A₂ are cross-sectional areas, α₁ is inlet guide vane angle, and β₂ is outlet blade angle.

Total Work and Efficiency

The total work rate is:

$$ \dot{W}_t = \dot{W}_s + \dot{W}_{ft} + \Delta p_v \dot{V} $$

where Ẇ_ft represents various friction losses (shock, whirl, wall friction), and Δp_vV̇ accounts for guide vane pressure drop. Turbine efficiency \(\eta = \dot{W}_s / \dot{W}_t\).

Turbine Design Algorithm

There is also available the runner design algorithm that can define all geometrical parameters based on the nominal parameters (net head, flow rate, power, speed). The algorithm determines: outlet blade angle β₂, runner radii R₁ and R₂, runner width w₁, and inlet blade angle β₁.

The turbine losses coefficients (k_ft1, k_ft2, k_ft3) can be also defined automatically. However, if some dynamic data from real turbine is available it is better to tune these parameters a bit more and use the defined values as a starting point.

Guide Vane Actuation

A model for servo that runs the guide vane opening is also available. A guide vane opening model relates actuator position Y to guide vane angle α₁ through geometric relationships. Furthermore it is possible to automatically generate all needed parameters for the servo, or simply specify them.

Low Load Performance

This mechanistic turbine model does not work really well for low loads (<10% guide vane opening). However there is parameters that could be tuned for low load regimes. These are u_min and k_ft4.

More Information

More info about the mechanistic turbine model can be found in: [Vytvytskyi2018] and [Vytvytskyi2019]. Additional details about the servo (and turbine model) are in: Resources/Documents/Turbines_model.pdf.

Parameters

Connectors

Components