Mathematic Modelling of a Reversible Hydropower System: Dynamic Effects in Turbine Mode
Abstract
:1. Introduction
2. Methods and Materials
2.1. Water Hammer Modelling
2.2. Boundary Conditions for MOC
2.3. Turbine Modelling
2.4. CFD Model Description and Mesh
3. Dynamic Effects in Hydropower Systems
3.1. Overspeed Effect Due to Full-Load Rejection
3.2. Influence of Other Characteristic Parameters
4. Comparisons between Modelling and Laboratory Tests for Different Type of Turbines and Specific Speeds
- -
- For Francis and Kaplan with fixed blades (propeller)
- -
- For radial PAT
- -
- For axial PAT
4.1. PAT Radial Runners—Low ns
4.2. PAT Axial Runners—High ns
4.3. Francis Turbine—Low
4.4. Kaplan Turbine—High
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
Wave celerity (m/s) | |
, , | Constants of a pump head curve |
Cross-sectional area of a pipe (m2) | |
Cross-sectional area of an orifice (m2) | |
Cross section of an air vessel (m2) | |
Constant computed in the initial condition of an air vessel | |
Relationship between , , and | |
Negative characteristic equation (-) | |
Positive characteristic equation (-) | |
Contraction coefficient (-) | |
Opening gate coefficient (-) | |
Negative characteristic constant (-) | |
Positive characteristic constant (-) | |
Runner’s rotational speed (-) | |
Constant of the turbulent eddy viscosity coefficient | |
Internal pipe diameter (m) | |
Friction factor (-) | |
Turbulent viscosity factor | |
Forces in x direction (N) | |
Gravitational acceleration (m/s2) | |
Iteration number (-) | |
Position in a pipe (-) | |
Relative turbine head (-) | |
Hydraulic grade line (m) | |
Piezometric head on section A (m) | |
Piezometric head on section B (m) | |
Barometric pressure (m) | |
Piezometric head at section P (m) | |
Pressure head at the end of an analyzed time step in an air vessel (m) | |
Piezometric head on an analyzed side of section P (m) | |
Piezometric head on the upstream side of section P (m) | |
Piezometric head on the downstream side of section P (m) | |
Turbine/pump rated head (m) | |
Water level of a reservoir (m) | |
Rotating mass inertia (kgm2) | |
Turbulent kinetic energy (m2/s2) | |
Valve head loss coefficient (-) | |
Pipe length (m) | |
Water mass (kg) | |
Relative rotating speed (-) | |
Rotational speed | |
Number of segments (-) | |
Turbine rated speed (-) | |
Runaway turbine rotating speed (r.p.m. | |
Nominal runner speed (r.p.m.) | |
Specific velocity (r.p.m.). | |
Pump power (kW) | |
Reference power (kW) | |
Pipe perimeter (m) | |
Pressure force (N) | |
Polytropic coefficient (-) | |
Discharge (m3/s) | |
Relative flow through the turbine (-) | |
Relative discharge (-) | |
Discharge at section A (m3/s) | |
Discharge at section B (m3/s) | |
Discharge at section P (m3/s) | |
Air vessel orifice discharge (m3/s) | |
Turbine/pump rated discharge (m3/s) | |
Turbine discharge at runaway speed (m3/s) | |
Time (s) | |
Turbine torque (N·m) | |
Electromagnetic resistance torque (N·m) | |
Actuating hydraulic torque (N·m | |
Guide-vane or valve closing time (s) | |
Elastic time constant (s) | |
Hydraulic turbine torque (N·m) | |
Start-up time of rotating masses (s) | |
Hydraulic inertia time (s) | |
Water velocity in a pipe (m/s) | |
Volume of the air enclosed in the vessel of an analyzed time step (m3) | |
Distance along the main direction of a pipe system (m) | |
Distance from the wall (m) | |
Initial elevation of the free surface inside an air vessel (m) | |
Free surface elevation at the end of the time step (m) | |
Inertial weight of rotating mass (N·m2) | |
Relative runaway discharge (-) | |
Relative runaway rotating speed (-) | |
Turbulent dissipation | |
Pipe slope (m/m) | |
Time step (s) | |
Overpressure (m) | |
Water density (kg/m3) | |
Water unit weight (N/m3) | |
Dynamic viscosity coefficient (-) | |
Turbulent eddy viscosity coefficient (-) | |
Shear stress (N/m2) | |
Viscous shear stress tensor | |
Reynolds-stress tensor | |
Kronecker delta function (-) | |
Angular speed (rad/s) | |
Unit rated efficiency (-) |
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Element | Scheme | Considerations and Additional Equations | Equations System | Notation |
---|---|---|---|---|
Constant-head reservoir | The boundary condition for a constant-head reservoir is computed neglecting entrance head losses as: | (10) | is the water level of a reservoir. | |
Pump | Based on pump characteristic curve for each time step as: | (11) | , , and are constants of a pump curve, and is the rotational speed. | |
Valve | The boundary condition in a regulating valve is obtained based on steady-state head loss equation as: | (12) | is the head loss coefficent obtained experiment20ally. | |
Air vessel | The equation of an air vessel is obtained considering the polytropic law , which needs to be solved simultaneously with the following five equations: | (13) | is the absolute pressure head at the end of an analysed time step, is the air volumne at the end of an analysed time step, is the air volume, is the flow through the orifice, is the cross-section of the orifice, is the discharge coefficient of the orifice, is the polytropic coefficient (usually takes a value of 1.2), is the initial elevation of the free surface, is the barometric pressure, is the constant computed in the initial condition of the air vessel, is the free surface elevation at the end of the time step, and is the cross-section of the air vessel. |
Mesh | Number of Fluid Mesh Cells | Number of Solid Mesh Cells | H | Error (%) | Duration |
---|---|---|---|---|---|
Mesh 1 | 35,888 | 25,378 | 9.8 | 0.15 h | |
Mesh 2 | 60,381 | 30,229 | 6.4 | 34% | 0.49 h |
Mesh 3 | 110,691 | 60,088 | 5.8 | 9% | 2.23 h |
Mesh 4 | 121,936 | 71,162 | 5.4 | 6% | 2.52 h |
Mesh 5 | 134,152 | 81,291 | 5.2 | 3% | 3.28 h |
Mesh 6 | 135,472 | 88,364 | 5.1 | 1% | 3.33 h |
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Ramos, H.M.; Coronado-Hernández, O.E.; Morgado, P.A.; Simão, M. Mathematic Modelling of a Reversible Hydropower System: Dynamic Effects in Turbine Mode. Water 2023, 15, 2034. https://doi.org/10.3390/w15112034
Ramos HM, Coronado-Hernández OE, Morgado PA, Simão M. Mathematic Modelling of a Reversible Hydropower System: Dynamic Effects in Turbine Mode. Water. 2023; 15(11):2034. https://doi.org/10.3390/w15112034
Chicago/Turabian StyleRamos, Helena M., Oscar E. Coronado-Hernández, Pedro A. Morgado, and Mariana Simão. 2023. "Mathematic Modelling of a Reversible Hydropower System: Dynamic Effects in Turbine Mode" Water 15, no. 11: 2034. https://doi.org/10.3390/w15112034