1. 2nd International Conference TSD’2018
Towards Sustainable Development
Skopje, Republic of Macedonia, November 2th-3th, 2018
MATHEMATICAL AND NUMERICAL MODEL OF THE STEAM TURBINE FOR SUSTAINABLE ENERGY SOLUTIONS
Vesna Antoska Knights , Faculty of Technology and Technical Sciences – Veles, University “St Kliment Ohridski”-Bitola, MK,
Olıvera Petrovska, Faculty of Technical Sciences, University Mother Teresa-Skopje, MK,

2. INTRODUCTION Turbomachinery is the key for sustainable development.
2nd International Conference TSD’2018
Towards Sustainable Development
Skopje, Republic of Macedonia, November 2th-3th, 2018
INTRODUCTION
Turbomachinery is the key for sustainable development.
From power plants and aircraft engines to refineries and the fans cooling our computers, turbomachinery is a part of everyday life. It is at the heart of many industry sectors: steam and gas turbines in power generation, pumps, compressors in refrigeration, turbochargers in transportation, aerospace, space and so on.
One way to contribute to the sustainability of the environment and its energy resources is in the implementation of industrial steam turbine solutions for alternative energy applications. In that way the objective is to classify the influenced factors which affect the efficiency of the work of the turbine stage at defined thermodynamics properties of the flow.

3. 2nd International Conference TSD’2018
Towards Sustainable Development
Skopje, Republic of Macedonia, November 2th-3th, 2018
MATHEMATICAL MODEL
The governing equations used to mathematical model are instantaneous Navies-Stokes equations time averaged and Standard k- turbulence model.
For a single species using Newtonian fluid, in a Cartesian coordinate system, the conservation equations for mass, momentum and energy are expressed in tensor form (CFX-TASCflow 2000, Theory Documentation) as:

4. MATHEMATICAL MODEL Conservation of mass equation:
2nd International Conference TSD’2018
Towards Sustainable Development
Skopje, Republic of Macedonia, November 2th-3th, 2018
MATHEMATICAL MODEL
Conservation of mass equation:
Conservation of momentum equation:
Conservation of energy equation:

5. 2nd International Conference TSD’2018
Towards Sustainable Development
Skopje, Republic of Macedonia, November 2th-3th, 2018
MATHEMATICAL MODEL
In the above equations represents the velocities in coordinate directions, P is the static pressure, H is the total enthalpy, is the density, is the molecular energy transport due to conduction, and the S terms are additional source terms.
The total enthalpy is:
The molecular fluxes are expressed in terms of velocity, temperature and concentration gradients:
m is dynamic viscosity of the fluid, l its conductivity Gk, hk, and Yk
are the molecular diffusion coefficient, static enthalpy and mass fraction of species k, respectively.

6. 2nd International Conference TSD’2018
Towards Sustainable Development
Skopje, Republic of Macedonia, November 2th-3th, 2018
MATHEMATICAL MODEL
The turbulent kinetics energy k, is determined from the solution of a semi-empirical transport equation. For a two equation model it is assumed that the length scale is a dissipation length scale, and when the turbulent scales are isotropic, Kolmogorov determined that
Where is the turbulent dissipation rate.
The turbulence model has been implemented in many CFD software packages and has a well-established regime of prediction capability. The standard implementation of the turbulence model employs wall functions to model the viscous near-wall layer.
The wall functions approach eliminates the necessity of numerically resolving the large gradients in the thin near-wall region, thus conserving variable computer recourses.

7. 2nd International Conference TSD’2018
Towards Sustainable Development
Skopje, Republic of Macedonia, November 2th-3th, 2018
NUMERICAL METHODS
As a solver in this work CFX-TASCflow is used. The code for solving equations for ideal gas used conservative forms of Navies Stokes Equations and
turbulence model.
A part of solver strategy is used in TASCflow is multigrid methods. The particular variant used in TASCflow is based on conservation principles already implicit in the Finite Volume discretisation and is called Additive Correction Multigrid. The linearised discrete algebraic equations that arise from most finite volume methods are sufficiently diagonally dominant to permit solution by simple relaxation methods, such as Gauss Seidel.

8. SOME SIMULATION RESULTS
2nd International Conference TSD’2018
Towards Sustainable Development
Skopje, Republic of Macedonia, November 2th-3th, 2018
SOME SIMULATION RESULTS
The numerical computations of this study were conducted at the Institute of Turbomachinery, Lodz Technical University, Poland.
The numerical computations were made for a two-stage turbine with the leaning stator blades are shown below.

9. SOME SIMULATION RESULTS
2nd International Conference TSD’2018
Towards Sustainable Development
Skopje, Republic of Macedonia, November 2th-3th, 2018
SOME SIMULATION RESULTS
Boundary conditions at the inlet and outlet, used in computation code CFX - TASCflow:
Inlet
Normal rotation speed 3000.
Total pressure over inlet Pa= Pa.
Inlet mass flow.
Absolute temperature 3150C.
Outlet.
Static pressure constant over outlet Pa.
Adiabatic wall.

10. SOME SIMULATION RESULTS - Pressure
2nd International Conference TSD’2018
Towards Sustainable Development
Skopje, Republic of Macedonia, November 2th-3th, 2018
SOME SIMULATION RESULTS - Pressure
Fig. 3 Total pressure distributions in control planes upstream and down- stream of the cascade.
The total pressure distribution allows one to observe the losses in the stage. The places with lower values of the total pressure show us about the higher losses in those places.
In Fig. 3 the distribution of the total pressure in a control section downstream, the stator blade row is shown. According to the predictions, the higher losses exist only in boundary regions, especially close to the hub in a plane downstream of the stator.

11. SOME SIMULATION RESULTS
2nd International Conference TSD’2018
Towards Sustainable Development
Skopje, Republic of Macedonia, November 2th-3th, 2018
SOME SIMULATION RESULTS
Fig. 4 The distribution of the total pressure along the meridional direction of the cascade.
In Fig. 4 the distribution of the total pressure along the meridional direction is shown. It is evident that the majority of the losses are generated at the outlet where the velocities are high and secondary flows developed.

12. SOME SIMULATION RESULTS
2nd International Conference TSD’2018
Towards Sustainable Development
Skopje, Republic of Macedonia, November 2th-3th, 2018
SOME SIMULATION RESULTS
Fig. 5 Graphical presentation of pressure around the blade in the middle of the blade channel (50% span).
In Fig. 5 The distribution of the static pressure around the blade in a 50% span is shown. This diagram points out to some possibilities of the profile refining. At the inlet, there are no significant pressure differences between the pressure and suction side of the blade. The suction side of the blade can be changed to get a more significant pressure difference between two sides of blade, however these changes should not cause the separation because they result in a higher loss level.

13. SOME SIMULATION RESULTS - Velocity
2nd International Conference TSD’2018
Towards Sustainable Development
Skopje, Republic of Macedonia, November 2th-3th, 2018
SOME SIMULATION RESULTS - Velocity
Fig. 6 Graphical presentation of the speed from the inlet to the outlet at section S1 located at a half of the blade height
The velocity vectors in section S1 located at a half of the blade length are shown in Fig 6. It is visible that the flow accelerates most significantly at a half of the channel length, which means that is the place where the flow direction is changed.

14. SOME SIMULATION RESULTS - Velocity
2nd International Conference TSD’2018
Towards Sustainable Development
Skopje, Republic of Macedonia, November 2th-3th, 2018
SOME SIMULATION RESULTS - Velocity
Fig. 7 View of the circumferential area averaged speed along the channel from inlet to outlet.
The view of the circumferential area averaged speed along the channel from inlet to outlet is more visible in diagram in Fig. 7, where the relation is shown as a function.
The computations were made for the whole stage as well.

15. SOME SIMULATION RESULTS - velocity
2nd International Conference TSD’2018
Towards Sustainable Development
Skopje, Republic of Macedonia, November 2th-3th, 2018
SOME SIMULATION RESULTS - velocity
Fig. 8 Axial and radial velocity distribution in the plane downstream of the stator
The distribution of the axial and radial velocities in the plane downstream of the stator are shown above. The wake tracks are clearly visible here and confirm the fact that the blades are leaned.

16. SOME SIMULATION RESULTS - Velocity
2nd International Conference TSD’2018
Towards Sustainable Development
Skopje, Republic of Macedonia, November 2th-3th, 2018
SOME SIMULATION RESULTS - Velocity
Fig. 9 Axial and radial velocity distribution in the plane downstream of the rotor.
In Fig 9 the distribution of the axial and radial velocities in the plane downstream of the rotor are presented. There are visible secondary flow effects observed in the regions close to the boundaries.

17. 2nd International Conference TSD’2018
Towards Sustainable Development
Skopje, Republic of Macedonia, November 2th-3th, 2018
ACKNOWLEDGMENTS
The numerical computations in this research were conducted at the Institute of Turbomachinery at the Technical University in Lodz, Poland.

18. 2nd International Conference TSD’2018
Towards Sustainable Development
Skopje, Republic of Macedonia, November 2th-3th, 2018
CONCLUSION
The mathematical model and numerical prediction methods have become an essential tool for the design and analysis of turbomachinery components over the last two decades. The continuous increase in turbine inlet pressure and temperature, and modification of the turbomachinery parts, definitely require reliable and accurate predictions of the main flow dynamics characteristics and the heat loads imposed on the blades.
The interest of the presented research is to contribute to the development of both numerical method and upgrading mathematical model of a flow in the turbine cascade.
The objective is to classify the influenced factors which affect the efficiency of the work of the turbine stage at defined thermodynamics properties of the flow.

19. 2nd International Conference TSD’2018
Towards Sustainable Development
Skopje, Republic of Macedonia, November 2th-3th, 2018
CONCLUSION
A thermodynamic model has been used for the prediction of the thermal performance of counter rotating turbines.
The challenge with any steam driven turbine is that the performances need to be defined before the construction phase; therefore parameters such as pressure and velocity need considering.
The results of investigations into the flow in turbine cascades can contribute to the development of both the numerical method and the upgrading of the mathematical model describing the physics of the flow in the turbine cascade.

20. 2nd International Conference TSD’2018
Towards Sustainable Development
Skopje, Republic of Macedonia, November 2th-3th, 2018
CONCLUSION
CFD simulation to deliver increasing efficiencies while improving reliability and reducing the cost of manufacturing. Additionally, optimisation tools that work across the various stages in the design process help to achieve performance improvements that were previously unattainable.
The turbomachinery industry is an ideal example of the benefits that can be realised in an integrated beginning-to-end process, thanks to high-level research and development into looking at the problem of a systemic level that has resulted in superior components and performance being available at lower cost to the consumer.

21. 2nd International Conference TSD’2018
Towards Sustainable Development
Skopje, Republic of Macedonia, November 2th-3th, 2018
CONCLUSION
One way to contribute to the sustainability of the environment and its energy resources is in the implementation of industrial steam turbine solutions for alternative energy applications.