1. Energy and Rotalpy Where: E 1 =Energy at the inlet of the turbine[J/kg] E 2 =Energy at the inlet of the turbine[J/kg] I 1 =Rotalpy at the inlet of the turbine[J/kg] I 2 =Rotalpy at the inlet of the turbine[J/kg] g=Gravity constant[m/s 2 ] H=Head[m]  =Efficiency[ - ]

2. Energy and Rotalpy w c cucu cmcm u

3. Rotalpy along the streamline

4. Absolute and relative acceleration We assume stationary flow and introduce relative and peripheral velocity: Where: w=relative velocity[m/s] u=peripheral velocity [m/s]  =angular velocity [rad/s] r=radius [m] Relative acceleration Centripetal acceleration Coriolis acceleration

5. Relative acceleration Centripetal acceleration Acceleration along a streamline

6. Forces acting in a rotating channel along a streamline

8. By inserting the equation And rearranging we obtain the following equation:

9. If we integrate the equation above we get the equation for rotalpy: Rotalpy

10. Centripetal acceleration Acceleration normal to a streamline Centripetal acceleration Coriolis acceleration

11. Forces acting in a rotating channel normal to a streamline

12. We insert the equation for the normal acceleration in to the equation above. We obtain the following equation:

13. We insert the equation the equations to the right in to the equation above. We obtain the following equation: We derive the rotalpy equation above with respect to the normal direction. We obtain the following equation:

14. We have the equation below from the derivation from the Rotalpy equation We have the equation below from the derivation from Newton’s second law If we rearrange the equations above we obtain the following equation:

15. Pump Pump-turbine in turbine mode Francis turbine

16. Pump 

17. Pump-turbine 

18. Francis turbine