Derivation of Power Density Equation of Hydro Turbine: Mathematical Basis of Osmosis Energy to Rotate a Reaction Turbine

Hydroelectric power from hydro-turbines is one of the most widely used renewable energy sources because it contributes to reducing greenhouse gas emissions. Power generation through a hydro-turbine needs to convert the water kinetic energy into electricity using rotors placed along free-flow paths. That is, there is a need for a continuous flow of water or pressure to rotate the turbine, and this pressure can be generated through what is known as osmosis energy. So, this paper will explore two main topics. The first will relate to one of the most important types of clean energies, which is called blue energy, that results from the convergence of two water sources with different salt concentrations. The resulting energy is invested in rotating the turbine, which in turn generates electricity. The mathematical basis and equations for the osmosis process will also be highlighted. The second topic will reveal the hydro turbines installed on river paths; the equations used to calculate the efficiency and energy of the turbine. The research derived an equation that calculates the energy generated by the turbine depending on the osmosis energy. The maximum power that can be generated from osmosis ranges between 0.75–14.1 kWh/m³ for low salinity currents. The best types of semi-permeable selective membranes are hollow fiber membranes because they are able to achieve a power density of 5.1 W/m². The Kaplan vertical axis reaction turbine was found to be one of the best types of turbines in low-head and flow rivers.