Consider a turbine operating under steady-flow conditions. The control volume is drawn around the turbine, with fluid entering at one point and exiting at another. The turbine extracts energy from the fluid, which performs mechanical work (shaft work).

For steady flow systems, the time derivative of the stored energy becomes zero since there is no energy accumulation within the control volume. This simplifies the energy equation to:

Since we focus on work done by the turbine shaft and assume negligible heat transfer, Q̇ ≈ 0. Therefore, the energy equation simplifies further to:

The flow is divided into two components: the fluid entering the turbine (inlet) and exiting the turbine (outlet). The mass flow rate at both points is constant, and the energy flux at the inlet and outlet is given by:

Where:

-ṁis the mass flow rate,

- ũ₁, V₁, and z₁ refer to the internal energy, velocity, and height at the inlet,

- ũ₂, V₂, and z₂ refer to the internal energy, velocity, and height at the outlet,

In a typical turbine, the fluid's internal and kinetic energy decreases as it passes through the turbine, which results in useful shaft work being done. Suppose a turbine operates with an inlet velocity of V₁=50 m/s, an outlet velocity of V₂= 20 m/s, an inlet height z₁=100 m, and an outlet height z₂=90m. Assume the internal energy change is small compared to the kinetic and potential energy changes, and the mass flow rate is 10 kg/s.

As a result, the turbine produces approximately 11.48 kW of mechanical power by extracting energy from the flowing fluid.