Conservation of Energy in Control Volume

The first law states that the rate of energy change in a control volume equals the net rate of heat and work entering it. For a control volume coinciding with a system, the net heat and work entering the system equal those entering the control volume. This principle forms the foundation for analyzing energy conservation in fluid systems, incorporating heat transfer, shaft work, and pressure forces at the control surface.

The Reynolds Transport Theorem relates the rate of energy change within a control volume to both the time rate of change in stored energy and the net energy flow through the boundary. It enables engineers to track energy entering and leaving the control volume, accounting for contributions from heat, shaft work from rotating components like turbines, and pressure work from fluid stresses.

Two primary work types occur: shaft work, produced by rotating components such as turbines, and pressure work, generated by fluid stresses and normal stresses at the control surface. The general energy equation combines both work types with stored energy and heat transfer to represent complete energy conservation in fluid systems.

In steady-flow systems, the time derivative of stored energy becomes zero since no energy accumulates within the control volume. This simplifies the energy equation significantly. When heat transfer is negligible and focus is on shaft work, the equation reduces to balancing energy flux at inlet and outlet points based on mass flow rate, internal energy, velocity, and height.

A turbine extracts mechanical power by reducing the fluid's internal and kinetic energy as it passes through. The energy equation shows that the difference between inlet and outlet energy flux, accounting for velocity and height changes, equals the useful shaft work produced. For example, a turbine with inlet velocity of 50 m/s and outlet velocity of 20 m/s at a 10 kg/s mass flow rate produces approximately 11.48 kW of mechanical power.

The control volume energy equation incorporates internal energy, kinetic energy from fluid velocity, and potential energy from height differences. These components, combined with heat transfer and work terms, form the complete energy balance. The energy flux at any point is determined by mass flow rate multiplied by the sum of internal energy, kinetic energy, and potential energy per unit mass.

Mass flow rate remains constant at inlet and outlet due to conservation of mass in steady-flow conditions. This principle ensures that the mass entering the control volume equals the mass leaving it. Constant mass flow rate allows engineers to directly relate energy changes to useful work output, simplifying calculations for turbine performance and mechanical power generation.