Mass and Energy Analysis for Control Volumes

The idea behind the turbine is that a fluid passes through and does work against the blades which are attached to the shaft. The shaft then rotates and the turbine produces work. Hydroelectric power plants use turbines to generate power, but turbines are also used in aircraft, ships, and many other large vehicles.

    Turbines are often operational for long periods of time so that they can be considered as steady-flow devices which undergo a steady-flow process. During a steady-flow process the mass, volume, and total energy content remain constant.

Heat transfer from turbines is usually negligible since they are typically well insulated. Potential energy chances are also negligible. Problem 5-55 in Thermodynamics: An Engineering Approach Sixth Edition (SI Units) is a good example of a typical mass and energy analysis problem.

    The problem states that argon gas enters an adiabatic turbine steadily at 900 kPa and 450°C with a velocity of 80 m/s and leaves at 150 kPa with a velocity of 150 m/s. The question wants to know the exit temperature of the argon if the power output of the turbine is 250 kW. When we rearrange equation 3 for the work out and remove potential energy and heat transfer we result in equation 4:

There are two things that we need to find before we can solve the problem. The first issue is that we aren’t given the mass flow rate. We can use the ideal gas law to solve for the specific volume and then find the mass flow rate using the definition of the mass flow rate.

Using the relationship that the change in enthalpy for an ideal gas can be expressed as the product of the specific heat at constant pressure and change in temperature we can now solve the problem.

For dimensional homogeneity, the change in kinetic energy needs to be converted to kJ/kg so the denominator becomes 2000 instead of 2. When you solve the equation you yield a temperature at the exit of: