Derivation of the Steady Flow Energy Equation (SFEE)

 

Derivation of the Steady Flow Energy Equation (SFEE)

The Steady Flow Energy Equation (SFEE) is an important concept in thermodynamics, particularly for analyzing the performance of various engineering systems like turbines, compressors, heat exchangers, and nozzles. The SFEE is derived from the first law of thermodynamics applied to a control volume under steady-state conditions. In this blog, we will go through a highly detailed derivation of the SFEE step by step.

Steady Flow System and Control Volume

Consider a control volume where fluid enters and exits at steady state. Steady state implies that the properties of the fluid within the control volume do not change with time. This means:

• The mass flow rate is constant.
• The energy of the system is constant over time.

Let’s define the variables:

• $\dot{m}$: Mass flow rate (kg/s)
• $\dot{Q}$: Rate of heat transfer into the system (W or J/s)
• $\dot{W}$: Rate of work done by the system (W or J/s)
• $h$: Specific enthalpy of the fluid (J/kg)
• $V$: Velocity of the fluid (m/s)
• $z$: Elevation above a reference level (m)
• $g$: Acceleration due to gravity (9.81 m/s²)

First Law of Thermodynamics for a Control Volume

The first law of thermodynamics for a control volume in steady state can be written as:

$\text{Rate of energy entering the control volume} = \text{Rate of energy leaving the control volume}$

Considering all forms of energy (internal, kinetic, potential, and flow work), we have:

$\dot{Q} + \dot{m} \left( h_1 + \frac{V_1^2}{2} + gz_1 \right) = \dot{W} + \dot{m} \left( h_2 + \frac{V_2^2}{2} + gz_2 \right)$

Here, the subscript 1 denotes the inlet and subscript 2 denotes the outlet of the control volume.

Simplifying the Energy Balance

Rearranging the above equation to isolate $\dot{Q}$ and $\dot{W}$:

$\dot{Q} - \dot{W} = \dot{m} \left[ \left( h_2 - h_1 \right) + \frac{V_2^2 - V_1^2}{2} + g \left( z_2 - z_1 \right) \right]$

This is the Steady Flow Energy Equation (SFEE). It states that the net energy transfer to the control volume as heat and work is equal to the change in the total energy (enthalpy, kinetic, and potential) of the fluid passing through the control volume.

Terms in the SFEE

1. Heat Transfer ($\dot{Q}$):
   • Heat added to or removed from the control volume.
2. Work Transfer ($\dot{W}$):
   • Work done by or on the fluid within the control volume.
3. Enthalpy ($h$):
   • Includes internal energy and flow work (pressure-volume work).
4. Kinetic Energy ($\frac{V^2}{2}$):
   • Energy due to the velocity of the fluid.
5. Potential Energy ($gz$):
   • Energy due to the elevation of the fluid in a gravitational field.

Application of SFEE to Different Devices

1. Nozzle/Diffuser:

   • In nozzles and diffusers, the change in enthalpy is converted to kinetic energy (no heat transfer and negligible work): $h_1 + \frac{V_1^2}{2} = h_2 + \frac{V_2^2}{2}$

2. Turbine:

   • In turbines, work is extracted from the fluid (usually adiabatic, no heat transfer): $\dot{W} = \dot{m} \left( h_1 - h_2 \right)$

3. Compressor/Pump:

   • In compressors and pumps, work is done on the fluid (usually adiabatic, no heat transfer): $\dot{W} = \dot{m} \left( h_2 - h_1 \right)$

4. Heat Exchanger:

   • In heat exchangers, heat is transferred between fluids without doing any work: $\dot{Q} = \dot{m} \left( h_2 - h_1 \right)$

5. Throttle Valve:

   • In throttle valves, there is no work done, and the process is adiabatic. The enthalpy remains constant: $h_1 = h_2$

Conclusion

The Steady Flow Energy Equation (SFEE) is a powerful tool in thermodynamics for analyzing various engineering devices and systems under steady-state conditions. By accounting for the energy changes in terms of enthalpy, kinetic energy, and potential energy, SFEE provides a comprehensive framework for understanding the performance and efficiency of turbines, compressors, heat exchangers, and other flow devices. The derivation and application of SFEE highlight the interplay between different forms of energy and their conservation in practical thermodynamic systems.