Energy Analysis of Pumps and Turbines

Hello! Welcome to your next lesson.

In our last session, we wrapped up our study of external fluid dynamics by analyzing how lift and drag forces act on immersed bodies like airfoils. We saw how these forces are quantified using dimensionless coefficients, a key concept in aerospace design.

Today, we'll shift our focus from external flows to internal flows through machinery. This lesson is the final one in our module on Fluid Flow Applications. We will apply the principle of energy conservation to analyze two of the most important devices in mechanical and aerospace engineering: pumps and turbines. Your learning outcome is to apply the steady flow energy equation to analyze pumps and turbines in a fluid system.

This topic is a perfect bridge between fluid mechanics and our next major subject, thermodynamics. Understanding how energy is transferred to or from a fluid in a machine is fundamental to analyzing everything from water supply systems to the jet engines that power aircraft.

1. The Energy Equation for Flowing Fluids

The foundation for analyzing pumps and turbines is the First Law of Thermodynamics, which states that energy is conserved. For a fluid flowing steadily through a device (a control volume), we can write an energy balance called the Steady Flow Energy Equation (SFEE).

This equation accounts for all the energy a fluid carries, transfers, and receives.

In its most common thermodynamic form, the SFEE relates heat transfer ($\dot{Q}$), shaft work rate (${\dot{W}}_s$), and the change in the fluid's enthalpy ($h$), kinetic energy ($\frac{1}{2}{V}^2$), and potential energy ($gz$) between the inlet (1) and outlet (2).

Here:

• ${\dot{Q}}_{in}$ is the rate of heat added to the fluid.
• ${\dot{W}}_{s,out}$ is the rate of shaft work done by the fluid (e.g., a turbine turning a shaft). Work done on the fluid (e.g., by a pump) is negative.
• $\dot{m}$ is the mass flow rate.
• $h$ is specific enthalpy ($h=u+Pv$), which combines internal energy ($u$) and flow work ($Pv$). Your background in electronics might lead you to think of enthalpy as analogous to the total energy carried by charge carriers, not just their kinetic energy.

2. The "Head" Form of the Energy Equation

For liquid systems (incompressible flow), it's often more practical to express the energy equation in terms of "head," where each term has units of length (meters). This is done by dividing the SFEE by $\dot{m}g$.

This gives us the extended Bernoulli equation, which includes terms for pumps, turbines, and frictional losses.

As the video explains, the final, most useful form of the head equation is:

Where:

• $\frac{P}{\rho g}$ is the pressure head.
• $\frac{V^2}{2g}$ is the velocity head.
• $z$ is the elevation head.
• $h_{pump,u}$ is the useful head added by a pump.
• $h_{turbine,e}$ is the extracted head removed by a turbine.
• $h_L$ is the total irreversible head loss due to friction.
• $\alpha$ is the kinetic energy correction factor (often assumed to be 1 for turbulent flow).

3. Analyzing Pumps

A pump is a device that does work on a fluid to add energy, typically to increase its pressure or move it to a higher elevation.

• Useful Power (Water Horsepower): The power actually transferred to the fluid.
  ${\dot{W}}_{useful}=\dot{m}g{h}_{pump,u}$
• Shaft Power (Brake Horsepower): The power that must be supplied to the pump's shaft from a motor. This is always greater than the useful power due to inefficiencies.
  ${\dot{W}}_{shaft}=\frac{{\dot{W}}_{useful}}{{\eta }_{pump}}$

Let's see this applied to a practical problem.

4. Analyzing Turbines and Compressors

A turbine does the opposite of a pump: it extracts energy from a high-energy fluid to produce shaft work. A compressor is functionally similar to a pump but is used for gases. Both are central to your aerospace interests, as they form the core of a jet engine.

For these devices, especially with compressible gases, it's more common to work with the enthalpy form of the SFEE. If we assume the process is adiabatic ($\dot{Q}\approx 0$) and changes in kinetic and potential energy are small, the equations simplify nicely:

• Turbine Power Output: ${\dot{W}}_{out}=\dot{m}(h_{inlet}-h_{outlet})$
• Compressor Power Input: ${\dot{W}}_{in}=\dot{m}(h_{outlet}-h_{inlet})$

For an ideal gas, the change in enthalpy is $\Delta h=C_p\Delta T$, where $C_p$ is the specific heat at constant pressure.

To give you a sense of the scale involved in aerospace, let's look at the power required to drive the compressor in a modern jet engine. This power is generated by the turbine section.

Conclusion

In this lesson, we have seen how a single, powerful principle—the conservation of energy, expressed as the Steady Flow Energy Equation—can be used to analyze the performance of essential engineering devices.

Key Takeaways:

• The Steady Flow Energy Equation (SFEE) is the primary tool for analyzing systems with flowing fluids, accounting for changes in enthalpy, kinetic/potential energy, heat, and work.
• For incompressible fluids like water, the head form of the energy equation is often used.
• Pumps add energy to a fluid, requiring shaft power input. Their performance is described by the useful head $h_{pump,u}$ and efficiency ${\eta }_{pump}$.
• Turbines extract energy from a fluid, producing shaft power output. Their analysis often uses the enthalpy form of the SFEE.
• The power involved in aerospace components like compressors and turbines is immense, highlighting the importance of efficient design.

Preview of the Next Module:
We have now completed our module on fluid mechanics applications. This lesson, with its focus on enthalpy, heat, and work, is a perfect segue into our next major topic: Thermodynamics. In the next module, we will formalize these concepts, starting with the definitions of thermodynamic systems, states, and properties, and then applying the First Law of Thermodynamics, which you've seen today as the SFEE.