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Notes by Category University Engineering

Mechanics & Stress Analysis*
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Steady Flow Processes

In this notes sheet:

  1. Modelling steady flow through a control volume

  2. The Steady Flow Energy Equation (SFEE)

  3. Applications of the SFEE

  4. Mass & volume flow rates

  5. The Rankine Cycle


The basic form of the first law, ΔQ – ΔW = ΔE, only applies to closed systems - no mass can transfer across the system boundary, only energy in the form of heat and work.

In reality, perfectly closed systems are quite rare (take a turbine, for example: air flows in as well as heat, shaft work and hot air flow out), and as such a different model is required:

The energy transfers across the control volume surface are:

  • Shaft work

  • Heat transfer

  • Energy in the working fluid (kinetic, potential, and internal energies)

To simplify the process, we only look at the energy inputs and outputs: what goes on inside the control volume is irrelevant.

We call where the working fluid enters and exits ports.

  • The control volume above has two ports: one inflow and one outflow port.

The Steady Flow Energy Equation (SFEE)

In order to solve problems involving steady flow through a control volume, we use the Steady Flow Energy Equation (SFEE) instead of the simple first law equation:

In a less mathematically intimidating form:

  • Left hand side: rate of energy transfer – rate of shaft work

  • Right hand side: sum of output mass energy flow rates – sum of input mass energy flow rates

Note: The dot above the letters means it is a rate of change with respect to time:

Thankfully, kinetic and potential energies can often be ignored, simplifying the equation immensely. However, it is important you start with the above form when solving problems, else you will likely forget parts.

See the derivation of the SFEE here

Mass Continuity Equation

We can use the SFEE in conjunction with the principle of conservation of mass. This means that the mass in the control volume must stay constant, so the total input mass flow rate = total output mass flow rate:

Therefore, for a two-port control volume where potential energies are negligible, the SFEE reduces to:

When kinetic energy too can be ignored, it becomes:

The point is, the SFEE in its full form may look difficult, but it really isn’t. Just be neat and methodical in your workings, and always start from the full form and then see which terms you can set to zero.

Applications of the SFEE

There are a number of very common examples where the SFEE is applied and simplified. These are worth knowing, as crop up in almost all problems involving steady flow.

This is also where some actual real-life engineering gets involved. Which is fun.

Each example has a standard block diagram symbol. These should be learned as they are used to describe multi-stage processes like the Rankine Cycle (see below).

Partly Closed Valve (Throttling)

This is the simplest form every term in the SFEE other than the enthalpies are negligible:

  • Heat transfer is negligible when the valve is well insulated

  • There is no shaft work

  • Kinetic energy change is negligible

  • Potential energy change is negligible

There are two ports, so the SFEE reduces to:

From this you can see that throttling (the process that takes place in a partly-closed valve) is isenthalpic.