Understanding Work Done in Steady Flow Processes

In thermodynamics, processes can be classified as either non-flow (closed system) or flow (open system). The formula for work done is different for these types of processes.

A steady flow process is one where mass and energy cross the boundaries of the system, but the conditions at any point within the system remain constant over time. Examples include flow through turbines, compressors, nozzles, and heat exchangers operating under steady conditions.

Non-Flow vs. Steady Flow Work

For a non-flow process (like a piston-cylinder expanding gas), the work done is given by the area under the pressure-volume (p-v) diagram, which is expressed as:

\(W_{non-flow} = \int\limits_1^2 p \, dv\)

This represents the boundary work done by the system on its surroundings.

For a steady flow process, energy interactions include not only boundary work but also flow work (the energy required to push the fluid into and out of the control volume), changes in kinetic energy, potential energy, and heat transfer. The Steady Flow Energy Equation (SFEE) accounts for these terms. A simplified form of SFEE (ignoring changes in kinetic and potential energy, and heat transfer) can lead to the expression for steady flow work.

Derivation of Steady Flow Work Formula

Considering flow through a control volume, the net work done in a steady flow process is the difference between the work done by the system and the flow work done on the system (to push mass in) and by the system (to push mass out).

The flow work at the inlet is \(p_1 v_1\) (where \(v_1\) is specific volume) and at the outlet is \(p_2 v_2\). The total energy change across the control volume in SFEE relates to heat transfer, work done, and changes in enthalpy, kinetic energy, and potential energy.

The differential work in a steady flow process can be related to the change in pressure and specific volume. For a reversible steady flow process, the differential work is given by:

\(dW_{steady flow} = -v \, dp\)

Integrating this expression from state 1 to state 2 gives the total work done in a steady flow process:

\(W_{steady flow} = \int\limits_1^2 dW_{steady flow} = \int\limits_1^2 -v \, dp\)

This formula \(-\int\limits_1^2 v \, dp\) represents the area to the left of the process curve on a p-v diagram.

Analyzing the Options

1. \( \frac{{{p_1}\;{v_1}\; - \;{p_2}\;{v_2}}}{{n - 1}} \) : This is the work done for a polytropic process \(pv^n = C\) in a non-flow system (closed system). It is not the formula for steady flow work.
2. \( -\int\limits_1^2 \) vdp : This is the correct formula for the work done in a steady flow process.
3. \( \int\limits_1^2 \) pdv : This is the formula for work done in a non-flow process (closed system).
4. pv : This term represents flow work (\(p \cdot (\text{volume}) = p \cdot (m/\rho) = p \cdot m \cdot v\), or \(pv\) per unit mass), which is part of the energy transported with the mass flow, but not the net shaft work done in a steady flow process itself.

Therefore, the steady flow process work done is given by \(-\int\limits_1^2 v \, dp\).