# Control Volume¶

A thermodynamic system where energy and matter is exchanged across its boundary.

## Key variables¶

### Mass Flow Rate¶

Mass flow rate, $\dot{m} = \frac{dm}{dt} = \rho\dot{V} = \rho AV$
Unit for mass flow rate: kg s-1

### Density¶

Density, $\rho = \frac{mass}{volume}$
Unit for density: kg m-3

### Specific Volume¶

Specific volume, $\nu = \frac{1}{\rho}$
Unit for specific volume: m3 kg-1

### Volumetric flow rate¶

Volumetric flow rate, $\dot{V} = AV$
Unit for volumetric flow rate: m3 s-1

## Conservation of Mass¶

At steady state, $\frac{dm_{cv}}{dt} = 0$.
$\therefore \sum\limits_{i}\dot{m}_{i} = \sum\limits_{e}\dot{m}_{e}$

where:

• $m_{cv}$ is the mass of the control volume; and
• $\dot{m}_{i}$ and and $\dot{m}_{e}$ are the mass flow rates at the inlets and exits.

## Work for a Control Volume¶

For a control volume with multiple inlets and exits, the work term $\dot{W}$ of a control volume is:

\dot{W} = \dot{W}_{cv}+\sum\limits_{e}\dot{m}_{e}\left(p_{e}\nu_{e}\right)-\sum\limits_{i}\dot{m}_{i}\left(p_{i}\nu_{i}\right)

where:

• $\dot{m}_{i}$ and and $\dot{m}_{e}$ are the mass flow rates at the inlets and exits;
• $\nu_{i}$ and $\nu_{e}$ are the specific volumes at the inlets and exits;
• $\dot{m}_{i}\left(p_{i}\nu_{i}\right)$ and $\dot{m}_{e}\left(p_{e}\nu_{e}\right)$ are the flow work at the inlets and exits; and
• $\dot{W}_{cv}$ is the shaft work across the boundary of the control volume.

### Flow and Shaft Work¶

• Flow work is due to the fluid pressure as the mass enters and exits the control volume.
• Shaft work occurs due to rotating shafts, displacement of boundary or electrical effects.

## Conservation of Energy for a Control Volume¶

For a control volume with multiple inlets and multiple exits, the energy rate balance is:

\begin{align*} \frac{dE_{cv}}{dt} &= \dot{Q}_{cv}-\dot{W}_{cv}+\sum\limits_{i}\dot{m}_{i}\left(u_{i}+p_{i}\nu_{i}+\frac{V_{i}^2}{2}+gz_{i}\right)-\sum\limits_{e}\dot{m}_{e}\left(u_{e}+p_{i}\nu_{i}+\frac{V_{e}^2}{2}+gz_{e}\right) \\ &= \dot{Q}_{cv}-\dot{W}_{cv}+\sum\limits_{i}\dot{m}_{i}\left(h_{i}+\frac{V_{i}^2}{2}+gz_{i}\right)-\sum\limits_{e}\dot{m}_{e}\left(h_{e}+\frac{V_{e}^2}{2}+gz_{e}\right) \\ &\because \text{Specific enthalpy, }h = u + p\nu \end{align*}

where:

• $\dot{E}_{cv}$ denotes the energy of control volume at time t;
• $\dot{Q}$ denotes the net rate of heat transfer at time t;
• $\dot{W}$ denotes the work across the boundary of the control volume at time t;
• $\dot{m}_{i}u_{i}$ and $\dot{m}_{e}u_{e}$ denotes the rate of transfer of internal energy of the multiple inlets and exits;
• $\dot{m}_{i}\frac{V_{i}^{2}}{2}$ and $\dot{m}_{e}\frac{V_{e}^{2}}{2}$ denotes the rate of transfer of kinetic energy of the multiple inlets and exits; and
• $\dot{m}_{i}gz_{i}$ and $\dot{m}_{e}gz_{e}$ denotes the rate of transfer of graviational potential energy of the multiple inlets and exits.

Assuming specifc heat capacity at constant pressure $c_{p}$ is independent of temperature:
h_{2} - h_{1} = c_{p}\left(T_{2}-T_{1}\right)

## Applications of Control Volumes¶

### Nozzles and Diffusers¶

• Nozzles increase the velocity of fluid in the direction of fluid flow.
• Diffusers decrease the velocity of fluid in the direction of fluid flow.
\frac{dE_{cv}}{dt} = \dot{Q}_{cv}-\dot{W}_{cv}+\sum\limits_{i}\dot{m}_{i}\left(h_{i}+\frac{V_{i}^2}{2}+gz_{i}\right)-\sum\limits_{e}\dot{m}_{e}\left(h_{e}+\frac{V_{e}^2}{2}+gz_{e}\right) \\ \frac{dE_{cv}}{dt} = \dot{Q}_{cv}-\dot{W}_{cv}+\sum\dot{m}\left[\left(h_{i}-h_{e}\right)+\frac{V_{i}^2-V_{e}^2}{2}+\left(gz_{i}-gz_{e}\right)\right] \\ \because \sum\limits_{i}\dot{m}_{i} = \sum\limits_{e}\dot{m}_{e} \ \text{at steady state}\\ \Rightarrow 0 = (h_{1}-h_{2})+\frac{V_{1}^{2}-V_{2}^{2}}{2}
• Net rate of energy $\frac{dE_{cv}}{dt} = 0$ at steady state;
• Net rate of heat transfer $\dot{Q}_{cv}$ is negligible;
• There is no shaft work $\dot{W}_{cv}$;
• There is only flow work; and
• Change in potential energy $\sum\dot{m}\left(gz_{i}-gz_{e}\right)$ is negligble.

### Turbines¶

• Turbines generate power when fluids pass through blades attached to a shaft, which can freely rotate.
\frac{dE_{cv}}{dt} = \dot{Q}_{cv}-\dot{W}_{cv}+\sum\limits_{i}\dot{m}_{i}\left(h_{i}+\frac{V_{i}^2}{2}+gz_{i}\right)-\sum\limits_{e}\dot{m}_{e}\left(h_{e}+\frac{V_{e}^2}{2}+gz_{e}\right) \\ \frac{dE_{cv}}{dt} = \dot{Q}_{cv}-\dot{W}_{cv}+\sum\dot{m}\left[\left(h_{i}-h_{e}\right)+\frac{V_{i}^2-V_{e}^2}{2}+\left(gz_{i}-gz_{e}\right)\right] \\ \Rightarrow \dot{W}_{cv} = \dot{m}\left(h_{1}-h{2}\right)
• Net rate of energy $\frac{dE_{cv}}{dt} = 0$ at steady state;
• Stray heat transfer $\dot{Q}_{cv}$ is negligible;
• Change in kinetic energy $\sum\dot{m}\left(\frac{V_{i}^2-V_{e}^2}{2}\right)$ is negligible; and
• Change in potential energy $\sum\dot{m}\left(gz_{i}-gz_{e}\right)$ is negligble.

### Compressors and Pumps¶

• Compressors do work on gases to change the state of gases.
• Pumps do work on liquids to change the state of liquids.
\frac{dE_{cv}}{dt} = \dot{Q}_{cv}-\dot{W}_{cv}+\sum\limits_{i}\dot{m}_{i}\left(h_{i}+\frac{V_{i}^2}{2}+gz_{i}\right)-\sum\limits_{e}\dot{m}_{e}\left(h_{e}+\frac{V_{e}^2}{2}+gz_{e}\right) \\ \frac{dE_{cv}}{dt} = \dot{Q}_{cv}-\dot{W}_{cv}+\sum\dot{m}\left[\left(h_{i}-h_{e}\right)+\frac{V_{i}^2-V_{e}^2}{2}+\left(gz_{i}-gz_{e}\right)\right] \\ \Rightarrow \dot{W}_{cv} = \dot{m}\left(h_{1}-h_{2}\right) \\ \because \dot{W}_{cv} < 0, \ \therefore \text{power input required}
• Net rate of energy $\frac{dE_{cv}}{dt} = 0$ at steady state;
• Stray heat transfer $\dot{Q}_{cv}$ is negligible;
• Change in kinetic energy $\sum\dot{m}\left(\frac{V_{i}^2-V_{e}^2}{2}\right)$ is negligible; and
• Change in potential energy $\sum\dot{m}\left(gz_{i}-gz_{e}\right)$ is negligble.

### Heat Exchangers¶

• Heat exchangers transfer heat between two or more fluids.
\frac{dE_{cv}}{dt} = \dot{Q}_{cv}-\dot{W}_{cv}+\sum\limits_{i}\dot{m}_{i}\left(h_{i}+\frac{V_{i}^2}{2}+gz_{i}\right)-\sum\limits_{e}\dot{m}_{e}\left(h_{e}+\frac{V_{e}^2}{2}+gz_{e}\right) \\ \frac{dE_{cv}}{dt} = \dot{Q}_{cv}-\dot{W}_{cv}+\sum\dot{m}\left[\left(h_{i}-h_{e}\right)+\frac{V_{i}^2-V_{e}^2}{2}+\left(gz_{i}-gz_{e}\right)\right] \\ \Rightarrow 0 = \dot{Q}_{cv} + \sum\limits_{i}\dot{m}_{i}h_{i}-\sum\limits_{e}\dot{m}_{e}h_{e}
• Net rate of energy $\frac{dE_{cv}}{dt} = 0$ at steady state;
• There is no shaft work $\dot{W}_{cv}$;
• There is only flow work;
• Change in kinetic energy $\sum\dot{m}\left(\frac{V_{i}^2-V_{e}^2}{2}\right)$ is negligible; and
• Change in potential energy $\sum\dot{m}\left(gz_{i}-gz_{e}\right)$ is negligible.

## Change in Specific Entropy for Ideal Gases¶

\Delta s_{1\rightarrow 2} = s_{2}-s_{1}=c_{v}\ln\left(\frac{T_{2}}{T_{1}}\right)-r \ln\left(\frac{V_{2}}{V_{1}}\right)\\ \Delta s_{1\rightarrow 2} = s_{2}-s_{1}=c_{p}\ln\left(\frac{T_{2}}{T_{1}}\right)-r \ln\left(\frac{P_{2}}{P_{1}}\right)\\ \Delta s_{1\rightarrow 2} = s_{2}-s_{1}=c_{p}\ln\left(\frac{V_{2}}{V_{1}}\right)-c_{v} \ln\left(\frac{P_{2}}{P_{1}}\right)\\