# 3-Port Constant Volume Chamber (G)

Rigid chamber with constant volume of gas

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• Simscape / Foundation Library / Gas / Elements

## Description

The 3-Port Constant Volume Chamber (G) block models mass and energy storage in a gas network. The chamber has three ports and a constant volume of gas. The enclosure can exchange mass and energy with the connected gas network, as well as exchange heat with the environment, allowing its internal pressure and temperature to evolve over time. The pressure and temperature evolve based on the compressibility and thermal capacity of this gas volume.

### Mass Balance

Mass conservation relates the mass flow rates to the dynamics of the pressure and temperature of the internal node representing the gas volume:

`$\frac{\partial M}{\partial p}\cdot \frac{d{p}_{\text{I}}}{dt}+\frac{\partial M}{\partial T}\cdot \frac{d{T}_{\text{I}}}{dt}={\stackrel{˙}{m}}_{\text{A}}+{\stackrel{˙}{m}}_{\text{B}}+{\stackrel{˙}{m}}_{\text{C}},$`

where:

• $\frac{\partial M}{\partial p}$ is the partial derivative of the mass of the gas volume with respect to pressure at constant temperature and volume.

• $\frac{\partial M}{\partial T}$ is the partial derivative of the mass of the gas volume with respect to temperature at constant pressure and volume.

• pI is the pressure of the gas volume. Pressure at port A is assumed equal to this pressure, pA = pI.

• TI is the temperature of the gas volume. Temperature at port H is assumed equal to this temperature, TH = TI.

• t is time.

• $\stackrel{˙}{m}$A is the mass flow rate at port A. Flow rate associated with a port is positive when it flows into the block.

• $\stackrel{˙}{m}$B is the mass flow rate at port B. Flow rate associated with a port is positive when it flows into the block.

• $\stackrel{˙}{m}$C is the mass flow rate at port C. Flow rate associated with a port is positive when it flows into the block.

### Energy Balance

Energy conservation relates the energy and heat flow rates to the dynamics of the pressure and temperature of the internal node representing the gas volume:

`$\frac{\partial U}{\partial p}\cdot \frac{d{p}_{\text{I}}}{dt}+\frac{\partial U}{\partial T}\cdot \frac{d{T}_{\text{I}}}{dt}={\Phi }_{\text{A}}+{\Phi }_{\text{B}}+{\Phi }_{\text{C}}+{Q}_{\text{H}},$`

where:

• $\frac{\partial U}{\partial p}$ is the partial derivative of the internal energy of the gas volume with respect to pressure at constant temperature and volume.

• $\frac{\partial U}{\partial T}$ is the partial derivative of the internal energy of the gas volume with respect to temperature at constant pressure and volume.

• ФA is the energy flow rate at port A.

• ФB is the energy flow rate at port B.

• ФB is the energy flow rate at port C.

• QH is the heat flow rate at port H.

### Partial Derivatives for Perfect and Semiperfect Gas Models

The partial derivatives of the mass M and the internal energy U of the gas volume with respect to pressure and temperature at constant volume depend on the gas property model. For perfect and semiperfect gas models, the equations are:

`$\begin{array}{l}\frac{\partial M}{\partial p}=V\frac{{\rho }_{I}}{{p}_{I}}\\ \frac{\partial M}{\partial T}=-V\frac{{\rho }_{I}}{{T}_{I}}\\ \frac{\partial U}{\partial p}=V\left(\frac{{h}_{I}}{ZR{T}_{I}}-1\right)\\ \frac{\partial U}{\partial T}=V{\rho }_{I}\left({c}_{pI}-\frac{{h}_{I}}{{T}_{I}}\right)\end{array}$`

where:

• ρI is the density of the gas volume.

• V is the volume of gas.

• hI is the specific enthalpy of the gas volume.

• Z is the compressibility factor.

• R is the specific gas constant.

• cpI is the specific heat at constant pressure of the gas volume.

### Partial Derivatives for Real Gas Model

For real gas model, the partial derivatives of the mass M and the internal energy U of the gas volume with respect to pressure and temperature at constant volume are:

`$\begin{array}{l}\frac{\partial M}{\partial p}=V\frac{{\rho }_{I}}{{\beta }_{I}}\\ \frac{\partial M}{\partial T}=-V{\rho }_{I}{\alpha }_{I}\\ \frac{\partial U}{\partial p}=V\left(\frac{{\rho }_{I}{h}_{I}}{{\beta }_{I}}-{T}_{I}{\alpha }_{I}\right)\\ \frac{\partial U}{\partial T}=V{\rho }_{I}\left({c}_{pI}-{h}_{I}{\alpha }_{I}\right)\end{array}$`

where:

• β is the isothermal bulk modulus of the gas volume.

• α is the isobaric thermal expansion coefficient of the gas volume.

### Variables

Use the Variables tab in the block dialog box (or the Variables section in the block Property Inspector) to set the priority and initial target values for the block variables prior to simulation. For more information, see Set Priority and Initial Target for Block Variables and Initial Conditions for Blocks with Finite Gas Volume.

### Assumptions and Limitations

• The chamber walls are perfectly rigid.

• There is no flow resistance between ports A, B, and C and the chamber interior.

• There is no thermal resistance between port H and the chamber interior.

## Ports

### Conserving

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Gas conserving port associated with one chamber inlet.

Gas conserving port associated with one chamber inlet.

Gas conserving port associated with one chamber inlet.

Thermal conserving port associated with the temperature of the gas inside the chamber.

## Parameters

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Volume of gas in the chamber. The chamber is rigid and its volume therefore constant during simulation. The chamber is assumed to be completely filled with gas at all times.

The cross-sectional area of chamber inlet A, in the direction normal to gas flow path.

The cross-sectional area of chamber inlet B, in the direction normal to gas flow path.

The cross-sectional area of chamber inlet C, in the direction normal to gas flow path.