# Positive-Displacement Compressor (G)

Positive displacement compressor in a gas network

Since R2022a

Libraries:
Simscape / Fluids / Gas / Turbomachinery

## Description

The Positive-Displacement Compressor (G) block represents a positive-displacement compressor, such as a reciprocating piston, rotary screw, rotary vane, or scroll, in a gas network. Port R and port C are mechanical rotational conserving ports associated with the compressor shaft and casing, respectively. When there is positive rotation at port R with respect to port C, gas flows from port A to port B. The block may not be accurate for reversed flow.

The figure shows the steps of a piston compressor on a P-V diagram, which has these states:

• a — The compressor cylinder is full at inlet pressure.

• b — The pressure inside the compressor exceeds that of the outlet, which results in fluid discharge.

• c — The compressor reaches the top of the piston stroke, and only the clearance volume remains in the cylinder.

• d — The pressure inside the cylinder drops below the inlet pressure, which results in fluid intake.

### Mass Flow Rate

The block calculates the mass flow rate as

`$\stackrel{˙}{m}={\eta }_{V}\omega \frac{{V}_{disp}}{{v}_{s}},$`

where:

• is the mass flow rate.

• ω is the angular velocity of port R relative to port C.

• vs is the specific volume at the inlet. The block calculates this value from the nominal inlet conditions.

• Vdisp is the displacement volume that the block uses.

### Displacement Volume

When you set Displacement specification to `Volumetric displacement`, the block uses the Displacement volume parameter as the value for Vdisp.

When you set Displacement specification to `Nominal mass flow rate and shaft speed`, the block calculates the displacement volume as

`${V}_{disp}=\frac{{\stackrel{˙}{m}}_{nominal}{v}_{s,nominal}}{{\omega }_{nominal}{\eta }_{{V}_{nominal}}},$`

where:

• nominal is the value of the Nominal mass flow rate parameter.

• ωnominal is the value of the Nominal shaft speed parameter.

• ηVnominal is the value of the Nominal volumetric efficiency parameter when the Efficiency specification parameter is `Analytical`. When the Volumetric efficiency parameter is `Tabulated`, the block uses the `tablelookup` function to interpolate ηVnominal as a function of the shaft speed and the pressure ratio.

### Volumetric Efficiency

You can parameterize the volumetric efficiency by using analytical values or a lookup table.

Analytical Volumetric Efficiency

When you set Efficiency specification to `Analytical`, the block calculates the volumetric efficiency by using analytical values. When the Thermodynamic model parameter is `Polytropic`, the volumetric efficiency is

`${\eta }_{V}=1+C-C{\left(\frac{{p}_{out}}{{p}_{in}}\right)}^{1}{n}},$`

where pin and pout are the inlet and outlet pressures, respectively, and n is the value of the Polytropic exponent parameter. The block calculates the clearance volume fraction, C, as

`$C=\frac{1-{\eta }_{{V}_{nominal}}}{{p}_{ratio}{}^{1/n}-1},$`

where ηVnominal is the value of the Nominal volumetric efficiency parameter and pratio is the value of the Nominal pressure ratio parameter.

When the Thermodynamic model parameter is `Isentropic`, the volumetric efficiency is

`${\eta }_{V}=1+C-C\left(\frac{{v}_{in}}{{v}_{out}}\right),$`

where vin and vout are the inlet and outlet specific volumes, respectively. The block calculates the clearance volume fraction, C, as

`$\text{C=}\frac{1-{\eta }_{Vnominal}}{v{r}_{nominal}-1}$`

where ${\text{vr}}_{nominal}={v}_{i{n}_{nominal}}/{v}_{ou{t}_{nominal}}$ is the nominal specific volume ratio. The block calculates this value from the nominal inlet conditions and the isentropic efficiency.

Tabulated Volumetric Efficiency

When you set Efficiency specification to `Tabulated`, the block calculates the volumetric efficiency by interpolating the values of the Volumetric efficiency table, eta_vol(pr,w) parameter as a function of the shaft speed and the pressure ratio.

### Continuity Equations

The block conserves mass such that

`${\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}=0,$`

where A and B are the mass flow rates at ports A and B, respectively. The block conserves energy such that

`${\varphi }_{A}+{\varphi }_{B}+{\stackrel{˙}{m}}_{A}\Delta {h}_{t}=0,$`

where Δht is the change in specific total enthalpy and AΔht is the fluid power, which is equal to the mechanical power, torque*ω.

When the Thermodynamic model parameter is `Polytropic`, the fluid power is

`${\stackrel{˙}{W}}_{c}=\omega \frac{n}{n-1}{\eta }_{V}{p}_{in}{V}_{disp}\left[{\left(\frac{{p}_{in}}{{p}_{out}}\right)}^{\frac{n-1}{n}}-1\right],$`

where the block uses the polytropic relationship ${\text{pv}}^{n}=\text{constant}$ to relate pin, pout, vin, and vout.

When the Thermodynamic model parameter is `Isentropic`, the fluid power is

`${\stackrel{˙}{W}}_{C}={\stackrel{˙}{m}}_{A}\Delta {h}_{t}.$`

The block calculates Δht from the isentropic efficiency, ηisen. When the Efficiency specification parameter is `Analytical`, ηisen is equal to the value of the Isentropic efficiency parameter. When the Efficiency specification parameter is `Tabulated`, the block calculates ηisen by interpolating the values of the Isentropic efficiency table, eta_isen(pr,w) parameter as a function of the pressure ratio and the shaft speed.

### Visualizing the Volumetric Efficiency

To visualize the block volumetric efficiency, right-click the block and select Fluids > Plot Volumetric Efficiency.

Each time you modify the block settings, click in the figure window.

When you set Efficiency specification to `Analytical` and Thermodynamic model to `Polytropic`, the block plots the compressor volumetric efficiency against the pressure ratio.

When you set Efficiency specification to `Analytical` and Thermodynamic model to `Isentropic`, the block plots the compressor volumetric efficiency against the pressure ratio at Tout nom, the calculated nominal outlet temperature.

When you set Efficiency specification to `Tabulated`, the block plots the compressor volumetric efficiency against the pressure ratio for each element in the Shaft speed vector, w parameter.

### Assumptions and Limitations

• The block may not be accurate for flow from port B to port A.

• The block assumes that the flow is quasi-steady. The compressor does not accumulate mass.

## Ports

### Conserving

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Gas conserving port associated with the compressor inlet.

Gas conserving port associated with the compressor outlet.

Mechanical rotational conserving port associated with the compressor case.

Mechanical rotational conserving port associated with the compressor shaft.

## Parameters

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### Displacement

Whether to specify the fluid displacement according to volume per cycle or nominal condition values.

Fluid volume displaced per cycle. The block defines a cycle as one compression stroke and one expansion stroke, which occur during a single crank shaft revolution. This volume is the difference between the maximum cylinder volume and the clearance volume, (VaVc). In general, the displacement volume is the volume swept out by the compressor over one shaft revolution.

#### Dependencies

To enable this parameter, set Displacement specification to ```Volumetric displacement```.

Nominal mass flow rate that the block uses to find the displacement volume.

#### Dependencies

To enable this parameter, set Displacement specification to ```Nominal mass flow rate and shaft speed```.

Nominal shaft speed that the block uses to find the displacement volume.

#### Dependencies

To enable this parameter, set Displacement specification to ```Nominal mass flow rate and shaft speed```.

### Efficiency

The table shows how the options for the Thermodynamic model and Efficiency specification parameters affect the availability of dependent efficiency parameters.

Thermodynamic model
`Polytropic``Isentropic`
Efficiency specificationEfficiency specification
`Analytical``Tabulated``Analytical``Tabulated`
Polytropic exponent Polytropic exponent Isentropic efficiency Isentropic efficiency table, eta_isen(pr,w)
Nominal volumetric efficiencyPressure ratio vector, prNominal volumetric efficiencyPressure ratio vector, pr
Shaft speed vector, wShaft speed vector, w
Volumetric efficiency table, eta_vol(pr,w)Volumetric efficiency table, eta_vol(pr,w)

Whether to parameterize the displacement volumetric efficiency by using analytical equations or tabulated data.

Whether to use a polytropic or isentropic thermodynamic model.

Exponent that the block uses when implementing the polytropic pressure-volume relationship.

#### Dependencies

To enable this parameter, set Thermodynamic model to `Polytropic`.

Constant efficiency the block uses when implementing the analytical isentropic pressure-volume relationship.

#### Dependencies

To enable this parameter, set Thermodynamic model to `Isentropic` and Efficiency specification to `Analytical`.

Efficiencies the block uses when implementing the tabulated isentropic pressure-volume relationship.

#### Dependencies

To enable this parameter, set Thermodynamic model to `Isentropic` and Efficiency specification to `Tabulated`.

Nominal volumetric efficiency to compute the clearance volume fraction. The nominal volumetric efficiency is the volumetric efficiency when the compressor operates at nominal conditions.

#### Dependencies

To enable this parameter, set Efficiency specification to `Analytical`.

Vector of pressure ratios that correspond to the compressor volumetric efficiency. Each element corresponds to a row of the Volumetric efficiency table, eta_vol(pr,w) parameter. This parameter must be a vector of size `N`, where `N` is the length of the Volumetric efficiency table, eta_vol(pr,w) parameter.

#### Dependencies

To enable this parameter, set Efficiency specification to `Tabulated`.

Vector of shaft speeds that correspond to the compressor volumetric efficiency. Each element corresponds to a column of the Volumetric efficiency table, eta_vol(pr,w) parameter. This parameter must be a vector of size `M`, where `M` is the width of the Volumetric efficiency table, eta_vol(pr,w) parameter.

#### Dependencies

To enable this parameter, set Efficiency specification to `Tabulated`.

Volumetric efficiency for a given shaft speed and pressure ratio. This parameter must be a matrix of size `N` by `M`, where `N` is the length of the Pressure ratio vector, pr parameter and `M` is the length of the Shaft speed vector, w parameter.

#### Dependencies

To enable this parameter, set Efficiency specification to `Tabulated`.

### Nominal Conditions

The table shows how the options for the Displacement specification, Thermodynamic model, and Efficiency specification parameters affect the availability of dependent nominal condition parameters.

This table shows the dependent parameters when you set the parameter to ```Volumetric displacement```:

When the parameter is `Volumetric displacement`
Efficiency specification
`Analytical``Tabulated`
Thermodynamic modelThermodynamic model
`Polytropic``Isentropic``Polytropic``Isentropic`
Nominal pressure ratioNominal pressure ratio Nominal pressure ratio
Nominal inlet pressureNominal inlet pressure
Nominal inlet temperatureNominal inlet temperature

When you set to `Nominal mass flow rate and shaft speed`, the parameters in Nominal Conditions do not depend on the Efficiency specification or Thermodynamic model parameters.

Nominal pressure ratio that the block uses to compute the clearance volume fraction.

#### Dependencies

To enable this parameter, select any parametrization except when Displacement specification is `Volumetric displacement`, Thermodynamic model is `Polytropic`, and Efficiency specification is `Tabulated`.

Nominal inlet pressure that the block uses the find the displacement volume.

#### Dependencies

To enable this parameter, either set Displacement specification to ```Nominal mass flow rate and shaft speed``` or set Displacement specification to ```Volumetric displacement``` and set Thermodynamic model to `Isentropic`.

Nominal inlet temperature that the block uses to find the displacement volume.

#### Dependencies

To enable this parameter, either set Displacement specification to ```Nominal mass flow rate and shaft speed``` or set Displacement specification to ```Volumetric displacement``` and set Thermodynamic model to `Isentropic`.

### Parameters

Ratio of the fluid power to the mechanical shaft power.

Inlet area associated with port A.

Outlet area associated with port B.

## References

[1] Mitchell, John W., and James E. Braun. Principles of Heating, Ventilation, and Air Conditioning in Buildings. Hoboken, NJ: Wiley, 2013.

## Version History

Introduced in R2022a

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