Positive-Displacement Compressor (2P)

Positive displacement compressor in a two-phase fluid network

Since R2022a

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Libraries:
Simscape / Fluids / Two-Phase Fluid / Fluid Machines

Description

The Positive-Displacement Compressor (2P) block represents a positive-displacement compressor, such as a reciprocating piston, rotary screw, rotary vane, or scroll, in a two-phase fluid 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, two-phase fluid flows from port A to port B. The block may not be accurate for reversed flow.

The figure shows the steps of a positive-displacement 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

$$\dot{m}={\eta }_V\omega \frac{V_{disp}}{v_s},$$

where:

• ṁ is the mass flow rate.

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

• v_s is the specific volume at the inlet. The block calculates this value based on the Nominal conditions specification parameter and specified nominal inlet conditions.

• V_disp 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 V_disp.

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

$$V_{disp}=\frac{{\dot{m}}_{nominal}{v}_{s,nominal}}{{\omega }_{nominal}{\eta }_{V_{nominal}}},$$

where:

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

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

• η_Vnominal is the value of the Nominal volumetric efficiency parameter when the Efficiency specification parameter is Analytical. When the Efficiency specification parameter is Tabulated, the block uses the tablelookup function to interpolate η_Vnominal 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

$${\eta }_V=1+C-C{\left(\frac{p_{out}}{p_{in}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.},$$

where p_in and p_out 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 η_Vnominal is the value of the Nominal volumetric efficiency parameter and p_ratio 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 v_in and v_out 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 depending on the choice for the Nominal conditions specification parameter.

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

$${\dot{m}}_A+{\dot{m}}_B=0,$$

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

$${\varphi }_A+{\varphi }_B+{\dot{m}}_A\Delta h_t=0,$$

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

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

$${\dot{W}}_c=\omega \frac{n}{n-1}{\eta }_V{p}_{in}{V}_{disp}\left[{\left(\frac{p_{in}}{p_{out}}\right)}^{\frac{n-1}{n}}-1\right],$$

where the block uses the polytropic relationship $${\text{pv}}^n=\text{constant}$$ to relate p_in, p_out, v_in, and v_out.

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

$${\dot{W}}_C={\dot{m}}_A\Delta h_t.$$

The block calculates Δh_t from the isentropic efficiency, η_isen. When Efficiency specification is Analytical, η_isen is equal to the value of the Isentropic efficiency parameter. When Efficiency specification is Tabulated, the block calculates η_isen 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 Reload Data 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 the block nominal conditions.

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.

• The block is designed to operate in superheated vapor. The block may not be accurate in two-phase mixture or subcooled liquid.

Examples

Liquid Air Energy Storage System

Models a grid-scale energy storage system based on cryogenic liquid air. When there is excess power, the system liquefies ambient air based on a variation of the Claude cycle. The cold liquid air is stored in a low-pressure insulated tank until needed. When there is high power demand, the system expands the stored liquid air to produce power based on the Rankine cycle.

• Open Model

Refrigeration Cycle (Air Conditioning)

A refrigeration cycle for a home air conditioning system. See Model a Refrigeration Cycle for the recommended steps to build this model in the two-phase fluid domain.

• Open Model

Vehicle HVAC System

Models the heating and cooling system of a passenger car. The cabin is represented as a volume of moist air exchanging heat with the external environment. The blower drives moist air through the evaporator, blend door, and heater core before returning to the cabin. The blend door controls the amount of air flow through the heater core. The recirculation door controls whether air is brought in from the external environment or from within the cabin.

• Open Model

Ports

Conserving

A — Compressor inlet

two-phase fluid

Two-phase fluid conserving port associated with the compressor inlet.

B — Compressor outlet

two-phase fluid

Two-phase fluid conserving port associated with the compressor outlet.

C — Compressor casing

mechanical rotational

Mechanical rotational conserving port associated with the compressor case.

R — Compressor shaft

mechanical rotational

Mechanical rotational conserving port associated with the compressor shaft.

Parameters

Displacement

Displacement specification — Option to specify displacement by volume or nominal conditions

Volumetric displacement (default) | Nominal mass flow rate and shaft speed

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

Displacement volume — Displacement volume

130 cm^3/rev (default) | positive scalar

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, (V_a — V_c). 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 — Nominal mass flow rate

0.01 kg/s (default) | positive scalar

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 — Nominal shaft speed

1000 rpm (default) | positive scalar

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 specification		Efficiency specification
Analytical	Tabulated	Analytical	Tabulated
Polytropic exponent	Polytropic exponent	Isentropic efficiency	Isentropic efficiency table, eta_isen(pr,w)
Nominal volumetric efficiency	Pressure ratio vector, pr	Nominal volumetric efficiency	Pressure ratio vector, pr
	Shaft speed vector, w		Shaft speed vector, w
	Volumetric efficiency table, eta_vol(pr,w)		Volumetric efficiency table, eta_vol(pr,w)

Efficiency specification — Option to parameterize volumetric efficiency analytically or by using tables

Analytical (default) | Tabulated

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

Thermodynamic model — Thermodynamic model parameterization option

Polytropic (default) | Isentropic

Whether to use a polytropic or isentropic thermodynamic model.

Polytropic exponent — Polytropic exponent

1.2 (default) | positive scalar

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

Dependencies

To enable this parameter, set Thermodynamic model to Polytropic.

Isentropic efficiency — Isentropic efficiency

0.65 (default) | positive scalar

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.

Isentropic efficiency table, eta_isen(pr,w) — Isentropic efficiency table

[.95, .9, .85, .8, .75; .925, .875, .825, .775, .725; .875, .825, .775, .725, .675; .75, .7, .65, .6, .55] (default) | matrix

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 — Nominal volumetric efficiency

0.95 (default) | positive scalar

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.

Pressure ratio vector, pr — Pressure ratio vector

[2, 4, 6, 8] (default) | vector

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.

Shaft speed vector, w — Shaft speed vector

[800, 1500, 2800, 3300, 4000] rpm (default) | vector

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 table, eta_vol(pr,w) — Volumetric efficiency

[.95, .9, .85, .8, .75; .925, .875, .825, .775, .725; .875, .825, .775, .725, .675; .75, .7, .65, .6, .55] (default) | matrix

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

These tables show how the options for the Displacement specification, Thermodynamic model, Efficiency specification, and Nominal conditions specification parameters affect the availability of dependent nominal condition parameters.

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

When the Displacement specification parameter is Volumetric displacement
Efficiency specification
Analytical				Tabulated
Thermodynamic model				Thermodynamic model
Polytropic		Isentropic		Polytropic		Isentropic
Nominal conditions specification		Nominal conditions specification				Nominal conditions specification
Nominal pressure	Nominal saturation temperatures	Nominal pressure	Nominal saturation temperatures			Nominal pressure	Nominal saturation temperatures
Nominal pressure ratio	Nominal evaporating temperature	Nominal pressure ratio	Nominal evaporating temperature			Nominal pressure ratio	Nominal evaporating temperature
	Nominal condensing temperature	Nominal inlet pressure	Nominal condensing temperature			Nominal inlet pressure	Nominal condensing temperature
		Nominal inlet temperature	Nominal evaporator superheat			Nominal inlet temperature

When you set Displacement specification to Nominal mass flow rate and shaft speed, the parameters in Nominal Conditions only depend on the Nominal conditions specification parameter:

When the Displacement specification parameter is Nominal mass flow rate and shaft speed
Nominal conditions specification
Nominal pressure				Nominal saturation temperatures
Nominal pressure ratio				Nominal evaporating temperature
Nominal inlet pressure				Nominal condensing temperature
Nominal inlet temperature				Nominal evaporator superheat

Nominal pressure ratio — Nominal pressure ratio

5 (default) | positive scalar

Nominal pressure ratio to compute the clearance volume fraction.

Dependencies

To enable this parameter, set Nominal conditions specification to Nominal pressure and:

• Set Displacement specification to Nominal mass flow rate and shaft speed.

• Set Displacement specification to Volumetric displacement and Thermodynamic model to Isentropic.

• Set Displacement specification to Volumetric displacement, Thermodynamic model to Polytropic, and Efficiency specification to Analytical.

Nominal conditions specification — Option to specify pressure ratio or refrigeration cycle saturation temperatures

Nominal pressure (default) | Nominal saturation temperatures

Whether to specify the nominal conditions that determine the displacement volume using the nominal outlet-to-inlet pressure ratio or the nominal evaporating and condensing temperatures.

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 — Nominal inlet pressure

0.101325 MPa (default) | positive scalar

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

Dependencies

To enable this parameter, set Nominal conditions specification to Nominal pressure and either:

• Set Displacement specification to Nominal mass flow rate and shaft speed.

• Set Displacement specification to Volumetric displacement and Thermodynamic model to Isentropic.

Nominal inlet temperature — Nominal inlet temperature

373.15 K (default) | positive scalar

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

Dependencies

To enable this parameter, set Nominal conditions specification to Nominal pressure and either:

• Set Displacement specification to Nominal mass flow rate and shaft speed.

• Set Displacement specification to Volumetric displacement and Thermodynamic model to Isentropic.

Nominal evaporating temperature — Nominal evaporating temperature

373.15 K (default) | positive scalar

Nominal evaporating temperature that the block uses the find the displacement volume. This value is the saturation temperature in the evaporator. The block assumes the nominal compressor inlet pressure is the pressure that corresponds to this saturation temperature.

The evaporating temperature is also known as the saturated suction temperature (SST).

Dependencies

To enable this parameter, set Nominal conditions specification to Nominal saturation temperature and any other parameter choice except when Displacement specification is Volumetric displacement, Thermodynamic model is Polytropic, and Efficiency specification is Tabulated.

Nominal condensing temperature — Nominal condensing temperature

398.15 K (default) | positive scalar

Nominal condensing temperature that the block uses to find the displacement volume. This value is the saturation temperature in the condenser. The block assumes that the nominal compressor outlet pressure is the pressure that corresponds to this saturation temperature.

The condensing temperature is also known as the saturated discharge temperature (SDT).

Dependencies

To enable this parameter, set Nominal conditions specification to Nominal saturation temperature and any other parameter choice except when Displacement specification is Volumetric displacement, Thermodynamic model is Polytropic, and Efficiency specification is Tabulated.

Nominal evaporator superheat — Nominal saturation excess temperature

10 deltaK (default) | scalar

Nominal excess temperature beyond the saturation point in the evaporator outlet, which the block assumes to be connected to the compressor inlet.

Dependencies

To enable this parameter, set Nominal conditions specification to Nominal saturation temperature and either:

• Set Displacement specification to Nominal mass flow rate and shaft speed.

• Set Displacement specification to Volumetric displacement, Thermodynamic model to Isentropic, and Efficiency specification to Analytical.

Parameters

Mechanical efficiency — Compressor mechanical efficiency

0.9 (default) | positive scalar

Ratio of the fluid power to the mechanical shaft power.

Inlet area at port A — Inlet area

0.01 m^2 (default) | positive scalar

Inlet area associated with port A.

Outlet area at port B — Outlet area

0.01 m^2 (default) | positive scalar

Outlet area associated with port B.

Report when fluid is not fully vapor — Two-phase fluid condition check

Warning (default) | None | Error

Action to perform when the fluid at port A does not meet vapor conditions.

References

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

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2022a

R2023b: Model a polytropic or isentropic process

You can now use the Positive-Displacement Compressor (2P) block to model the positive displacement process as polytropic or isentropic. Previously, the block assumed a polytropic process. Use the new Thermodynamic model parameter to select the efficiency model.