Translational Mechanical Converter (G)

Interface between gas and mechanical translational networks

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Libraries:
Simscape / Foundation Library / Gas / Elements

Description

The Translational Mechanical Converter (G) block models an interface between a gas network and a mechanical translational network. The block converts gas pressure into mechanical force and vice versa. It can be used as a building block for linear actuators.

The converter contains a variable volume of gas. The pressure and temperature evolve based on the compressibility and thermal capacity of this gas volume. The Mechanical orientation parameter lets you specify whether an increase in the gas volume results in a positive or negative displacement of port R relative to port C.

Port A is the gas conserving port associated with the converter inlet. Port H is the thermal conserving port associated with the temperature of the gas inside the converter. Ports R and C are the mechanical translational conserving ports associated with the moving interface and converter casing, respectively.

Mass Balance

Mass conservation equation is similar to that for the Constant Volume Chamber (G) block, with an additional term related to the change in gas volume:

$$\frac{\partial M}{\partial p}\cdot \frac{d{p}_I}{dt}+\frac{\partial M}{\partial T}\cdot \frac{d{T}_I}{dt}+{\rho }_I\frac{dV}{dt}={\dot{m}}_A$$

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.

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

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

• ρ_I is the density of the gas volume.

• V is the volume of gas.

• t is time.

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

Energy Balance

Energy conservation equation is also similar to that for the Constant Volume Chamber (G) block. The additional term accounts for the change in gas volume, as well as the pressure-volume work done by the gas on the moving interface:

$$\frac{\partial U}{\partial p}\cdot \frac{d{p}_I}{dt}+\frac{\partial U}{\partial T}\cdot \frac{d{T}_I}{dt}+{\rho }_I{h}_I\frac{dV}{dt}={\Phi }_A+Q_H$$

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.

• Ф_A is the energy flow rate at port A.

• Q_H is the heat flow rate at port H.

• h_I is the specific enthalpy of the gas volume.

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:

• ρ_I is the density of the gas volume.

• V is the volume of gas.

• h_I is the specific enthalpy of the gas volume.

• Z is the compressibility factor.

• R is the specific gas constant.

• c_pI 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.

Gas Volume

The gas volume depends on the displacement of the moving interface:

$$V=V_{dead}+S_{\mathrm{int}}{x}_{\mathrm{int}}{\epsilon }_{\mathrm{int}}$$

where:

• V_dead is the dead volume.

• S_int is the interface cross-sectional area.

• x_int is the interface displacement.

• ε_int is the mechanical orientation coefficient. If Mechanical orientation is Pressure at A causes positive displacement of R relative to C, ε_int = 1. If Mechanical orientation is Pressure at A causes negative displacement of R relative to C, ε_int = –1.

If you connect the converter to a Multibody joint, use the physical signal input port p to specify the displacement of port R relative to port C. Otherwise, the block calculates the interface displacement from relative port velocities. The interface displacement is zero when the gas volume is equal to the dead volume. Then, depending on the Mechanical orientation parameter value:

• If Pressure at A causes positive displacement of R relative to C, the interface displacement increases when the gas volume increases from dead volume.

• If Pressure at A causes negative displacement of R relative to C, the interface displacement decreases when the gas volume increases from dead volume.

Force Balance

Force balance across the moving interface on the gas volume is

$$F_{\mathrm{int}}=\left(p_{env}-p_I\right)S_{\mathrm{int}}{\epsilon }_{\mathrm{int}}$$

where:

• F_int is the force from port R to port C.

• p_env is the environment pressure.

Variables

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

Nominal values provide a way to specify the expected magnitude of a variable in a model. Using system scaling based on nominal values increases the simulation robustness. Nominal values can come from different sources, one of which is the Nominal Values section in the block dialog box or Property Inspector. For more information, see Modify Nominal Values for a Block Variable.

Assumptions and Limitations

• The converter casing is perfectly rigid.

• There is no flow resistance between port A and the converter interior.

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

• The moving interface is perfectly sealed.

• The block does not model mechanical effects of the moving interface, such as hard stop, friction, and inertia.

Examples

Pneumatic Actuation Circuit

How the Foundation Library gas components can be used to model a controlled pneumatic actuator. The Directional Valve is a masked subsystem created from Variable Local Restriction (G) blocks to model the opening and closing of the flow paths. The Double-Acting Actuator is a masked subsystem created from Translational Mechanical Converter (G) blocks to model the interface between the gas network and the mechanical translational network.

• Open Model

Gamma Stirling Engine

Model a Gamma Stirling engine using gas, thermal, and mechanical Simscape™ components and domains.

• Open Model

Ports

Input

p — Displacement of port R relative to port C, m

physical signal

Input physical signal that passes the position information from a Simscape™ Multibody™ joint. Connect this port to the position sensing port p of the joint. For more information, see Connecting Simscape Networks to Simscape Multibody Joints.

Dependencies

To enable this port, set the Interface displacement parameter to Provide input signal from Multibody joint.

Conserving

A — Converter inlet

gas

Gas conserving port associated with the converter inlet.

H — Temperature inside converter

thermal

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

R — Rod

mechanical translational

Mechanical translational conserving port associated with the moving interface.

C — Case

mechanical translational

Mechanical translational conserving port associated with the converter casing.

Parameters

Mechanical orientation — Select the converter orientation

Pressure at A causes positive displacement of R relative to C (default) | Pressure at A causes negative displacement of R relative to C

Select the relative orientation of the converter with respect to the converter gas volume:

• Pressure at A causes positive displacement of R relative to C — Increase in the gas volume results in a positive displacement of port R relative to port C.

• Pressure at A causes negative displacement of R relative to C — Increase in the gas volume results in a negative displacement of port R relative to port C.

Interface displacement — Select method to determine relative port positions

Calculate from velocity of port R relative to port C (default) | Provide input signal from Multibody joint

Select method to determine displacement of port R relative to port C:

• Calculate from velocity of port R relative to port C — Calculate displacement from relative port velocities, based on the block equations. This is the default method.

• Provide input signal from Multibody joint — Enable the input physical signal port p to pass the displacement information from a Multibody joint. Use this method only when you connect the converter to a Multibody joint by using a Translational Multibody Interface block. For more information, see How to Pass Position Information.

Initial interface displacement — Translational offset of port R relative to port C at the start of simulation

0 m (default)

Translational offset of port R relative to port C at the start of simulation. A value of 0 corresponds to an initial gas volume equal to Dead volume.

Dependencies

Enabled when the Interface displacement parameter is set to Calculate from velocity of port R relative to port C.

• If Mechanical orientation is Pressure at A causes positive displacement of R relative to C, the parameter value must be greater than or equal to 0.

• If Mechanical orientation is Pressure at A causes negative displacement of R relative to C, the parameter value must be less than or equal to 0.

Interface cross-sectional area — The area on which the gas exerts pressure to generate the translational force

0.01 m^2 (default)

The area on which the gas exerts pressure to generate the translational force.

Dead volume — Volume of gas when the interface displacement is 0

1e-5 m^3 (default)

Volume of gas when the interface displacement is 0.

Cross-sectional area at port A — Area normal to flow path at the converter inlet

0.01 m^2 (default)

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

Environment pressure specification — Select a specification method for the environment pressure

Atmospheric pressure (default) | Specified pressure

Select a specification method for the environment pressure:

• Atmospheric pressure — Use the atmospheric pressure, specified by the Gas Properties (G) block connected to the circuit.

• Specified pressure — Specify a value by using the Environment pressure parameter.

Environment pressure — Pressure outside the converter

0.101325 MPa (default)

Pressure outside the converter acting against the pressure of the converter gas volume. A value of 0 indicates that the converter expands into vacuum.

Dependencies

Enabled when the Environment pressure specification parameter is set to Specified pressure.

Extended Capabilities

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

Version History

Introduced in R2016b