Translational Mechanical Converter (2P)

Interface between two-phase fluid and mechanical translational networks

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Libraries:
Simscape / Foundation Library / Two-Phase Fluid / Elements

Description

The Translational Mechanical Converter (2P) block models an interface between two-phase fluid and mechanical translational networks. The interface converts pressure in the fluid network into force in the mechanical translational network and vice versa.

This block enables you to model a linear actuator powered by a two-phase fluid system. It does not, however, account for mass, friction, or hard stops, common in linear actuators. You can model these effects separately using Simscape™ blocks such as Mass, Translational Friction, and Translational Hard Stop.

Port A represents the inlet through which fluid enters and exits the converter. Ports C and R represent the converter casing and moving interface, respectively. Port H represents the wall through which the converter exchanges heat with its surroundings.

Force Direction

The force direction depends on the mechanical orientation of the converter. If the Mechanical Orientation parameter is set to positive, then a positive flow rate through the inlet tends to translate the moving interface in the positive direction relative to the converter casing.

Positive Mechanical Orientation

If the Mechanical Orientation parameter is set to negative, then a positive mass flow rate through the inlet tends to translate the moving interface in the negative direction relative to the converter casing.

Negative Mechanical Orientation

The flow resistance between port A and the converter interior is assumed negligible. Pressure losses between the two is approximately zero. The pressure at port A is therefore equal to that in the converter:

$$p_A=p_I,$$

where:

• p_A is the pressure at port A.

• p_I is the pressure in the converter.

Similarly, the thermal resistance between port H and the converter interior is assumed negligible. The temperature gradient between the two is approximately zero. The temperature at port H is therefore equal to that in the converter:

$$T_H=T_I,$$

where:

• T_H is the temperature at port H.

• T_I is the temperature in the converter.

Fluid Volume

The volume of fluid in the converter is the sum of the dead and displaced fluid volumes. The dead volume is the amount of fluid left in the converter at a zero interface displacement. This volume enables you to model the effects of dynamic compressibility and thermal capacity even when the interface is in its zero position.

The displacement volume is the amount of fluid added to the converter due to translation of the moving interface. This volume increases with the interface displacement. The total volume in the converter as a function of the interface displacement is

$$V=V_{dead}+S_{int}{x}_{int}{ϵ}_{or},$$

where:

• V is the total volume of fluid in the converter.

• V_dead is the dead volume of the converter.

• S_int is the cross-sectional area of the interface, assumed equal to that of the inlet.

• x_int is the displacement of the moving interface.

• ∊_or is the mechanical orientation of the converter (1 if increase in fluid pressure causes positive displacement of R relative to C, -1 if increase in fluid pressure causes negative displacement of R relative to C).

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, according to the block equations. The interface displacement is zero when the fluid 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 fluid volume increases from dead volume.

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

Force Balance

At equilibrium, the internal pressure in the converter counteracts the external pressure of its surroundings and the force exerted by the mechanical network on the moving interface. This force is the reverse of that applied by the fluid network. The force balance in the converter is therefore

$$p_I{S}_{int}=p_{atm}{S}_{int}-F_{int}{ϵ}_{or},$$

where:

• p_atm is the environmental pressure outside the converter.

• F_int is the magnitude of the force exerted by the fluid network on the moving interface.

Energy Balance

The total energy in the converter can change due to energy flow through the inlet, heat flow through the converter wall, and work done by the fluid network on the mechanical network. The energy flow rate, given by the energy conservation equation, is therefore

$$\dot{E}={\varphi }_A+{\varphi }_H-p_I{S}_{int}{\dot{x}}_{int}{ϵ}_{or},$$

where:

• E is the total energy of the fluid in the converter.

• ϕ_A is the energy flow rate into the converter through port A.

• ϕ_H is the heat flow rate into the converter through port H.

Taking the fluid kinetic energy in the converter to be negligible, the total energy of the fluid reduces to:

$$E=M{u}_I,$$

where:

• M is the fluid mass in the converter.

• u_I is the specific internal energy of the fluid in the converter.

Mass Balance

The fluid mass in the converter can change due to flow through the inlet, represented by port A. The mass flow rate, given by the mass conservation equation, is therefore

$$\dot{M}={\dot{m}}_A,$$

where:

• $${\dot{m}}_A$$ is the mass flow rate into the converter through port A.

A change in fluid mass can accompany a change in fluid volume, due to translation of the moving interface. It can also accompany a change in mass density, due to an evolving pressure or specific internal energy in the converter. The mass rate of change in the converter is then

$$\dot{M}=\left[{\left(\frac{\partial \rho }{\partial p}\right)}_u{\dot{p}}_I+{\left(\frac{\partial \rho }{\partial u}\right)}_p{\dot{u}}_I\right]V+\frac{S_{int}{\dot{x}}_{int}{ϵ}_{or}}{v_I},$$

where:

• $${\left(\frac{\partial \rho }{\partial p}\right)}_u$$ is the partial derivative of density with respect to pressure at constant specific internal energy.

• $${\left(\frac{\partial \rho }{\partial u}\right)}_p$$ is the partial derivative of density with respect to specific internal energy at constant pressure.

• v_I is the specific volume of the fluid in the converter.

The block blends the density partial derivatives of the various domains using a cubic polynomial function. At a vapor quality of 0–0.1, this function blends the derivatives of the subcooled liquid and two-phase mixture domains. At a vapor quality of 0.9–1, it blends those of the two-phase mixture and superheated vapor domains.

The smoothed density partial derivatives introduce into the original mass conservation equation undesirable numerical errors. To correct for these errors, the block adds the correction term

$${ϵ}_M=\frac{M-V/v_I}{\tau },$$

where:

• ∊_M is the correction term.

• τ is the phase-change time constant—the characteristic duration of a phase change event. This constant ensures that phase changes do not occur instantaneously, effectively introducing a time lag whenever they occur.

The final form of the mass conservation equation is

$$\left[{\left(\frac{\partial \rho }{\partial p}\right)}_u{\dot{p}}_I+{\left(\frac{\partial \rho }{\partial u}\right)}_p{\dot{u}}_I\right]V+\frac{D_{vol}{\dot{\theta }}_{int}{ϵ}_{or}}{v_I}={\dot{m}}_A+{ϵ}_M.$$

The block uses this equation to calculate the internal pressure in the converter given the mass flow rate through the inlet.

Assumptions and Limitations

• The converter walls are rigid. They do not deform under pressure.

• The flow resistance between port A and the converter interior is negligible. The pressure is the same at port A and in the converter interior.

• The thermal resistance between port H and the converter interior is negligible. The temperature is the same at port H and in the converter interior.

• The moving interface is perfectly sealed. No fluid leaks across the interface.

• Mechanical effects such as hard stops, inertia, and friction, are ignored.

Examples

Cavitation in Two-Phase Fluid

How two-phase fluid components can be used to simulate cavitation. The model is a translational mechanical converter driven by an oscillating pressure source. During the negative portion of the pressure source cycle, the fluid cavitates, reducing the force produced by the converter. As a result, the converter displacement drifts and does not return to the starting position.

• Open Model

Calculate Energy Required to Boil Water in the Two-Phase Fluid Domain

Calculate the amount of energy needed to lift the lid of a container filled with water by boiling the water. First, you solve for the solution analytically, and then build a Simscape™ model to represent the physical scenario and validate your results.

• Open Live Script

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

two-phase fluid

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

H — Temperature inside converter

thermal

Thermal conserving port associated with the temperature of the liquid 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

Main

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 alignment of moving interface with respect to the converter fluid volume:

• Pressure at A causes positive displacement of R relative to C — Increase in the fluid 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 fluid 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 mass balance 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) | scalar

Translational offset of port R relative to port C at the start of simulation. A value of 0 corresponds to a total fluid volume in the converter 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 fluid exerts pressure to generate the translational force

0.01 m^2 (default) | positive scalar

The area on which the fluid exerts pressure to generate the translational force. This area does not have to be the same as the inlet area. Pressure losses due to changes in flow area inside the converter are ignored.

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

1e-5 m^3 (default) | positive scalar

Volume of fluid when the interface displacement is 0.

Cross-sectional area at port A — Flow area of the converter inlet

0.01 m^2 (default) | positive scalar

Flow area of the converter inlet, represented by port A. This area does not have to be the same as the interface cross-sectional area. Pressure losses due to changes in flow area inside the converter are ignored.

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

Atmospheric pressure (default) | Specified pressure

Select a specification method for the pressure outside the converter:

• Atmospheric pressure — Use the atmospheric pressure, specified by the Two-Phase Fluid Properties (2P) 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) | positive scalar

Pressure outside the converter acting against the pressure of the converter fluid 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.

Effects and Initial Conditions

Initial fluid energy specification — Thermodynamic variable to use in defining initial conditions

Temperature (default) | Vapor quality | Vapor void fraction | Specific enthalpy | Specific internal energy

Thermodynamic variable in terms of which to define the initial conditions of the block.

The value for the Initial fluid energy specification parameter limits the available initial states for the two-phase fluid. When Initial fluid energy specification is:

• Temperature — Specify an initial state that is a subcooled liquid or superheated vapor. You cannot specify a liquid-vapor mixture because the temperature is constant across the liquid-vapor mixture region.

• Vapor quality — Specify an initial state that is a liquid-vapor mixture. You cannot specify a subcooled liquid or a superheated vapor because the liquid mass fraction is 0 and 1, respectively, across the whole region. Additionally, the block limits the pressure to below the critical pressure.

• Vapor void fraction — Specify an initial state that is a liquid-vapor mixture. You cannot specify a subcooled liquid or a superheated vapor because the liquid mass fraction is 0 and 1, respectively, across the whole region. Additionally, the block limits the pressure to below the critical pressure.

• Specific enthalpy — Specify the specific enthalpy of the fluid. The block does not limit the initial state.

• Specific internal energy — Specify the specific internal energy of the fluid. The block does not limit the initial state.

Initial pressure — Absolute pressure at start of simulation

0.101325 MPa (default) | positive scalar

Pressure in the chamber at the start of simulation, specified against absolute zero.

Initial temperature — Absolute temperature at start of simulation

293.15 K (default) | positive scalar

Temperature in the chamber at the start of simulation, specified against absolute zero.

Dependencies

To enable this parameter, set Initial fluid energy specification to Temperature.

Initial vapor quality — Mass fraction of vapor at start of simulation

0.5 (default) | unitless scalar between 0 and 1

Mass fraction of vapor in the chamber at the start of simulation.

Dependencies

To enable this parameter, set Initial fluid energy specification to Vapor quality.

Initial vapor void fraction — Volume fraction of vapor at start of simulation

0.5 (default) | unitless scalar between 0 and 1

Volume fraction of vapor in the chamber at the start of simulation.

Dependencies

To enable this parameter, set Initial fluid energy specification to Vapor void fraction.

Initial specific enthalpy — Specific enthalpy at start of simulation

1500 kJ/kg (default) | positive scalar

Specific enthalpy of the fluid in the chamber at the start of simulation.

Dependencies

To enable this parameter, set Initial fluid energy specification to Specific enthalpy.

Initial specific internal energy — Specific internal energy at start of simulation

1500 kJ/kg (default) | positive scalar

Specific internal energy of the fluid in the chamber at the start of simulation.

Dependencies

To enable this parameter, set Initial fluid energy specification to Specific internal energy.

Phase change time constant — Characteristic duration of a phase-change event

0.1 s (default) | positive scalar

Characteristic time to equilibrium of a phase-change event taking place in the chamber. Increase this parameter to slow the rate of phase change or decrease it to speed the rate.

Extended Capabilities

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

Version History

Introduced in R2015b