# Rotational Mechanical Converter (G)

Interface between gas and mechanical rotational networks

**Library:**Simscape / Foundation Library / Gas / Elements

## Description

The Rotational Mechanical Converter (G) block models an interface between a gas network and a mechanical rotational network. The block converts gas pressure into mechanical torque and vice versa. It can be used as a building block for rotary 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 rotation 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 rotational 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 rotation of the moving interface:

$$V={V}_{dead}+{D}_{\mathrm{int}}{\theta}_{\mathrm{int}}{\epsilon}_{\mathrm{int}}$$

where:

*V*_{dead}is the dead volume.*D*_{int}is the interface volume displacement.*θ*_{int}is the interface rotation.*ε*_{int}is the mechanical orientation coefficient. If**Mechanical orientation**is`Pressure at A causes positive rotation of R relative to C`

,*ε*_{int}= 1. If**Mechanical orientation**is`Pressure at A causes negative rotation of R relative to C`

,*ε*_{int}= –1.

If you connect the converter to a Multibody joint, use the physical signal input
port **q** to specify the rotation of port **R**
relative to port **C**. Otherwise, the block calculates the
interface rotation from relative port angular velocities. The interface rotation 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 rotation of R relative to C`

, the interface rotation increases when the gas volume increases from dead volume.If

`Pressure at A causes negative rotation of R relative to C`

, the interface rotation decreases when the gas volume increases from dead volume.

### Torque Balance

Torque balance across the moving interface on the gas volume is

$${\tau}_{\mathrm{int}}=\left({p}_{env}-{p}_{I}\right){D}_{\mathrm{int}}{\epsilon}_{\mathrm{int}}$$

where:

*τ*_{int}is the torque 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.

### 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.

## Ports

### Input

### Conserving

## Parameters

## Extended Capabilities

## Version History

**Introduced in R2016b**