Mechanical converter (rotational motion) (P)#

The Mechanical converter (rotational motion) (P) block simulates the conversion of gas pressure to mechanical force and vice versa for rotational motion systems.

Pressure and temperature depend on the compressibility and heat capacity of the gas volume.

1.Equations of mass conservation(1):

(1)#\[\dot{m}=\frac{\delta M}{\delta p}\cdot\frac{dp}{dt}+\frac{\delta M}{\delta T} \cdot \frac{dT}{dt}+\rho \cdot \frac{dV}{dt}\]

where:

\(\dot m\) - mass flow rate through the pneumatic port, [kg/s];

\(\delta M/ \delta p\) - partial derivative of gas mass by pressure at constant temperature and volume;

\(\delta M/ \delta T\) - partial derivative of gas mass by temperature at constant pressure and temperature;

p - pressure of the medium in the chamber, [Pa];

T - medium temperature in the chamber, [℃];

\(ρ\) - density of the medium in the chamber, [kg/m\(^3\)];

V - chamber volume, [m\(^3\)].

For an ideal gas (2), (3):

(2)#\[\delta M/ \delta p=V \cdot \frac {\rho}{p}\]

(3)#\[\delta M/ \delta T=-V \cdot \frac {\rho}{T}\]

2. Energy balance

Law of conservation of energy (4):

(4)#\[\phi+Q=\frac {\delta U}{\delta p}\cdot\frac{dp}{dt}+\frac {\delta U}{\delta T}\cdot\frac{dT}{dt}+\rho \cdot h \cdot \frac{dV}{dt}\]

where:

\(\frac{\delta U}{\delta p}\) - partial derivative of the internal energy of a gas by pressure at constant temperature and volume;

\(\frac{\delta U}{\delta T}\) - partial derivative of the internal energy of a gas by temperature at constant pressure and temperature;

Q - heat flux through the heat port, [W];

h - specific enthalpy of gas;

\(\phi\) - energy fluxe through input port.

For an ideal gas (5), (6):

(5)#\[\delta U/ \delta p=V \cdot (\frac {h}{Z\cdot R \cdot T}-1)\]

(6)#\[\delta U/ \delta T=V \cdot \rho \cdot (c_p - \frac {h}{T})\]

where:

Z - compressibility factor;

R - specific gas constant;

h - enthalpy of the medium in the chamber;

\(c_p\) - isobaric mass heat capacity.

Piston angular velocity (7):

(7)#\[\omega=\frac{d\theta}{dt}\]

Hydraulic chamber volume (8):

(8)#\[V=V_{dead} + D \cdot \theta\]

where:

\(V_{dead}\) - dead volume, [m\(^3\)];

\( heta\) - rotation angle, [rad].

Torque equation (9) :

(9)#\[\tau = D \cdot (p-p_{env})\]

where:

p - pressure in the chamber, [Pa];

p\(_{env}\) - environment pressure, [Pa].

1. Properties#

Property	Default value	Description	Units of measurement
Initial rotation	0.1	-	rad
Volume displacement	0.001	-	m\(^3\)/rad
Dead volume	1e-6	-	m\(^3\)
Input port cross-sectional area	0.01	-	m\(^2\)
Environment gauge pressure specification: Atmosphere Pressure; User Pressure	Atmosphere Pressure	-	-
Environment gauge pressure, available when `Environment gauge pressure specification` - `User Pressure`	10000	-	Pa
Initial gas temperature in the chamber	20.0	-	°C
Initial gas gauge pressure in the chamber	0.0	-	Pa

2. Inputs and outputs#

Parameters	Input/Output	Description	Units of measurement	Port type
Mass flow	Input	-	kg/s	Pneumatics
Heat port	Input	-	-	Heat exchange
Torque	Output	-	N \(\cdot\) m	Rotational motion

In order to see the connection capabilities of the block, you need to go to page Description of block ports