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Interface between moist air and mechanical rotational networks

**Library:**Simscape / Foundation Library / Moist Air / Elements

The Rotational Mechanical Converter (MA) block models an interface between a moist air network and a mechanical rotational network. The block converts moist air pressure into mechanical torque and vice versa. You can use it as a building block for rotary actuators.

The converter contains a variable volume of moist air. The pressure and temperature
evolve based on the compressibility and thermal capacity of this moist air volume.
Liquid water condenses out of the moist air volume when it reaches saturation. The
**Mechanical orientation** parameter lets you specify whether an
increase in the moist air volume inside the converter results in a positive or negative
rotation of port **R** relative to port **C**.

The block equations use these symbols. Subscripts `a`

,
`w`

, and `g`

indicate the properties of dry air, water
vapor, and trace gas, respectively. Subscript `ws`

indicates water vapor at
saturation. Subscripts `A`

, `H`

, and `S`

indicate the appropriate port. Subscript `I`

indicates the properties of
the internal moist air volume.

$$\dot{m}$$ | Mass flow rate |

Φ | Energy flow rate |

Q | Heat flow rate |

p | Pressure |

ρ | Density |

R | Specific gas constant |

V | Volume of moist air inside the converter |

c_{v} | Specific heat at constant volume |

h | Specific enthalpy |

u | Specific internal energy |

x | Mass fraction
(x_{w} is specific humidity,
which is another term for water vapor mass fraction) |

y | Mole fraction |

φ | Relative humidity |

r | Humidity ratio |

T | Temperature |

t | Time |

The net flow rates into the moist air volume inside the converter are

$$\begin{array}{l}{\dot{m}}_{net}={\dot{m}}_{A}-{\dot{m}}_{condense}+{\dot{m}}_{wS}+{\dot{m}}_{gS}\\ {\Phi}_{net}={\Phi}_{A}+{Q}_{H}-{\Phi}_{condense}+{\Phi}_{S}\\ {\dot{m}}_{w,net}={\dot{m}}_{wA}-{\dot{m}}_{condense}+{\dot{m}}_{wS}\\ {\dot{m}}_{g,net}={\dot{m}}_{gA}+{\dot{m}}_{gS}\end{array}$$

where:

$$\dot{m}$$

_{condense}is the rate of condensation.*Φ*_{condense}is the rate of energy loss from the condensed water.*Φ*_{S}is the rate of energy added by the sources of moisture and trace gas. $${\dot{m}}_{wS}$$ and $${\dot{m}}_{gS}$$ are the mass flow rates of water and gas, respectively, through port**S**. The values of $${\dot{m}}_{wS}$$, $${\dot{m}}_{gS}$$, and*Φ*_{S}are determined by the moisture and trace gas sources connected to port**S**of the converter.

Water vapor mass conservation relates the water vapor mass flow rate to the dynamics of the moisture level in the internal moist air volume:

$$\frac{d{x}_{wI}}{dt}{\rho}_{I}V+{x}_{wI}{\dot{m}}_{net}={\dot{m}}_{w,net}$$

Similarly, trace gas mass conservation relates the trace gas mass flow rate to the dynamics of the trace gas level in the internal moist air volume:

$$\frac{d{x}_{gI}}{dt}{\rho}_{I}V+{x}_{gI}{\dot{m}}_{net}={\dot{m}}_{g,net}$$

Mixture mass conservation relates the mixture mass flow rate to the dynamics of the pressure, temperature, and mass fractions of the internal moist air volume:

$$\left(\frac{1}{{p}_{I}}\frac{d{p}_{I}}{dt}-\frac{1}{{T}_{I}}\frac{d{T}_{I}}{dt}\right){\rho}_{I}V+\frac{{R}_{a}-{R}_{w}}{{R}_{I}}\left({\dot{m}}_{w,net}-{x}_{w}{\dot{m}}_{net}\right)+\frac{{R}_{a}-{R}_{g}}{{R}_{I}}\left({\dot{m}}_{g,net}-{x}_{g}{\dot{m}}_{net}\right)+{\rho}_{I}\dot{V}={\dot{m}}_{net}$$

where $$\dot{V}$$ is the rate of change of the converter volume.

Finally, energy conservation relates the energy flow rate to the dynamics of the pressure, temperature, and mass fractions of the internal moist air volume:

$${\rho}_{I}{c}_{vI}V\frac{d{T}_{I}}{dt}+\left({u}_{wI}-{u}_{aI}\right)\left({\dot{m}}_{w,net}-{x}_{w}{\dot{m}}_{net}\right)+\left({u}_{gI}-{u}_{aI}\right)\left({\dot{m}}_{g,net}-{x}_{g}{\dot{m}}_{net}\right)+{u}_{I}{\dot{m}}_{net}={\Phi}_{net}-{p}_{I}\dot{V}$$

The equation of state relates the mixture density to the pressure and temperature:

$${p}_{I}={\rho}_{I}{R}_{I}{T}_{I}$$

The mixture specific gas constant is

$${R}_{I}={x}_{aI}{R}_{a}+{x}_{wI}{R}_{w}+{x}_{gI}{R}_{g}$$

The converter 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
moist air volume inside the converter 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 moist air volume increases from dead volume.If

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

, the interface rotation decreases when the moist air volume increases from dead volume.

The torque balance on the mechanical interface 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.

Flow resistance and thermal resistance are not modeled in the converter:

$$\begin{array}{l}{p}_{A}={p}_{I}\\ {T}_{H}={T}_{I}\end{array}$$

When the moist air volume reaches saturation, condensation may occur. The specific humidity at saturation is

$${x}_{wsI}={\phi}_{ws}\frac{{R}_{I}}{{R}_{w}}\frac{{p}_{wsI}}{{p}_{I}}$$

where:

*φ*_{ws}is the relative humidity at saturation (typically 1).*p*_{wsI}is the water vapor saturation pressure evaluated at*T*_{I}.

The rate of condensation is

$${\dot{m}}_{condense}=\{\begin{array}{ll}0,\hfill & \text{if}{x}_{wI}\le {x}_{wsI}\hfill \\ \frac{{x}_{wI}-{x}_{wsI}}{{\tau}_{condense}}{\rho}_{I}V,\hfill & \text{if}{x}_{wI}{x}_{wsI}\hfill \end{array}$$

where *τ*_{condense} is the value of the
**Condensation time constant** parameter.

The condensed water is subtracted from the moist air volume, as shown in the conservation equations. The energy associated with the condensed water is

$${\Phi}_{condense}={\dot{m}}_{condense}\left({h}_{wI}-\Delta {h}_{vapI}\right)$$

where *Δh*_{vapI} is the specific enthalpy of
vaporization evaluated at *T*_{I}.

Other moisture and trace gas quantities are related to each other as follows:

$$\begin{array}{l}{\phi}_{wI}=\frac{{y}_{wI}{p}_{I}}{{p}_{wsI}}\\ {y}_{wI}=\frac{{x}_{wI}{R}_{w}}{{R}_{I}}\\ {r}_{wI}=\frac{{x}_{wI}}{1-{x}_{wI}}\\ {y}_{gI}=\frac{{x}_{gI}{R}_{g}}{{R}_{I}}\\ {x}_{aI}+{x}_{wI}+{x}_{gI}=1\end{array}$$

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

The converter casing is perfectly rigid.

Flow resistance between the converter inlet and the moist air volume is not modeled. Connect a Local Restriction (MA) block or a Flow Resistance (MA) block to port

**A**to model pressure losses associated with the inlet.Thermal resistance between port

**H**and the moist air volume is not modeled. Use Thermal library blocks to model thermal resistances between the moist air mixture and the environment, including any thermal effects of a chamber wall.The moving interface is perfectly sealed.

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