Local Restriction (MA)

Restriction in flow area in moist air network

Libraries:
Simscape / Foundation Library / Moist Air / Elements

Description

The Local Restriction (MA) block models the pressure drop due to a localized reduction in flow area, such as a valve or an orifice, in a moist air network. Choking occurs when the restriction reaches the sonic condition.

Ports A and B represent the restriction inlet and outlet. The input physical signal at port AR specifies the restriction area. Alternatively, you can specify a fixed restriction area as a block parameter.

The block icon changes depending on the value of the Restriction type parameter.

Restriction Type Block Icon

Restriction Type	Block Icon
`Variable`
`Fixed`

Variable

Fixed

The restriction is adiabatic. It does not exchange heat with the environment.

The restriction consists of a contraction followed by a sudden expansion in flow area. The moist air accelerates during the contraction, causing the pressure to drop. The moist air separates from the wall during the sudden expansion, causing the pressure to recover only partially due to the loss of momentum.

Local Restriction Schematic

Caution

Moist air flow through this block can choke. If a Mass Flow Rate Source (MA) block or a Controlled Mass Flow Rate Source (MA) block connected to the Local Restriction (MA) block specifies a greater mass flow rate than the possible choked mass flow rate, the simulation generates an error. For more information, see Choked Flow.

The block equations use these symbols.

$\dot{m}$	Mass flow rate
Φ	Energy flow rate
p	Pressure
ρ	Density
R	Specific gas constant
S	Cross-sectional area
C_d	Discharge coefficient
h	Specific enthalpy
c_p	Specific heat at constant pressure
T	Temperature

Subscripts a, w, d, and g indicate the properties of dry air, water vapor, water droplets, and trace gas, respectively. Subscripts lam and tur indicate the laminar and turbulent regime, respectively. Subscripts A and B indicate the appropriate port. Subscript R indicates the restriction.

Mass balance:

$\begin{array}{l} {\dot{m}}_{A} + {\dot{m}}_{B} = 0 \\ {\dot{m}}_{w A} + {\dot{m}}_{w B} = 0 \\ {\dot{m}}_{g A} + {\dot{m}}_{g B} = 0 \\ {\dot{m}}_{d A} + {\dot{m}}_{d B} = 0 \end{array}$

Energy balance:

$Φ_{A} + Φ_{B} = 0$

When the flow is not choked, the mixture mass flow rate (positive from port A to port B) in the turbulent regime is

$\begin{array}{l} {\dot{m}}_{t u r} = C_{d} S_{R} (p_{A} - p_{B}) \sqrt{\frac{2 ρ_{R}}{| p_{A} - p_{B} | K_{t u r}}} \\ K_{t u r} = (1 + \frac{S_{R}}{S}) (1 - \frac{ρ_{R}}{ρ_{i n}} \frac{S_{R}}{S}) - 2 \frac{S_{R}}{S} (1 - \frac{ρ_{R}}{ρ_{o u t}} \frac{S_{R}}{S}) \end{array}$

Subscripts in and out indicate the inlet and outlet, respectively. If p_A ≥ p_B, the inlet is port A and the outlet is port B; otherwise, they are reversed. The cross-sectional area S is assumed to be equal at ports A and B. S_R is the area at the restriction.

The mixture mass flow rate equation is derived by combining the equations from two control volume analyses:

Momentum balance for flow area contraction from the inlet to the restriction
Momentum balance for sudden flow area expansion from the restriction to the outlet

In the analysis for the flow area contraction, pressure p_in acts on the area at the inlet, S, and pressure p_R acts on the area at the restriction, S_R. The pressure acting on the area outside the restriction, S − S_R, is assumed to be (p_inS + p_RS_R)/(S + S_R).

In the analysis for the flow area expansion, the pressure acting on both the area at the restriction, S_R, and the area outside the restriction, S − S_R, is assumed to be p_R, because of flow separation from the restriction. The pressure acting on the area at the outlet , S, is equal to p_out.

The mixture mass flow rate (positive from port A to port B) in the laminar regime is linearized with respect to the pressure difference:

${\dot{m}}_{l a m} = C_{d} S_{R} (p_{A} - p_{B}) \sqrt{\frac{2 ρ_{R}}{Δ p_{t h r e s h o l d} {(1 - \frac{S_{R}}{S})}^{2}}}$

where the threshold for transition between the laminar and turbulent regime is defined based on the laminar flow pressure ratio, B_lam, as

$Δ p_{t h r e s h o l d} = (\frac{p_{A} + p_{B}}{2}) (1 - B_{l a m})$

When $| p_{A} - p_{B} | \geq Δ p_{t h r e s h o l d}$ , the flow is assumed to be turbulent and therefore ${\dot{m}}_{u n c h o k e d} = {\dot{m}}_{t u r}$ .

When $| p_{A} - p_{B} | < Δ p_{t h r e s h o l d}$ , ${\dot{m}}_{u n c h o k e d}$ smoothly transitions to ${\dot{m}}_{l a m}$ .

When the flow is choked, the velocity at the restriction is equal to the speed of sound and cannot increase any further. Assuming the flow is choked, the mixture mass flow rate is

${\dot{m}}_{c h o k e d} = C_{d} S_{R} p_{R} \sqrt{\frac{γ_{R}}{R T_{R}}}$

where $γ_{R} = c_{p R} / (c_{p R} - R)$ . Therefore, the actual mixture mass flow rate is equal to ${\dot{m}}_{u n c h o k e d}$ , but is limited in magnitude by ${\dot{m}}_{c h o k e d}$ :

${\dot{m}}_{A} = {\begin{array}{l} - {\dot{m}}_{c h o k e d}, & if {\dot{m}}_{u n c h o k e d} \leq - {\dot{m}}_{c h o k e d} \\ {\dot{m}}_{u n c h o k e d}, & if - {\dot{m}}_{c h o k e d} < {\dot{m}}_{u n c h o k e d} < {\dot{m}}_{c h o k e d} \\ {\dot{m}}_{c h o k e d}, & if {\dot{m}}_{u n c h o k e d} \geq {\dot{m}}_{c h o k e d} \end{array}$

The expression for the pressure at the restriction is obtained by considering the momentum balance for flow area contraction from the inlet to the restriction only.

$p_{R} = p_{i n} - \frac{1}{2 ρ_{R}} {(\frac{{\dot{m}}_{A}}{C_{d} S_{R}})}^{2} (1 + \frac{S_{R}}{S}) (1 - \frac{ρ_{R}}{ρ_{i n}} \frac{S_{R}}{S})$

The local restriction is assumed adiabatic, so the mixture specific total enthalpies are equal. Therefore, the changes in mixture specific enthalpies are:

$\begin{array}{l} h_{A} - h_{R} = (\frac{1}{ρ_{R}^{2} S_{R}^{2}} - \frac{1}{ρ_{A}^{2} S^{2}}) \frac{{\dot{m}}_{A}^{2}}{2 C_{D}^{2}} \\ h_{B} - h_{R} = (\frac{1}{ρ_{R}^{2} S_{R}^{2}} - \frac{1}{ρ_{B}^{2} S^{2}}) \frac{{\dot{m}}_{B}^{2}}{2 C_{D}^{2}} \end{array}$

Assumptions and Limitations

The restriction is adiabatic. It does not exchange heat with the environment.
This block does not model supersonic flow.

Examples

Vehicle HVAC System

Models moist air flow in a vehicle heating, ventilation, and air conditioning (HVAC) system. The vehicle cabin is represented as a volume of moist air exchanging heat with the external environment. The moist air flows through a recirculation flap, a blower, an evaporator, a blend door, and a heater before returning to the cabin. The recirculation flap selects flow intake from the cabin or from the external environment. The blender door diverts flow around the heater to control the temperature.

Open Model

Aircraft Environmental Control System

Models an aircraft environmental control system (ECS) that regulates pressure, temperature, humidity, and ozone (O3) to maintain a comfortable and safe cabin environment. Cooling and dehumidification are provided by the air cycle machine (ACM), which operates as an inverse Brayton cycle to remove heat from pressurized hot engine bleed air. Some hot bleed air is mixed directly with the output of the ACM to adjust the temperature. Pressurization is maintained by the outflow valve in the cabin. This model simulates the ECS operating from a hot ground condition to a cold cruise condition and back to a cold ground condition.

Open Model

Medical Ventilator with Lung Model

Models a positive-pressure medical ventilator system. A preset flow rate is supplied to the patient. The lungs are modeled with the Translational Mechanical Converter (MA), which converts moist air pressure into translational motion. By setting the Interface cross-sectional area to unity, displacement in the mechanical translational network becomes a proxy for volume, force becomes a proxy for pressure, spring constant becomes a proxy for respiratory elastance, and damping coefficient becomes a proxy for respiratory resistance.

Open Model

Pneumatic Actuator with Humidity

How the Simscape™ Foundation Library moist air components can be used to model a pneumatic actuator operating in a humid environment. The Directional Valve is a subsystem composed of four Variable Local Restriction (MA) blocks, and the Double-Acting Actuator is a subsystem composed of two Translational Mechanical Converter (MA) blocks in opposite mechanical orientation.

Open Model

Ports

Input

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AR — Restriction area control signal, m^2
physical signal

Input physical signal that controls the air flow restriction area. The signal saturates when its value is outside the minimum and maximum restriction area limits, specified by the block parameters.

Dependencies

This port is visible only if you set the Restriction type parameter to Variable.

Conserving

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A — Inlet or outlet
moist air

Moist air conserving port associated with the inlet or outlet of the local restriction. This block has no intrinsic directionality.

B — Inlet or outlet
moist air

Moist air conserving port associated with the inlet or outlet of the local restriction. This block has no intrinsic directionality.

Parameters

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Restriction type — Specify whether restriction area can change during simulation
`Variable` (default) | `Fixed`

Select whether the restriction area can change during simulation:

Variable — The input physical signal at port AR specifies the restriction area, which can vary during simulation. The Minimum restriction area and Maximum restriction area parameters specify the lower and upper bounds for the restriction area.
Fixed — The restriction area, specified by the Restriction area block parameter value, remains constant during simulation. Port AR is hidden.

Minimum restriction area — Lower bound for the restriction cross-sectional area
`1e-10 m^2` (default)

Lower bound for the restriction cross-sectional area. You can use this parameter to represent the leakage area. The input signal AR saturates at this value to prevent the restriction area from decreasing any further.

Dependencies

Enabled when the Restriction type parameter is set to Variable.

Maximum restriction area — Upper bound for the restriction cross-sectional area
`0.005 m^2` (default)

Upper bound for the restriction cross-sectional area. The input signal AR saturates at this value to prevent the restriction area from increasing any further.

Dependencies

Enabled when the Restriction type parameter is set to Variable.

Restriction area — Area normal to flow path at the restriction
`1e-3 m^2` (default)

Area normal to flow path at the restriction.

Dependencies

Enabled when the Restriction type parameter is set to Fixed.

Cross-sectional area at ports A and B — Area normal to flow path at the ports
`0.01 m^2` (default)

Area normal to flow path at ports A and B. This area is assumed to be the same for the two ports.

Discharge coefficient — Ratio of actual mass flow rate to theoretical mass flow rate through the restriction
`0.64` (default)

Ratio of actual mass flow rate to the theoretical mass flow rate through the restriction. The discharge coefficient is an empirical parameter that accounts for nonideal effects.

Laminar flow pressure ratio — Pressure ratio at which air flow transitions between laminar and turbulent regimes
`0.999` (default)

Pressure ratio at which the moist air flow transitions between laminar and turbulent regimes. The pressure loss is linear with respect to mass flow rate in the laminar regime and quadratic with respect to mass flow rate in the turbulent regime.

Extended Capabilities

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

Version History

Introduced in R2018a

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R2024b: Model water droplets suspended in moist air flow

Blocks in the moist air domain can now model water droplets suspended in a moist air flow. To model water droplets, select Enable entrained water droplets in the Moist Air Properties (MA) block connected to your moist air network.

Local Restriction (MA)